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Title Pathogen transmission models in clonal plant population : Analysis on the effects of superinfection and seedreproduction
Author(s) 酒井, 佑槙
Citation 北海道大学. 博士(環境科学) 甲第12487号
Issue Date 2016-12-26
DOI 10.14943/doctoral.k12487
Doc URL http://hdl.handle.net/2115/64730
Type theses (doctoral)
File Information Yuma_Sakai.pdf
Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP
Pathogen transmission models
in clonal plant population
– Analysis on the effects of superinfection
and seed reproduction –
Yuma Sakai
Graduate School of Environmental Science
Hokkaido University
2016
CONTENTS
Summary 4
Chapter 1 Introduction 10
1.1 Clonal plant & pathogen . . . . . . . . . . . . . . . . . . . 10
1.2 Mathematical models of pathogen propagation . . . . . . . 13
1.3 Approximation methods . . . . . . . . . . . . . . . . . . . 16
1.4 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Chapter 2 Superinfection model 24
2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.1 1-strain model . . . . . . . . . . . . . . . . . . . . . 32
2.2.2 Multiple-strain models . . . . . . . . . . . . . . . . 40
2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Chapter 3 Seed propagatin model 62
3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.1 Single population . . . . . . . . . . . . . . . . . . . 69
3.2.2 Optimal proportion of vegetative propagation (Mixed
population) . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Chapter 4 Conclusion 89
Acknowledgements 97
Referances 98
Appendix 109
A. Simplification of the master equation . . . . . . . . . . . . 109
B. Analysis in superinfection model . . . . . . . . . . . . . . . 112
B.1 Mean-field approximation . . . . . . . . . . . . . . 112
B.2 Pair approximation . . . . . . . . . . . . . . . . . . 114
C. Analysis in seed propagation model . . . . . . . . . . . . . 122
C.1 Mena-field approximaation . . . . . . . . . . . . . . 122
C.2 Extinction phase in pair approximation . . . . . . . 123
Summary
Many clonal plants have two breeding systems, vegetative and seed
propagation. In seed propagation, plants reproduce genetically differ-
ent offspring that have a high mortality rate because their long-distance
dispersal and lack of a physical connection does not allow them to be
supported by their parents. In vegetative propagation, plants reproduce
genetically identical offspring that have lower mortality rates because re-
sources are supplied to the offspring from other individuals through inter-
connected ramets. However, vegetative propagation assists the pathogen’
s spread because the vascular system in the ramets acts as a transmission
pathway to other ramets. Thus, the disease becomes epidemic in a colony
of vegetative propagules. Increasing seed propagation is an effective de-
fensive behavior against the spread of pathogens because the plants will
reproduce in areas distant from the infection site.
Pathogens take several actions to increase their fitness within plant
populations. A diversity of infections, a superinfection, influences the
ability of pathogens to spread. A superinfection involves different pathogens
infecting (secondary infections) already infected individuals, either se-
4
quentially or concurrentl, and it leads to an increase in pathogen fitness
levels relative to a single infection. In this thesis, the focus is on two
characteristic phenomena, superinfection of pathogens, which increase
the fitness of the pathogen, and seed propagation in plants, which is an
effective defensive behavior against pathogen spread. Thus, two mod-
els, superinfection and seed propagation, were constructed applying the
two-stage contact process on lattice space to explore the effects of spa-
tial structures and each phenomenon on plant and pathogen populations.
The two-stage contact process expresses the dynamics of the contact in-
fection process simply on a graph and is applied to express the plant
reproduction process. However the analysis of the model is too complex,
thus we adopted two approximation methods, mean-field approximation
and pair approximation, to analyze the model analytically. Additionally,
we examined the effect of spatial structure through the comparison of the
result among the approximation methods and Monte Carlo simulation.
In the superinfection model, the effects of superinfection events on
the genetic diversity of pathogens using several models, including the
1-strain and multiple-strain models, were examined. In the analysis of
5
the 1-strain model, an equilibrium value was derived using the mean-
field approximation and pair approximation, and its local stability using
the Routh–Hurwitz stability criterion. In the multiple-strain models, the
dynamics using numerical simulations and Monte Carlo simulation were
analyzed. Through these analyses, the effects of parameter values, such
as the density of individuals, transition of a dominant pathogenic strain,
and competition between plants and pathogens, on the dynamics of the
models were shown. As a result, the superinfection event is one of the
important factors to maintenance of genetic diversity of pathogens. (i)
The strain with an intermediate cost became dominant, when both the
superinfection and growth rates were low; (ii) The competition among
strains occurred in the coexistence of various strains phase; (iii) Too high
a growth rate led to occupation by the strain with the lowest cost. Thus,
competition between the strain and the hosts occurred, and, therefore,
the host population decreased in all of the models; (iv) Pathogens easily
maintained their genetic diversity when there was a low superinfection
rate. However, if they did not superinfect, such maintenance became dif-
ficult; and (v) When the growth rate of a plant was low, an individual at
6
a local site was strongly interconnected by distant individuals.
In the seed propagation model, the dynamics of plant reproduction and
pathogen propagation, and the effects of seed propagation on the defense
responses to pathogen spread in single and mixed (coexistence of sev-
eral plant types) plant populations were examined. Thus, the change of
relative merit in the breeding system caused by the invasion of a plant
population by systemic pathogens was expressed. In the analysis, the
equilibrium and its local stability were derived using pair approximation
in the case of single populations. Additionally, using the Monte Carlo
simulation, the effects of spatial structure through a comparison with the
results of the pair approximation was examined. In mixed populations,
two situations were assumed, infected and uninfected populations, and
they were analyzed using only the Monte Carlo simulation because other
analyses of the model are too complex to obtain analytical results, having
too many variables,. The efficacy of seed propagation on the suppression
of epidemic infections was examined by comparing the results in the two
situations. As a result, seed propagation is an effective defensive behavior
against systemic pathogens. Generally, when the pathogen infectivity is
7
low, relative to plant fecundity, plants can escape from infected individ-
uals through the vegetative propagation, and the feature is expressed in
the presented model. However, the effect of pathogen abilities (infectiv-
ity and virulence) on the optimal balance of breeding systems becomes
decrease with the increase in the fecundity of pathogen in analysis of the
model. Thus, the adjustment of the breeding systems has an important
role to block of pathogen transmission when the plants have low fecundity.
In Chapter 4, the results of Chapter 2 and Chapter 3 were collected
to discuss the effect of the focal phenomena and the spatial structure on
plant reproduction and pathogen spread within clonal plant population.
As a result, the intrinsic fecundity of plant makes a major impact on the
evolution of pathogen within plant population and on the optimal balance
of breeding systems in clonal plant population, and the both the adjust-
ment of the balance of breeding systems and the selection of virulence
level through superinfection have influence on each other. Additionally,
the spatial structure impacts the dynamics of a plant population infected
by a systemic pathogen. In particular, when the plant ’s growth rate
is slower than the pathogen ’s infection rate, then the influence of the
8
spatial structures increases as indicated by the increasing quantitative
discrepancy between the approximation method and Monte Carlo sim-
ulation. However, the pair approximation can analyze the qualitative
characteristics of dynamics well.
9
Chapter 1 Introduction
1.1 Clonal plant & pathogen
Approximately 70% of terrestrial plants [1] and most aquatic plants,
such as sea grass [2], are clonal. Many clonal plants have two breeding
systems, vegetative and seed propagation [Fig. 1]. In seed propagation,
plants reproduce genetically different offspring that have a high mortality
rate because their long-distance dispersal and lack of a physical connec-
tion do not allow them to be supported by their parents. Thus, plants
maintain their genetic diversity and increase their habitat range, even
though the seedlings have a greater mortality rate. By contrast, in veg-
etative propagation, plants reproduce genetically identical offspring that
have lower mortality rates because resources are supplied to the offspring
from other individuals through interconnected ramets [3, 4]. However, if
a systemic pathogen invades the population, then it spreads rapidly and
the plants suffer serious damage because the vegetative propagules are
growing so close together [5].
According to Stuefer et al. (2004), systemic pathogens have diverse
10
Fig 1. Breeding systems of clonal plants. Plants can widely
disperse their offspring through seed propagation. Through vegetative
propagation, plants reproduce physically interconnected offspring. The
vegetatively propagated offspring can share resources through
interconnected ramets; however, pathogens can also spread through the
vascular system of the ramets.
11
negative effects on plants, which result in severe damage or death. For
example, they can lead to leaf deformations [6], growth rate reductions [7,
8, 9], growth-form changes [10, 11] and reduced reproduction [12, 13, 14].
Pathogens take several actions to increase their fitness within plant pop-
ulations. A diversity of infections, a superinfection, influences the ability
of pathogens to spread [15, 16, 17]. A superinfection involves differ-
ent pathogens infecting (secondary infections) already infected individ-
uals, either sequentially or concurrentl [16], and it leads to an increase
in pathogen fitness levels relative to a single infection [18]. Additionally,
the selection of pathogen’s virulence level also increases their fitness, de-
pending on the plant’s life cycle. For instance, if the host plant has a
long lifespan, then a low level of virulence is beneficial to the pathogen
because the host survives for a long period. However, if the plant repro-
duces vegetatively, then many susceptible individuals are produced. In
this situation, a high virulence level is beneficial because there are plenty
of other plants to infect. Thus, the methods of plant reproduction and
pathogen propagation influence each other.
Plants have diverse defense responses to systemic pathogens [19], such
12
as (i) deliberately detaching the infected ramets or tissues [20], (ii) in-
creasing their clonal growth rate [1, 10, 11, 21, 22], and (iii) limiting the
infection risk and pathogen spread by severing the physical connections
of ramets [9, 23] or by long-distance dispersal through seed propagation.
The detaching action blocks the spread of the pathogen in a popula-
tion, although the benefits of vegetative propagules decrease due to the
reduction in the genet size. The increase in the growth rate of vege-
tative propagation is an effective escape behavior from the pathogen’s
spread. However, vegetative propagation assists the pathogen’s spread
because the vascular system in the ramets acts as a transmission path-
way to other ramets. Thus, the disease becomes epidemic in a colony of
vegetative propagules [5]. Increasing seed propagation is an effective de-
fensive behavior against the spread of pathogens because the plants will
reproduce in areas distant from the infection site.
1.2 Mathematical models of pathogen propagation
Spatial structures play important roles in the evolution of both plants
and pathogens. The interactions between a plant and a pathogen de-
13
pend on the spatiotemporal dynamics, such as pathogen dispersal and
the spatial positioning of ramets [24, 19]. According to Koubek and Her-
ben (2008), features of the host assist local pathogen transmission and
the evolution of the pathogen towards lower virulence levels [25] because
clonal growth increases the probability of finding susceptible hosts in the
vicinity of the initially infected host. To explain these dynamics, mod-
els were constructed based on the contact process (CP) [26], especially
the two-stage contact process (TCP) [27, 28], described in mathemat-
ics. These are simple models that graphically express the dynamics of
the contact infection process. Thus, the models represent the spatiotem-
poral dynamics of pathogen propagation depending on the state of the
connected vertices.
In the CP, a vertex of the graph can represent either state, healthy or
infected, and the state of the vertex transitions to the other state proba-
bilistically with time. A vertex representing the healthy state transitions
to the infected state at a rate of nIβ, which indicates infection rate (β)
proportional to the number of connected vertices of the infected state (nI)
(infection), and a vertex representing the infected state transitions to the
14
healthy state at a rate of 1 (recovery) at the next time t + ∆t [Fig. 2].
The transition process is represented by a master equation as follows:
PSS =2 (PSI − nIβIPISS) ,
PSI =nIβIPISS + PII − nIβIPISI − PSI,
PII =2(nIβIPISI − PII).
(1)
Here, let Pσiσj(t) be the probability that two randomly chosen connected
sites are of state σi and state σj at time t (Pσiσj= Pσjσi
). Pσiσjσk(t) is the
probability that a randomly chosen site is of state σj and two randomly
chosen connected sites are of state σi and σk at time t (Pσiσjσk= Pσkσjσi
).
The positive and negative terms indicate transitions from one state and
to any other state. For instance, PSS transitions to PSI because of infection
and transitions from PSI because of recovery. The model is applied to
express the plant reproduction process, specifically, the names of the two
states are changed from“ healthy”and ”infected”to ”empty”and
”occupied”, respectively.
The TCP assumes three states, empty, healthy and infected. The
empty state transitions to the healthy state at a rate proportional to
15
the number of vertices having the healthy state (reproduction) and from
the healthy state or the infected state at a rate of 1 (death) [Fig. 3]. The
healthy state transitions to the infected state at a rate proportional to
the number of vertices having the healthy state (infection), and the in-
fected state does not transition to the healthy state (no recovery), which
is different from the CP. Thus, healthy individuals reproduce new off-
spring into neighboring open areas, and infected individuals increase only
by the infection’s transmission into neighboring healthy individuals. In-
fected individuals do not recover to the healthy state. The plant offspring
remain close to the parents, and the pathogens transmit to close individ-
uals. Therefore, this model is suited to describe the features of both the
vegetative and pathogen propagation processes.
1.3 Approximation methods
Eq. (1) is not closed because the probability relevant to a triple vertex is
necessary to express the dynamics of the probability relevant to a pair of
vertices. Additionally, to express the dynamics of the probability relevant
to a triple vertex, the probability relevant to a quad vertex is necessary.
16
t t+ t rate
S I nIβ
I S 1
S I S
S S S
1
tt ∆+
t I S I
I I I
β2
S S I
S I I
β
Fig 2. Dynamics of the contact process. (a) transition rule, (b and
c) reproduction process, (d) death process. The symbols 0 and S
indicate empty and occupied by an individual, respectively. Parameter
β indicates fecundity and nS indicates the number of individuals in the
nearest-neighbor sites of the empty area.
Thus, to analyze the Eq. (1) exactly, the probability relevant to infinite
connected vertices is necessary. Therefore, neither an explicit solution for
the equilibrium nor a threshold for the phase transition were obtained in
the CP and TCP models.
Several approximation methods have been applied to analytically dis-
cuss the behavior of the system, such as the mean-field approximation
(MA) and the pair approximation (PA). The MA is the simplest approx-
imation method, and it does not take into account the effects of other
sites. Thus, the dynamics of each vertex are independent in this method.
17
t t+ t rate
0 S nS βS
S I nI βI
S,I 0 1
S 0 I
0 0 0
1
tt ∆+
t S S I
S I I
Iβ
S 0 I
S S I
Sβ
Fig 3. Dynamics of the two-stage contact process. (a) transition
rule, (b) reproduction process, (c) infected process, and (d)death
process. The symbols 0, S and I indicate empty, healthy and infected,
respectively. The parameters are: βS, fecundity; βS, infectivity; nS, the
number of individuals in the nearest-neighbor sites of the empty area;
and nS, the number of infected individuals in the nearest-neighbor sites
of healthy individuals.
18
Thus, Pσiσjσkis approximated to ρσi
ρσjρσk
. Here, ρσi(t) is the probabil-
ity that a randomly chosen site is of the state σi at time t) [Fig. 4 (a)].
The PA assumes that the effects of distant sites will be less important
than those of the nearest neighbor sites. Thus, Pσiσjσkis approximated
to PσiσjPσjσk
/ρσj[Fig. 4 (b)]. Specifically, the probability relevant to the
connected triple vertex is expressed by the multiplication of the probabil-
ity relevant to the two pairs of vertices. Here, in the multiplication, the
chosen probability of the center vertex (in the triplet vertex) is included
in both probabilities relevant to the pairs. Thus, the probability relevant
to the triplet vertex is approximated by dividing the multiplication of
the probability relevant to the pairs of vertices by the chosen probability
of the center vertex. Therefore, this is a valid approximation method
for analyzing the effects of local connections. Additionally, the analyzing
the model on lattice space is suited to express plant reproduction pro-
cesses. The mathematical model on lattice space has been analyzed in
mathematics, physics and ecology (including the Ising model, percolation
model and contact process). The model create discrete spaces, and the
framework (the configurations of sites and distances between each site)
19
of the lattice does not change. Plants are distributed discretely in space
and plants cannot move from the established place during their lifetime.
Thus, the lattice model is better suited to express plant dispersal.
0 S S
S0 S S0 S
S0
S S
Fig 4. Definitions of the approximation methods. (a) mean-field
approximation, (b) pair approximation. These methods do not take into
account the effects of distant sites.
1.4 Purpose
In this thesis, the focus is on two characteristic phenomena, superin-
fection of pathogens, which increase the fitness of the pathogen, and seed
propagation in plants, which is an effective defensive behavior against
pathogen spread. There are several theoretical studies pertinent to the
evolution of pathogen virulence through superinfection [29, 30, 31, 32,
25, 33, 34], and of the optimal balance between seed and vegetative prop-
20
agation [35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Previous superinfection
models did not consider spatial structure or the dynamics of host repro-
duction, and the previous seed propagation models did not examine the
effects of pathogen spread on the optimal propagation balance. Thus, in
this thesis, two models, superinfection and seed propagation, were con-
structed applying the TCP on lattice space to explore the effects of spatial
structures and each phenomenon on plant and pathogen populations.
In the superinfection model, the pathogen propagation process related
to superinfections in vegetatively propagated populations (seed propa-
gation was not considered) and the effects of superinfection events on
the genetic diversity of pathogens using several models, including the
1-strain and multiple-strain models, were examined. In the analysis of
the 1-strain model, an equilibrium value was derived using the MA and
PA, and its local stability using the Routh–Hurwitz stability criterion.
In the multiple-strain models, the dynamics using numerical simulations,
including the MA, PA and the Monte Carlo simulation (MCS), were an-
alyzed. Through these analyses, the effects of parameter values, such as
the density of individuals, transition of a dominant pathogenic strain,
21
and competition between plants and pathogens, on the dynamics of the
models were shown.
In the seed propagation model, the dynamics of plant reproduction and
pathogen propagation, and the effects of seed propagation on the defense
responses to pathogen spread in single and mixed (coexistence of sev-
eral plant types) plant populations were examined. Thus, the change of
relative merit in the breeding system caused by the invasion of a plant
population by systemic pathogens was expressed. In this model, the
superinfection process was not included because the focus was on de-
termining the optimal balance of the breeding system against pathogen
propagation. Additionally, the genetic diversity of pathogens has a less di-
rect impact than the reproductive distance on the strategy was assumed.
In the analysis, the equilibrium and its local stability were derived using
pair approximations in the case of single populations. Additionally, using
the MCS, the effects of spatial structure through a comparison with the
results of the PA was examined, and the case of a mixed population was
analyzed. In mixed populations, two situations were assumed, infected
and uninfected populations, and they were analyzed using only the MCS
22
because other analyses of the model are too complex to obtain analytical
results, having too many variables,. The efficacy of seed propagation on
the suppression of epidemic infections was examined by comparing the
results in the two situations.
23
Chapter 2 Superinfection model
There are several theoretical studies pertinent to the evolution of pathogen
virulence during superinfections that use an ordinary differential equa-
tion [29, 30, 31, 32, 25, 33, 34]. These studies defined the already infected
individual as superinfected and as being taken over by a second pathogen
strain of higher virulence. Thus, the strains do not permanently share
the host. Additionally, most of the studies assumed a trade-off between
the infection rate and the virulence of the pathogen. They explored the
evolution of virulence within a host population using the host–parasite
model, and they analyzed the model in the cases where a host is either
infected by only one strain of a pathogen (single infection) or by sev-
eral strains (superinfection) of a pathogen. According to these studies,
the virulence of the superinfection model evolves towards a higher value
compared with that of a single infection. However, they did not consider
the effects of spatial structures.
In contrast, there are several studies using the lattice model [45, 46, 47].
Sato et al. (1994) analyzed the TCP in detail using the PA; however,
they could not analytically determine the stability of the epidemic equi-
24
librium (coexistence of healthy and infected individuals). Satulovsky and
Tome (1994) studied a predator–prey system using an improved TCP
and obtained analytical results with respect to the coexistence equilib-
rium, although they assumed only one predatory species. Haraguchi and
Sasaki (2000) considered mutants of pathogens, which have a different
mortality rate, based on the TCP and analyzed the evolutionarily stable
strategy (ESS) of the mortality and transmission rates using computer
simulations. However, they did not include the superinfection events in
the propagation processes of the pathogens. In summary, the models in
previous studies are insufficient to examine the effects of superinfection
events on pathogen spread within clonal plant populations, which are
affected by spatial structures. Thus, further modifications of the mod-
els are necessary to express a pathogen spread process that incorporates
superinfection events.
2.1 Model
A model of plant growth and pathogen propagation processes, includ-
ing superinfection, was constructed. In the model, a single plant species
25
and multiple pathogen strains were assumed. It was also assumed that a
healthy individual plant is infected by a pathogen strain and an already
infected plant is superinfected by other pathogen strains. The model’s
dynamics is a continuous Markov process on a lattice space. The state of
each site, and the transition and the mortality rates of each state are rep-
resented by a vector Ω = (σ0, σ1, σ2, · · · , σn), B = (βσ0 , βσ1 , βσ2 , · · · , βσn)
and D = (dσ0 , dσ1 , dσ2 , · · · , dσn), respectively, where the total number of
states is n+1. ρσi(t) is the probability that a randomly chosen site is of
the state σi at time t. Thus, ρσi(t) indicates the global density of the site
at the state σi. σ0 = ”0”, σ1 = ”S”, and σi+1 = ”Ii”(i = 1, 2, · · · , n− 1)
represent empty, susceptible (healthy) individuals and individuals in-
fected with i pathogen strains, respectively. In addition, it was assumed
that the already infected individuals having i-strains (”Ii”) are superin-
fected (and taken over) by the more virulent j-strain, because the strain
with higher virulence often wins within-host competitions [25, 48]. Based
on the definitions of d0 and β0, d0 = 0 and β0 =1n
∑ni=1 dσi
because the
empty site does not die, and a transition to ”0” indicates the deaths of
healthy and infected individuals.
26
Four demographic processes were configured: (i) plant growth, (ii) first
infection, (iii) superinfection and (iv) death. The growth process is repre-
sented by a transition from state ”0” to ”S”, which indicates that plants
grow their ramets into an open area. Thus, empty sites are occupied by
healthy individuals. The first infection process is represented by a transi-
tion from state ”S” to ”Ii”, which indicates that healthy individuals have
been infected by pathogens of the i-strain. The superinfection process is
represented by a transition from state ”Ii” to ”Ij” (i > j). The death
process is represented by a transition from ”S” or ”Ii” to ”0”, which rep-
resents the deaths of healthy and infected individuals from natural causes
and the pathogen, respectively. In addition, infected individuals are not
able to recover to a healthy one. Thus, these processes are described
using the following notation, which is often used to explain the TCP:
(i) 0 → S at rateβSn (S)
z
(ii) S → Ii at rateβIi
n (Ii)
z
(iii) Ii → Ij at rate sβIj
n (Ij)
z
(iv) S, Ii → 0 at rate dS, dIi
(TP)
27
Parameter n(σi) is the number of σi-sites among the nearest neighbors
of the focal sites, z is the number of nearest-neighbor sites (e.g. z = 4
for a von Neumann neighborhood on the two-dimensional square lattice),
and s is the superinfection rate. Thus, sβIjdescribes the ratio of the
superinfection to the first infection [30]. Growth and infection events
occur at a rate proportional to the number of the healthy and infected
states, respectively, among the nearest-neighbor sites.
Here, let qσj/σi(t) be the conditional probability that a randomly chosen
nearest neighbor of a σi-site is a σj-site. In particular, qσi/σirepresents the
local density of σi-sites. Pσiσj(t) is the probability that a randomly chosen
site is of the state σi and a randomly chosen nearest-neighbor site is of the
state σj at time t. These variables have the following relationship [49, 45]:
Pσiσj= ρσi
qσj/σi. (2)
Thus, using the above dynamics, the following describes a set of master
equations, which is referred to as the general model (GM):
28
P00 =2
(n−1∑i=1
dIiPIk0 + dSPS0 −
βS (z − 1) qS/00z
P00
),
PS0 =βS (z − 1) qS/00
zP00 + dSPSS −
βS
zPS0 −
βS (z − 1) qS/0Sz
PS0
− dSPS0 +n−1∑i=1
[dIiPSIi
−βIi
(z − 1) qIi/S0z
PS0
],
PSS =2
(βS
zPS0 +
βS (z − 1) qS/0Sz
PS0 −n−1∑i=1
βIi(z − 1) qIi/SS
zPSS − dSPSS
),
˙PIi0 =βIi
(z − 1) qIi/S0z
PS0 + dSPSIi−
βS (z − 1) qS/0Iiz
PIi0 +n−1∑j=1
dIjPIiIj
− dIiPIi0
+ s
(n−1∑
j=i+1
βIi(z − 1) qIi/Ij0
zPIj0 −
i−1∑j=1
βIj(z − 1) qIj/Ii0
zPIi0
),
˙PSIi=βS (z − 1) qS/0Ii
zP0Ii +
βIi(z − 1) qIi/SS
zPSS −
βIi
zPSIi
−n−1∑j=1
βIj(z − 1) qIj/SIi
zPSIi
− (dS + dIi)PSIi
+ s
(n−1∑
j=i+1
βIi(z − 1) qIi/IjS
zPSIj
−i−1∑j=1
βIj(z − 1) qIj/IiS
zPSIi
),
˙PIiIj=βIi
(z − 1) qIi/SIjz
PSIj+
βIj(z − 1) qIj/SIi
zPIiS
−(dIi
+ dIj
)PIiIj
+ s
(n−1∑
k=i+1
βIi(z − 1) qIi/IkIj
zPIkIj
+n∑
k=j+1
βIj(z − 1) qIj/IkIi
zPIiIk
−i−1∑k=1
βIk(z − 1) qIk/IiIj
zPIiIj
−j−1∑k=1
βIk(z − 1) qIk/Ij Ii
zPIiIj
−βIj
zPIiIj
)(i > j) ,
˙PIiIi=2
[βIi
zPSIi
+βIi
(z − 1) qIi/SIiz
PSIi− dIi
PIiIi
+s
(n−1∑
j=i+1
(βIi
zPIiIj
+βIi
(z − 1) qIi/Ij Iiz
PIj Ii
)−
i−1∑j=1
βIj(z − 1) qIj/IiIi
zPIiIi
)].
(3)
29
Here, for example, in the right side of the fifth equation in the set (the
differential equation of PSIi), the first term describes the transition from
P0Ii to PSIi, which indicates that healthy individuals’ offspring take up
empty sites (state 0 → S). In this term, the transition rate is determined
by TP(i), and n(S) is (z − 1) qS/0Ii . The transition begins from P0Ii ; there-
fore, one of the nearest-neighbor sites of the 0-state site is of the state
Ii, and at least one of the other nearest-neighbor sites of the 0-state site
(in (z − 1) sites) should be S for a transition from 0 to S. Thus, the
probability is qS/0Ii , and the expectation of n (S) is equal to (z − 1) qS/0Ii .
In subsequent terms, the n (σ) (σ ∈ Ω) is obtained in a similar process,
except for the third term.
The second and third terms indicate that healthy individuals are in-
fected by i-strains of pathogens (state S → Ii). These terms describe
the transition from PSS to PSIiand from PSIi
to PIiIi, respectively, and the
transition rate of these terms is determined by the TP(ii). In particular,
the value of n(Ii) in the third term is equal to 1. The transition begins
from PSIi; therefore, there is already a Ii-state site among the nearest-
neighbor sites of the S-state site. Here, the state of the other sites among
30
the nearest-neighbor sites is also Ii, which is included in the fourth term.
The fourth term indicates that healthy individuals are infected by j-
strains of pathogens (state S → Ij). This term describes the transition
from PSIito PIj Ii
(i ∈ j), and sums the transition rates for all strains (from
I1 to In−1 ).
The fifth term indicates the death of healthy or infected individuals
(state S, Ii → 0). The term describes the transition from PSIito P0Ii or
PS0, and the transition rate of each process is determined by the TP(iv).
The last term indicates that the already infected individuals are super-
infected by other strains of the pathogen (e.g. state I2 → I1). The first
and second terms in parentheses describe the transition from PSIjto PSIi
and from PSIito PSIj
(j 6= i), respectively. The transition rates of these
terms are determined by the TP(iii). The first term indicates that the
infecting pathogens superinfect already infected individuals with another
strain, and the second term indicates that an already infected individual
is superinfected by another strain of the pathogen. Thus, the range of
the summation in the first term is from i+ 1 to n− 1, and in the second
term it is from 1 to i− 1, based on our assumptions.
31
2.2 Results
In this part, a new parameter mi (mortality cost) was introduced and
defined as βIi/dIi
for n− 1 strains(mi := βIi/dIi
, m1 < m2 < · · · < mn−1),
and set dIi= 1 (∀Ii ∈ Ω) was used to standardize the parameter, for
ease of analysis. The mortality cost represents the expectation of the
number of newly infected individuals produced during the lifetime of an
infected individual. Thus, a more highly virulent strain has a lower mor-
tality cost. Consequently, already infected individuals are superinfected
by strains with lower mortality costs. In addition, dS ≈ 0, because the
plant mortality is generally less than the plant growth rate in long-lived
clonal plants.
2.2.1 1-strain model
Initially, the simplest case (n = 2) for the GM was analyzed using
the MA and PA. The state of each site was denoted by σi ∈ S ≡
0, S, I (I := I1) from the assumption of only one pathogen strain. There-
fore, the following set of master equations to rewrite Eq. (3) was obtained:
32
P00 =2PI0 − 2βS (z − 1) qS/00
zP00,
PS0 =PSI +βS (z − 1) qS/00
zP00
−[βS
z+
βS (z − 1) qS/0Sz
+mI (z − 1) qI/S0
z
]PS0,
PI0 =PII +mI (z − 1) qI/S0
zPS0 −
[βS (z − 1) qS/0I
z+ 1
]PI0,
PSS =2
[βS
z+
βS (z − 1) qS/0Sz
]PS0 − 2
[mI (z − 1) qI/SS
z
]PSS,
PSI =βS (z − 1) qS/0I
zP0I +
mI (z − 1) qI/SSz
PSS
−[mI
z+
mI (z − 1) qI/SIz
+ 1
]PSI,
PII =2
[mI
z+
mI (z − 1) qI/SIz
]PSI − 2PII.
(4)
In this model, superinfection does not occur because there is only one
pathogen strain.
2.2.1.1 Mean-field Approximation
To close the set of Eqs. (4), several variables, qσi/σj≈ ρσi
(e.g. PS0 =
ρSq0/S ≈ ρSρ0) and qσi/σjσk≈ ρσi
, were approximated using the MA. In
addition, the equations were simplified, using the definition of variables
from Eq. (13) (Appendix. A.).
33
ρ0 = 1− ρ0 − ρS (1 + βSρ0) ,
ρS = ρS (βSρ0 −mI (1− ρ0 − ρS)) .
(5)
The system has three equilibrium states.
EM ≡ (ρ∗0 , ρ∗S , ρ
∗I ) = (1, 0, 0) ,
EM ≡ (ρ∗0 , ρ∗S , ρ
∗I ) = (0, 1, 0) ,
EM ≡ (ρ∗0 , ρ∗S , ρ
∗I )
=
(mI − 1
βS +mI
,1
mI
,βS (mI − 1)
mI (βS +mI)
).
EM, EM and EM indicate the states of extinction, disease-free and epi-
demic, respectively. From a local stability analysis of the each equilibrium
(see Appendix.B.1), EM is always unstable, which means that plants do
not become extinct at the positive parameter range in the system. When
mI < 1, EM is stable, then the pathogen is not able to survive if it has a
low mortality cost. By contrast, when mI exceeds 1, EM becomes unsta-
34
ble and EM is always stable, the epidemic occurs because the pathogen
spreads within the plant population. Thus, mI = 1 is the threshold value
for stability shifting. In conclusion, the stability of the equilibrium states
and the equilibrium density of healthy individuals (ρ∗S ) in the epidemic
state depended only on mI, regardless of βS in the MA.
2.2.1.2 Pair Approximation
To close the set of Eqs. (4) and consider the effect of local connec-
tions on the dynamics, several variables were approximated using the PA,
qS/0σ ≈ qS/0 and qI/Sσ ≈ qI/S. In addition, the equations were simplified by
the definitions of the variables (see Appendix.A.), and the following three
equilibrium states were obtained:
35
EP ≡(ρ∗0 , ρ
∗S , ρ
∗I , q
∗0/0, q
∗S/0, q
∗I/0
)=(1, 0, 0, 1, 0, 0) ,
EP ≡(ρ∗0 , ρ
∗S , ρ
∗I , q
∗0/S, q
∗S/S, q
∗I/S
)=(0, 1, 0, 0, 1, 0) ,
EP ≡(ρ∗0 , ρ
∗S , ρ
∗I , q
∗0/0, q
∗S/0, q
∗I/0, q
∗0/S, q
∗S/S, q
∗I/S, q
∗0/I, q
∗S/I, q
∗I/I
)=(see B.2) .
EP indicates that the plants become extinct. Therefore, qσ/S and qσ/I are
non-existent because ρS and ρI are equal to 0. EP represents the disease-
free state. Thus, qσ/0 and qσ/I are non-existent because ρ0 and ρI are equal
to 0. EP represents the epidemic state, at which all 12 variables exist and
have positive values. The local stability of each equilibrium state was ex-
amined using the Routh–Hurwitz stability criterion (see Appendix. B.2).
From the stability analysis, the three stable-equilibrium phases, disease-
free, epidemic and periodic oscillation [Fig. 5], were obtained. In par-
ticular, EP is always unstable (plants do not become extinct), and two
36
thresholds, epidemic and bifurcation, were derived. The stability of EP
and EP depends on whether the parameter values exceed each thresh-
old, especially the epidemic condition that depends only on the mortality
cost, irrespective of the growth rate because the epidemic threshold is
(mI)c = z/ (z − 1), which is similar to that of the MA. Thus, if mI is low,
then pathogens become extinct [panels (a) and (b) in Fig. 6], because the
low mI means a high virulence or low infection rate. Therefore, pathogens
die within the infected hosts before infecting other hosts. In addition, a
large mI leads to a decrease in both healthy and infected individuals. βS
affects the equilibrium value in the epidemic phase. For example, a large
βS leads to an increase in the equilibrium density of infected individuals
[panel (c) in Fig. 6], because pathogens can spread within the hosts sup-
plied by the fast growth rate. In the epidemic phase, when βS is large,
the equilibrium density of healthy individuals (ρ∗S ) decreases. Thus, a
slow growth rate had an advantage over a high growth rate for plants in
this phase. In addition, a Hopf bifurcation occurs when the parameter
values exceeded the bifurcation threshold [panel (b) in Fig. 6]. Thus, the
stability of EP shifted from stable to unstable, which was different from
37
under the MA.
0 20 40 60 80 100
0
20
40
60
80
100
Epidemic
Oscillation
Disease-free
Mortality cost (mI)
Growth
rate
(βS)
Bifurcation threshold
Epidemic
threshold
Fig 5. Phase diagram of the pair approximation. This figure
shows the three phases of the equilibrium state. In the epidemic phase,
plants and pathogens coexist and the equilibrium is stable. In the
oscillation phase, plants and pathogens coexist but the equilibrium is
unstable, and oscillation is observed. Therefore, Hopf bifurcation
occurs. The solid and dashed lines indicates the bifurcation and the
epidemic thresholds, respectively. In the disease-free phase, pathogens
become extinct through a too low mortality cost.
To check the validity of each approximation method, the equilibrium
values of the MA, PA and MCS were compared. The MCS was conducted
100 times for each given parameter set: (a) βS = 10 and mI = 0 ∼ 30, (b)
βS = 30 and mI = 0 ∼ 30, and (c) mI = 15 and βS = 0 ∼ 25 in Fig. 6. A
two-dimensional square lattice torus was used, and the average values of
38
(a). βS = 10 (b). βS = 30 (c). mI = 15
Den
sity
of
healthyindividual(ρ∗ S)
Den
sity
of
infected
individual(ρ∗ I)
Variance
ofρ∗ I
Mortality cost (mI) Grwoth rate (βS)
(i)
(ii)
(iii)
Fig 6. Comparisons among the numerical simulation of the
mean-field approximation (MA) and the pair approximation
(PA) and Monte Carlo simulation (MCS), which include the
mortality cost and growth rate, βS(= 10; 30; 50). (i) equilibrium
densities of healthy individuals (ρ∗S)(ii) equilibrium densities of infected
individuals (ρ∗I ). The dotted, solid and dashed lines represent the results
of the MA, PA and MCS, respectively. These lines show that when mI
is low, both approximation methods express similar trends to that of
the MCS, although the equilibrium value is overestimated. (iii) the
variance among 100 trials using the MCS. The high variance indicates
that the oscillatory solution is observed.
39
100 trials at each parameter value were calculated [Fig. 6]. There were
several discrepancies with respect to the equilibrium and threshold val-
ues. These included that the MA and PA overestimated the equilibrium
values, and that the periodic solution, were observed in the PA under
higher parameter values compared with those of the MCS. In Fig. 6, the
discrepancies between the threshold values determined by the MCS and
MA/PA were great when the mortality cost was large ((a) , (b)) or the
growth rate was low ((c)). The use of these approximation methods,
which neglected the effects from remote sites, may be the cause of the
discrepancies. However, a periodic solution was observed using the re-
sults of the PA and MCS [47], which were different from that of the MA.
Thus, the PA could explain the basic behavior of the system better than
the MA.
2.2.2 Multiple-strain models
Four sub-models (n = 3, 4, 11 and 26) of the GM (Eq. (3)) were exam-
ined; however, the analyses were too complex, having too many variables
to obtain an analytical result (the MA and PA require at least n and
40
Σni=1i−1 variables, respectively). Thus, the equilibrium value was derived
using the MA in the 2- and 3-strain models, and all of the models were
analyzed using computer simulations (the MCS and numerical simula-
tions of the MA and PA). In the simulations, mi = mi−1+∆mi,i−1 (i = 2,
3, . . ., n − 1), s = 0, 0.5, 1.0 and 1.5 and the value of βS varied. In this
paper, it was assumed that ∆m := ∆mi,i−1 was constant to maintain the
simplicity of the model. As a result of the numerical simulations of the
MA and PA, healthy individuals did not become extinct in all of the mod-
els. However, the healthy and infected individuals did become extinct in
the MCS, especially when n was large and βS was small. A comparison
among the MA, PA and MCS, indicated that the discrepancies increased
as the number of strains increased.
2.2.2.1 The 2-strain model (n = 3)
Five equilibrium states were obtained using the MA (Table 1 in detail):
E1: extinction (ρ∗0 = 1), E2: disease-free (ρ∗I1 = 0, ρ∗I2 = 0), E3: occu-
pation of a strain with high cost (ρ∗I1 = 0, ρ∗I2 > 0), E4: occupation of
a strain with low cost (ρ∗I1 > 0, ρ∗I2 = 0), and E5: coexistence (ρ∗I1 > 0,
ρ∗I2 > 0). In particular, in the (equilibrium) phase of coexistence (E5 is
41
stable), the dominant strain changed depending on the plant growth rate.
Table 1. Equilibria of the 2-strain model in MA.
E1 (1, 0, 0, 0)
E2 (0, 1, 0, 0)
E3
(m2−1βS+m2
, 1m2
, 0, βS(m2−1)m2(βS+m2)
)E4
(m1−1βS+m1
, 1m1
, βS(m1−1)m1(βS+m1)
, 0)
E5
(m2−m1
sβSm1, m1(sβS+1)−m2
βS(m2+m1(s−1)), sβSm1(m2−1)−(m2−m1)(βS+m2)
sβSm1(m2+m1(s−1)), m1(m2−m1)+βS(m2−m1−sm1(m1−1))
sβSm1(sm1+m2−m1)
)E ≡
(ρ∗0, ρ
∗S, ρ
∗I1, ρ∗I2
).
When s > 0, Figs. 7 and 8 show that the equilibrium density of each
state depended on the growth rate (βS) of a given superinfection rate (s)
and that the difference in mortality cost (∆m) was simulated by the MA
[panels (a)], PA [panels (b)] and MCS [panels (c)], as well as the tran-
sition of the equilibrium phase [panels (d)]. As a result, when βS was
small, the strain with a high cost was dominant. Then, the dominant
strain shifted to the strain with a low cost as βS increased, and the strain
occupied the pathogen population at a large βS. The increase in s and
the decrease in ∆m led to lower threshold values of the phase transitions
[panels (d) in Figs. 7 and 8]. Thus, when s was small or ∆m was large,
the parameter range of the phase of coexistence increased. Notably, when
∆m was large (moderate competition), multiple strains were more likely
42
to coexist , as in the ’limiting similarity of niche’ proposed by the com-
petition theory [50, 51]. For healthy individuals, their densities increased
at the coexistence phase and decreased at other phases in the PA. Thus,
if the growth rate was out of the range value of the coexistence phase,
the healthy population did not increase.
When s = 0 (no superinfection), the shift of the equilibrium phase
along the gradient of βS was different from the case of s > 0 in the MCS.
For example, when βS was large, the strain with the highest mortality cost
occupied and coexisted with healthy plants (strains with lower costs be-
come extinct). In addition, the range of the coexistence phase decreased
compared with when s > 0 [Fig. 9 (i)]. In the MA and PA, the strain
with the higher cost always occupied the host, regardless of the values of
the other parameters [panels (a) and (b) in Fig. 9 (i)]. Thus, when s = 0
and βS was large, the pathogen population was occupied by the strain
with highest cost, contrary to when s > 0.
In a comparison of the numerical simulations (the MA and PA) with
the MCS, when βS was small, the result was at extreme variance with the
MCS. Thus, in the parameter range, the approximation method could not
43
0 20 40 60 80 100
0.5
1
0 20 40 60 80 100
0.5
1
0 20 40 60 80 100
0.5
1
0 20 40 60 80 100
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
s = 0.5(a)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2
(b)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2
(c)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2
(d)
MA
E5
E4
E3
PA E4 E5 E3
MCS E4 E5
s = 1.0(a)
ρ∗0
ρ∗S
ρ∗I1ρ∗I2
(b)
ρ∗0ρ∗S
ρ∗I1ρ∗I2
(c)
ρ∗0ρ∗S
ρ∗I1ρ∗I2ρ∗I1
(d)
MA E4 E5 E3
PA E4 E5 E3
MCS
E2
E3 E5 E4
E5
E3
s = 1.5(a)
ρ∗0
ρ∗S
ρ∗I1ρ∗I2
(b)
ρ∗0
ρ∗S
ρ∗I1ρ∗I2
(c)
ρ∗0
ρ∗S
ρ∗I1ρ∗I2
(d)MA E4
E5
E3
PA E4 E3
MCS E3 E5 E3
Growth rate βS
Equilibrium
Density
ofeach
strain
Fig 7. The equilibrium value of each state and the transition
from the equilibrium phase depends on the superinfection rate
in the 2-strain model. m1 = 5 and ∆m = 5. The I, II and III differ
in their values of s (=0.5, 1.0 and 1.5, respectively). (a-c) the variation
of the equilibrium density of each state (”0”, ”S”, ”I1” and ”I2”.
Σσρσ = 1) with the growth rate in each simulation: (a) MA, (b) PA and
(c) MCS. (d) the transition from the equilibrium phase with βS in the
MA, PA and MCS.44
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0 2 4 6 8 10 12 14
0.5
1
0 2 4 6 8 10 12 14
0.5
1
0 2 4 6 8 10 12 14
0.5
1
0 2 4 6 8 10
0 2 4 6 8 10 12 14
0.5
1
0 2 4 6 8 10 12 14
0.5
1
0 2 4 6 8 10 12 14
0.5
1
0 2 4 6 8 10 12 14
∆m = 5(a)
ρ∗0ρ∗S
ρ∗I1ρ∗I2
(b)
ρ∗0ρ∗S
ρ∗I1ρ∗I2
(c)
ρ∗0
ρ∗Sρ∗I1ρ∗I2
(d)
MA E4
E5
E3
PA E4 E3
MCS E3 E5 E3
∆m = 10(a)
ρ∗0ρ∗S
ρ∗I1ρ∗I2
(b)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2
(c)
ρ∗0 ρ∗Sρ∗I1 ρ∗I1ρ∗I2
(d)
MA E4E5 E3
PA E4 E5 E3
MCS E3 E5 E4
E5
E3
∆m = 15(a)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2
(b)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2
(c)
ρ∗0
ρ∗S
ρ∗S
ρ∗I1ρ∗I1 PP ρ∗I2
ρ∗I2
(d)
MA E4 E5 E3
PA E4 E5 E3
MCS E3 E5
E3
E2 E4 E5 E3
Growth rate βS
Equilibrium
Density
ofeach
strain
Fig 8. The equilibrium value of each strain and transition from
the equilibrium phase depends on the mortality cost in the
2-strain model. mI1 = 5 and ∆m = 5 in the figures. The I, II and III
differ in their values of s (=0.5, 1.0 and 1.5, respectively). The (a) MA,
(b) PA, (c) MCS and (d) the transition from equilibrium phase.
45
0 5 10 15 20 25
0.5
1
0 5 10 15 20 25
0.5
1
0 5 10 15 20 25
0.5
1
0 2 4 6 8 10
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
(i). 2-strain (n = 3)(a)
ρ∗0
ρ∗S ρ∗I2
(b)
ρ∗0
ρ∗S
ρ∗I2
(c)
ρ∗0ρ∗S
ρ∗I2
(d)
MA E4
PA E5
MCS
E2
E3
E2
E4
E5
E4
(ii). 3-strain(n = 4)(a)
ρ∗0
ρ∗S ρ∗I2
(b)
ρ∗0
ρ∗S
ρ∗I2
(c)
ρ∗0
ρ∗S
ρ∗I2
ρ∗I3
(d)
MA E5
PA E5
MCS
E2
E3
E6 E4 E8 E5
Growth rate βS
Equilibrium
Density
ofeach
strain
Fig 9. No superinfection (s = 0) in the 2-strain and 3-strain
models using the MA, PA and MCS. The global density of each
strain of the pathogen is plotted at the equilibrium state depending on
βS. The values of the parameters are: (a) n = 3, mI1 = 5, ∆m = 10, and
(b) n = 4, mI1 = 5, ∆m = 5.
46
be applied. When βS was large enough, the discrepancy between them
increased as s [Fig. 7] and ∆m decreasd [Fig. 8]. However, the MA and
PA could explain the transition process of the equilibrium phase [panels
(d) in Figs. 7 and 8]. In addition, the oscillatory solution, in which the
solution oscillates for a long time, although unproven, was not observed
in the coexistence phase [Fig. 10]. However, in the occupation phase of a
strain, the oscillatory solution was observed because the behavior of the
model then followed that of the 1-strain model.
5 10 15 20 25 30
0.1
0.2
0.3
0.4
0.5
0.6
5 10 15 20 25 30
0.1
0.2
0.3
0.4
0.5
5 10 15 20 25 30
0.1
0.2
0.3
0.4
0.5
20 40 60 80 100
0.2
0.4
0.6
0.8
1.0
20 40 60 80 100
0.2
0.4
0.6
0.8
20 40 60 80 100
0.2
0.4
0.6
0.8
Time (t)
Global
density
ofeach
state
I.(a)
II.(a)
(b)
(b)
(c)
(c)
Fig 10. Time series in the 2-strain model. The equilibrium value
of the global density of each state is plotted. In the figures, s = 1.0 and
∆m = 15. I. mI1 = 5 and II. mI1 = 30 with (a) βS = 10, (b) βS = 15 and
(c) βS = 25.
2.2.2.2 The 3-strain model (n = 4 )
Nine equilibrium states were obtained using the MA (Table 2 in detail):
47
E1: extinction, E2: disease-free, E3−−5: occupation of a strain (ρ∗Ii > 0,
ρ∗Ij = 0, ρ∗Ik = 0), E6−−8: coexistence of two strains (ρ∗Ii > 0, ρ∗Ij > 0,
ρ∗Ik = 0), and E9: coexistence of all strains (ρ∗Ii > 0, ρ∗Ij > 0, ρ∗Ik > 0).
Table 2. Equilibria of the 3-strain model in MA.
E1 (1, 0, 0, 0, 0)
E2 (0, 1, 0, 0, 0)
E3
(m1−1βS+m1
, 1m1
, βS(m1−1)m1(βS+m1)
, 0, 0)
E4
(m2−1βS+m2
, 1m2
, 0, βS(m2−1)m2(βS+m2)
, 0)
E5
(m3−1βS+m3
, 1m3
, 0, 0, βS(m3−1)m3(βS+m3)
)E6
(m2−m1
sβSm1, m1(sβS+1)−m2
βS(m2+m1(s−1)), sβSm1(m2−1)−(m2−m1)(βS+m2)
sβSm1(m2+m1(s−1)), m1(m2−m1)+βS(m2−m1−sm1(m1−1))
sβSm1(sm1+m2−m1), 0)
E7
(m3−m1
sβSm1, m1(sβS+1)−m3
βS(m3+m1(s−1)), sβSm1(m3−1)−(m3−m1)(βS+m3)
sβSm1(m3+m1(s−1)), 0, m1(m3−m1)+βS(m3−m1−sm1(m1−1))
sβSm1(sm1+m3−m1)
)E8
(m3−m2
sβSm2, m2(sβS+1)−m3
βS(m3+m2(s−1)), 0, sβSm2(m3−1)−(m3−m2)(βS+m3)
sβSm2(m3+m2(s−1)), m2(m3−m2)+βS(m3−m2−sm2(m2−1))
sβSm2(sm2+m3−m2)
)E9
(m1m3−m2
βSm2+m1m3, m2
m1m3, βSm2(s(m1m3−m2)−m3+m2)−m1m3(m3−m2)
sm1m3(βSm2+m1m3),
m2(m3−m1)−sm1(m2+m1m3)sm1m3(βSm2+m1m3)
, m1m3(sβSm1−m2+m1)−βSm2(sm1+m2−m1)sm1m3(βSm2+m1m3)
)E ≡
(ρ∗0, ρ
∗S, ρ
∗I1, ρ∗I2 , ρ
∗I3
).
When s > 0, Figs. 11 and 12 show the equilibrium densities of each state
based on βS being in a given s and the difference in the mortality costs
(∆m), respectively, using the MA [panel (a)], PA [panel (b)] and MCS
[panel (c)]. The occupation phase of a single strain and the coexistence
phase of two strains showed generally similar responses to the parameter
values in the single-strain and 2-strain models. The effects of s and∆m on
48
equilibrium values, threshold values and discrepancies among simulation
results were similar to those of the 2-strain model. Thus, the increase or
decrease in the density of healthy individuals depended on βS in the PA,
which was different from in the MA, in which the growth rate negatively
affected the density of the healthy individuals. However, when s and
∆m were both large, the transition of the equilibrium phase in the MA
(PA) was different from in the MCS, because the strain with the highest
cost (I3) became extinct in the MCS. In addition, the oscillatory solution
was observed in a parameter range [Fig. 13], and the strain with the
middle cost (I2) was dominant, which was similar to the results of previous
studies [29, 30, 31, 32, 25][Figs. 11 and 12]. In addition, when s = 0
[Fig. 9(ii)], the response to βS was the same as in the 2-strain model (the
range of the coexistence phase decreased and the pathogen population
was occupied by the strain with the highest cost at a large βS).
2.2.2.3 Multiple-strain models
In the multiple-strain models, there are many equilibrium states: ex-
tinction, disease-free, occupation of a strain, coexistence of various strains,
and coexistence of all strains.
49
0 10 20 30 40 50 60 70
0.5
1
0 10 20 30 40 50 60 70
0.5
1
0 10 20 30 40 50 60 70
0.5
1
0 5 10 15 20 25
0.5
1
0 5 10 15 20 25
0.5
1
0 5 10 15 20 25
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
s = 0.5(a)
ρ∗0ρ∗S
ρ∗I1ρ∗I2
ρ∗I3
ρ∗I3
(b)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2
ρ∗I3
ρ∗I3
(c)
ρ∗0
ρ∗S ρ∗I1ρ∗I2
s = 1.0(a)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2 ρ∗I3ρ∗I3
(b)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2
ρ∗I3
ρ∗I3
(c)
ρ∗0ρ∗S
ρ∗I1ρ∗I2
ρ∗I3
s = 1.5(a)
ρ∗0ρ∗S
ρ∗I1ρ∗I2
ρ∗I3
(b)
ρ∗0
ρ∗S
ρ∗I1ρ∗I2 ρ∗I3ρ∗I3
(c)
ρ∗0
ρ∗S
ρ∗I1ρ∗I2
ρ∗I3P
Growth rate βS
Equilibrium
Density
ofeach
strain
Fig 11. The equilibrium value of each strain depends on the
superinfection rate in the 3-strain model. mI1 = 5 and ∆m = 5 in
the figures. The I, II and III differ in their values of s (=0.5, 1.0 and
1.5, respectively). (a) MA, (b) PA and (c) MCS
50
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 2 4 6 8 10
0.5
1
0 5 10 15 20 25 30
0.5
1
0 5 10 15 20 25 30
0.5
1
0 5 10 15 20 25 30
0.5
1
0 5 10 15 20 25 30
0.5
1
0 5 10 15 20 25 30
0.5
1
0 5 10 15 20 25 30
0.5
1
∆m = 5(a)
ρ∗0ρ∗S
ρ∗I1ρ∗I2
ρ∗I3
(b)
ρ∗0
ρ∗S
ρ∗I1ρ∗I2 ρ∗I3ρ∗I3
(c)
ρ∗0
ρ∗S
ρ∗I1ρ∗I2
ρ∗I3PP
∆m = 10(a)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2ρ∗I3 ρ∗I3
(b)
ρ∗0ρ∗S
ρ∗I1ρ∗I2 ρ∗I3
ρ∗I3
(c)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2 ρ∗I3
∆m = 15(a)
ρ∗0ρ∗S
ρ∗I1ρ∗I2
ρ∗I3
ρ∗I3
(b)
ρ∗0ρ∗S
ρ∗I1ρ∗I2
ρ∗I3ρ∗I3
(c)
ρ∗0 ρ∗S
ρ∗I1ρ∗I2 ρ∗I3
Growth rate βS
Equilibrium
Density
ofeach
strain
Fig 12. The equilibrium value of each strain depends on the
mortality cost in the 3-strain model. mI1 = 5 and s = 1.5 in the
figures. The I, II and III differ in their values of ∆m (=5, 10, 15). (a)
MA, (b) PA and (c) MCS.
51
200 400 600 800 1000
0.2
0.4
0.6
0.8
1.0
200 400 600 800 1000
0.2
0.4
0.6
0.8
200 400 600 800 1000
0.2
0.4
0.6
0.8
200 400 600 800 1000
0.2
0.4
0.6
0.8
1.0
200 400 600 800 1000
0.2
0.4
0.6
0.8
200 400 600 800 1000
0.2
0.4
0.6
0.8
Time (t)
Global
density
ofeach
state
I.(a)
II.(a)
(b)
(b)
(c)
(c)
Fig 13. Time series for the global density of each state in the
3-strain model. The equilibrium value of the global density of each
state is plotted. The parameter values are set as: I. s = 1.0, mI1 = 10,
∆m = 15 and II. s = 1.5, mI1 = 10, ∆m = 20 with (a) βS = 3, (b)
βS = 5 and (c) βS = 7. In the 3-strain model, the oscillatory solution is
observed in a particular parameter range during the coexistence phase
[panel (b)].
The results of the 10-strain [Fig. 14] and 25-strain models [Fig. 15]
were plotted at ∆m = 5 with varying βS and s values. As a result, a
smaller value of βS or s led to the dominance of the strain with the higher
cost [Figs. 14 and 15], and the cost of the dominant strain shifted to a
lower value as these parameter values increased, which was similar to the
results of the 2-strain and 3-strain models. In addition, when the βS was
small, the oscillatory solution was observed [Fig. 16], and when the s was
also small, there was a possibility of extinction in the MCS when the n
was too large [Fig. 15].
52
Number of strain
Globaldensity
I. s = 0.5, βS = 25
II.s = 1.0, βS = 25
III.s = 1.5, βS = 25
IV.s = 1.5, βS = 5
MA PA MCS
Fig 14. The equilibrium density distribution of strains in the
10-strain model. The left, center and right panels show the results of
simulations using the MA, PA and MCS, respectively. mI1 = 5 and
∆m = 5. The other parameter values are: I. s = 0.5, βS = 25, II.
s = 1.0, βS = 25, III. s = 1.5, βS = 25 and IV. s = 1.5, βS = 5.
53
Number of strain
Globaldensity
I. s = 0.5, βS = 25
II.s = 1.0, βS = 25
III.s = 1.5, βS = 25
IV.s = 1.5, βS = 5
MA PA MCS
Fig 15. The equilibrium density distribution of strains in the
25-strain model. The left, center and right panels show the results of
simulations using the MA, PA and MCS, respectively. mI1 = 5 and
∆m = 5. The other parameter values are: I. s = 0.5, βS = 5, II. s = 0.5,
βS = 25, III. s = 1.0, βS = 25 and IV. s = 1.5, βS = 25.
200 400 600 800 1000
0.2
0.4
0.6
0.8
20 40 60 80 100
0.2
0.4
0.6
0.8
time t
Globalden
sity
Fig 16. Time series in the 25-strain model. The equilibrium value
of the global density of each state is plotted. In the figures, mI1 = 5,
∆m = 5 and βS = 5. Additionally, (a) s = 1.5 and (b) s = 0.5.
54
2.3 Discussion
Superinfection events increase the fitness of pathogens by widely spread-
ing them, through the additional associated transmission pathways, within
a plant population. Several studies have analyzed the process of pathogen
spread by approximating the lattice space [45, 47, 46] and the superinfec-
tion process using mathematical models without spatial structures [30].
Sato et al. (1994) studied the case of dS = 1 in our single-strain model.
Their model had three equilibrium states, (disease-free, endemic and epi-
demic), and they derived equilibrium values at two states (disease-free
and endemic) along with their explicit local stability conditions. How-
ever, they did not obtain the epidemic equilibrium value and its local
stability. Haraguchi and Sasaki (2000) considered that there is no trade-
off between the infection rate and the virulence of a pathogen, and they
assumed that multiple pathogens have different virulence levels. They
examined the ESS of the infection rate using a numerical simulation and
discussed the evolution of the pathogen’s infection rate. Their simulation
suggested that pathogens evolve to an intermediate infection rate. Sat-
ulovsky and Tome (1994) considered a transition rule similar to that in
55
our model and assumed a correlation between the transition rates of each
state (βS +mI + dI = 1). According to their PA and MCS results, there
are four equilibrium phases (three stationary and one oscillation phase),
and if the transition rate to state 0 (dI) is small, then the Hopf bifurca-
tion occurs. Nowak and May (1994) examined the superinfection events
using an ordinary differential equation. As a result, superinfection leads
to the maintenance of the pathogen strain’s polymorphism, and the os-
cillatory solution (competition among plants and pathogens) is observed
when there is more than one strain. However, they did not consider spa-
tial structures or host reproductive dynamics. They assumed that the
host constantly increases.
To consider the effects of spatial structures, plant reproduction and
pathogen propagation dynamics during superinfection in a clonal plant
population on a lattice space were analyzed. Five models (the single-
strain model and four multiple-strain models) that included interactions
among the host plant and several strains of pathogen, were analyzed, and
the MA and PA were adopted to analyze the dynamics of the models. In
addition, the validity of the approximation methods in comparison with
56
the MCS was checked.
In the single-strain model (n = 2), the value of the epidemic thresh-
old depends only on the mortality cost, which means that establishing a
pathogen within a plant population depends only on the pathogen’s abil-
ity, regardless of the plant. In addition, the density of healthy individuals
decreases with increasing growth rates and mortality costs [panel (i) in
Fig. 6]. By contrast, the density of infected individuals increases with the
growth rate and decreases with an increase in the mortality cost [panel
(ii) in Fig. 6]. Therefore, plants should not increase their growth rate to
maintain a large population size when infected by a systemic pathogen,
and the pathogen should evolve a low mortality cost that is higher than
the epidemic threshold to maintain their population and that of their
hosts.
Additionally, if the parameter values exceed the bifurcation thresh-
old, then the Hopf bifurcation occurs and the periodic solution, in which
plants and pathogens continue to compete forever, is observed using the
PA and MCS, but not the MA. Thus, the effects of local interactions are
important when expressing the dynamics of the pathogen propagation
57
process. However, the MA (which neglects the local interactions) is use-
ful for analyzing the dynamics within a measured situation, such as the
number of equilibrium states in the model.
In the multiple-strain models (n > 2), it was assumed that the multi-
ple pathogen strains had different mortality costs, and that the already
infected individuals were superinfected by strains with lower mortality
costs. The analytical results of the MA showed that there are a lot of
equilibrium states: extinction, disease-free, occupation of a strain, co-
existence of various strains, and coexistence of all strains (Table 1 and
2). Based on the MA, PA and MCS results, the equilibrium phase and
the dominance of a strain in the coexistence phase depend on parameter
values, and the oscillatory solution is observed in the coexistence phases,
except for in the 2-strain model using the PA and MCS [Figs. 10 and 13].
In addition, the genetic diversity of a pathogen is maintained by a
decrease in superinfection events. In fact, the parameter range of the
coexistence phase increases with a decrease in superinfection rates, even
when the difference in the mortality cost is small. This is because a su-
perinfection event is conducive to a strong competition among strains,
58
which is caused by additional transmission routes that decrease the dif-
ferences in infection rates among the strains. However, if superinfection
does not occur (s = 0), then the range of the coexistence phase decreases
[Fig. 9]. Thus, superinfection is important to maintain genetic diversity.
Additionally, when the plant growth rate increases, the pathogen popu-
lation is eventually occupied by a strain, regardless of the superinfection
rate [Figs. 7, 9 and 11]. Thus, the increase in the plant growth rate
causes a decrease in the genetic diversity of the pathogen. For healthy
individuals, too high of a growth rate provides them no benefit. Healthy
individuals can increase their abundance through the growth rate in the
coexistence phase of several strains. However, if the growth rate is too
high, an equilibrium phase shifts to the phase of occupation by one strain,
and the density of healthy individuals then decreases as the growth rate
increases. The dynamics follow those of the single-strain model in this
phase.
The results are summarized as follows: (i) The strain with an inter-
mediate cost became dominant, similar to the results of previous stud-
ies [29, 30, 31], when both the superinfection and growth rates were low.
59
However, a high superinfection or growth rate led to the dominance of
the strain with the lowest cost in our model. Additionally, the pathogen
received more benefit due to a low mortality cost when the hosts grow
rapidly [5]; (ii) The competition among strains occurred in the coexis-
tence of various strains phase when using the PA and MCS in the n > 3
models; (iii) Too high a growth rate led to occupation by the strain with
the lowest cost. Thus, competition between the strain and the hosts oc-
curred, and, therefore, the host population decreased in all of the models;
(iv) Pathogens easily maintained their genetic diversity when there was a
low superinfection rate. However, if they did not superinfect, such main-
tenance became difficult; and (v) When the growth rate of a plant was
low, an individual at a local site was strongly interconnected by distant
individuals because the MA and PA did not apply in this case.
In conclusion, pathogens maintain their genetic diversity through su-
perinfection events and a moderate mortality cost relative to growth rate.
Thus, their mortality costs and superinfection rates evolve based on the
host plant’s ability to maintain their populations. By contrast, the num-
ber of healthy individuals (plants) increases in (all and several strains)
60
coexistence phases with the growth rate. Thus, when systemic pathogens
invade the plant population, the plant’s growth rate evolves to be slightly
lower than the threshold value at which the equilibrium phase shifts to
the phase of occupation by a strain to increase their population.
61
Chapter 3 Seed propagatin model
Many clonal plants have two breeding systems, vegetative and seed
propagation [Fig. 1]. The seed-propagated offspring (seedlings) have a
higher mortality rate because of their long-distance dispersal, which does
not allow them to be supported by their parents (no physical connec-
tion). The vegetatively propagated offspring (vegetative propagules) have
a lower mortality rate because they supply resources among themselves
through interconnected ramets [3, 4]. However, if systemic pathogens
invade the population, the interconnected ramets become pathways of
pathogen spread. Thus, the balance between the two breeding systems
has an important role in the defensive behavior against systemic pathogens.
The balance between the breeding systems, vegetative and seed prop-
agation, has been studied experimentally [52, 53, 54, 55]. According
to these studies, the balance is determined by several functions, such as
resource allocation, competitive ability and colonization capacity. For
instance, if the resource is distributed heterogeneously in space, then
vegetative propagation has an advantage over seed propagation because
the vegetative propagules can be supplied resources from local colonies
62
of clones [56]. On the contrary, if the resource is distributed homoge-
neously, then seed propagation has an advantage over vegetative propa-
gation because seed propagation can spread the offspring long distances
and distribute them over an entire habitat [57].
Several mathematical models, such as transition matrix model [35],
reaction-diffusion equation model [36], lattice model [37, 38, 39] and
individual-based model [40, 41, 42, 43, 44], are used to express the plant
reproductive process. They analyze the optimal balance of the breed-
ing systems depending on several functions, such as resource distribu-
tion [41, 56], distance from the parents [39] and the density of individu-
als [35, 44]. Harada et al. (1996) analyzed the plant reproduction process
with seed propagation using the TCP on lattice space. They assumed that
the optimal balance of the breeding systems depends on the distance from
the parents. Ikegami et al. (2012) examined the effects of plant density
and mortality on the adoption of a breeding system using a computer
simulation on lattice space. However, these studies did not consider the
effects of pathogen propagation on the optimal breeding system balance.
63
3.1 Model
A model that included the plant reproductive process (both vegetative
and seed propagation) and the pathogen transmission process was con-
structed. The dynamics of the model is a continuous Markov process on
a lattice space. The states of each site are presented as empty (”0”),
susceptible (healthy) individual (”S”), infected individual (”I”), the in-
trinsic reproduction rate of the plant by mS, the intrinsic transmission
rate of the pathogen by mI, the proportion of vegetative propagation by
α, and the mortality rate of individuals of each state by 1. Additionally,
ρσ (t) (σ ∈ 0, S, I) is the probability that a randomly chosen site has
state σ at time t. Thus, ρσ (t) indicates the global density of the site with
state σ.
Four demographic processes were configured: (i) vegetative propaga-
tion (VP); (ii) seed propagation (SP); (iii) infection (IP); and (iv) death
(DP). The plant reproductive processes (VP and SP) are represented by
transitions from state ”0” to ”S”, which indicates that plants reproduce
offspring by either breeding system into an open area (an empty site is
then occupied by a healthy individual). IP is represented by the transi-
64
tion from state ”S” to ”I”, which indicates that healthy individuals are
infected by pathogens. DP is represented by the transition from ”S” or
”I” to ”0”, which represents the death of healthy or infected individuals,
respectively, from natural causes and the virulence of the pathogen, re-
spectively. In addition, infected individuals can not return to health and
reproduce their offspring.
Plant offspring inhabit distant and close open areas through seed and
vegetative propagation, respectively. Thus, an empty site becomes oc-
cupied by a healthy individual through reproduction from a randomly
chosen healthy site by seed propagation or from nearest-neighbor healthy
sites by vegetative propagation. Additionally, the pathogens transmit
from infected individual to surrounding healthy individual. Thus, in the
vegetative propagation and infection processes, the transition rate de-
pends on the states of nearest-neighbor sites. For instance, a healthy
individual is likely to become infected if the individual is surrounded by
infected individuals. These processes were described using the following
notation that is often used to explain the CP:
65
(i) 0 → S at rate (1− α)mSρS
(ii) 0 → S at rateαmSn (S)
z
(iii) S → I at ratemIn (I)
z
(iv) S, I → 0 at rate 1
(TP)
Parameter n(σ) is the number of σ-sites in the nearest neighbors of the
focal sites, z (= 2) is the number of nearest-neighbor sites (e.g. z = 4 for
a von Neumann neighborhood on a two-dimensional square lattice). The
vegetative propagation and infection events occur at rates proportional
to the number of the healthy and infected states in the nearest-neighbor
sites, respectively.
The above dynamics were described using a master equation that incor-
porates the additional variables Pσiσjand Pσiσjσk
(σi, σj, σk ∈ Σ), which
are referred to as pair density and triplet density, respectively. The vari-
ables represent the probability that the randomly chosen two or three
neighboring sites have the state σi, σj and σk at time t. Thus, Pσiσj=
Pσjσiand Pσiσjσk
= Pσkσjσi(master equation) Here, the positive and neg-
ative terms indicate transition probabilities from any state and to any
66
other state, respectively.
P00 =2
− (1− α)mSρSP00︸ ︷︷ ︸(i)SP
−αmS (z − 1)
zPS00︸ ︷︷ ︸
(ii)VP
+PS0 + PI0︸ ︷︷ ︸(iv)DP
P0S = (1− α)mSρS (P00 − P0S)︸ ︷︷ ︸
(i)
+αmS
[(z − 1)
z(PS00 − PS0S)−
1
zP0S
]︸ ︷︷ ︸
(ii)
−(z − 1)mI
zPIS0︸ ︷︷ ︸
(iii)IP
+(PSS − P0S) + PSI︸ ︷︷ ︸(iv)
PI0 =− (1− α)mSρSPI0︸ ︷︷ ︸(i)
−αmS (z − 1)
zPS0I︸ ︷︷ ︸
(ii)
+mI (z − 1)
zPIS0︸ ︷︷ ︸
(iii)
+PIS + (PII − PI0)︸ ︷︷ ︸(iv)
PSS =2
(1− α)mSρSPS0︸ ︷︷ ︸(i)
+αmS
(1
zPS0 +
(z − 1)
zPS0S
)︸ ︷︷ ︸
(ii)
+mI (z − 1)
zPISS︸ ︷︷ ︸
(iii)
−PSS︸︷︷︸(iv)
PIS = (1− α)mSρSPI0︸ ︷︷ ︸
(i)
+αmS (z − 1)
zPS0I︸ ︷︷ ︸
(ii)
+mI
[(z − 1)
z(PISS − PISI)−
1
zPIS
]︸ ︷︷ ︸
(iii)
−PIS − PIS︸ ︷︷ ︸(iv)
PII =2
mI
1
zPIS +
(z − 1)
zPISI
︸ ︷︷ ︸
(iii)
−PII︸︷︷︸(iv)
(6)
Here, the positive and negative terms indicate transition probabilities
67
from any state and to any other state, respectively.
3.2 Result
The two models, single population and mixed population models, were
analyzed in homogeneous environment to ease the analysis although the
merit of vegetative propagation through resource sharing disappears (i.e.
we can examine the effect of long distance dispersal through seed propaga-
tion on pathogen spread directly). In the single population model, there
are one type of plant and pathogen, and the plant adjusted the breed-
ing system’s balance to block the spread of the pathogen. In contrast,
in the mixed population, there are several types of plants and one type
of pathogen, and it was expected that the optimal balance of the breed-
ing systems is different from that of the single population model because
of competition among the plant types and the block to the pathogen’s
spread. The system was analyzed using the MA, PA (in single population
model) and MCS (in both models).
The MCS was conducted 100 times at each parameter set in a two-
dimensional square lattice torus (whose size is 100×100), and the average
68
value of the 100 trials was calculated.
As a result, the master equation using the MA does not depend on
the parameter α (Appendix. C.1). Thus, the PA and MCS were used
in subsequent analyses. As a result of the PA, the system has three
equilibrium states (extinction, disease-free and epidemic), similarly to a
previous study [45] (corresponding to the case of α = 1 in our model),
and two thresholds [Fig. 17(a)], which are referred to as the extinction
and epidemic threshold, were derived from the local stability analysis
(Appendix. C.1). Additionally, the results of the PA and MCS were com-
pared to examine the effects of the distant sites (i.e. spatial structure).
Here, the new parameter µ = mI/mS was introduced, and it represented
the relative scale to plant fecundity of pathogen infectivity.
3.2.1 Single population
3.2.1.1 Extinction phase
In this phase, both the healthy and infected plants went to extinction.
Thus, all of the sites converted to the empty state; ρ∗0 = 1, ρ∗S = 0, and
ρ∗I = 0. The extinction equilibrium became stable when the values of the
69
parameters were below the extinction threshold [Fig. 17(a)]. If the mS
was large enough, then the extinction equilibrium was always unstable,
regardless of α. Thus, plants with low fecundity could prevent extinction
by increasing the proportion of seed propagation in their reproductive
strategy. Using the PA, the following extinction threshold was analyti-
cally derived (Appendix C.2):
(mS)c =α− z +
√(z + α)2 − 4zα2
2α (1− α)
The threshold did not depend on mI; therefore, the infectivity of the
pathogen was irrelevant to the extinction of the host plants.
In a comparison of the MCS with the PA [Fig. 18], the discrepancy in
the equilibrium value increased with an increase in α and a decrease inmS.
Thus, the importance of the spatial structure (the effects of distant sites)
increased with an increase in the proportion of vegetative propagation
and a decrease in fecundity.
3.2.1.2 Disease-free phase
In this phase, the pathogens disappeared with the infected individuals
70
0.0 0.2 0.4 0.6 0.8 1.0
1.0
1.1
1.2
1.3
1.4(a)
Extin tion
Disease-free
Extin tion threshold
Proportion of vegetative propagation ()
F
e
u
n
d
i
t
y
(
m
S
)
0.0 0.2 0.4 0.6 0.8 1.0
0.35
0.36
0.37
0.38
0.39
0.40(b)
Disease-free
Epidemi
Epidemi threshold
Proportion of vegetative propagation()
P
r
o
p
o
r
t
i
o
n
o
f
i
n
f
e
t
i
v
i
t
y
t
o
f
e
u
n
d
i
t
y
(
)
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.3
0.4
0.5
0.6 (c)mS = 4
mS = 5
mS = 6
Proportion of vegetative propagation(α)
Proportion
ofinfectivityto
fecundity(µ)
Fig 17. The extinction and epidemic thresholds in the pair
approximation. (a) The solid line indicates the extinction threshold.
When the values of the parameters are below the threshold, the
extinction equilibrium is stable, and when the value exceeds that of the
threshold, it becomes unstable, and the disease-free equilibrium is
stable. Thus, if the fecundity of the plant is low, then the plant becomes
extinct. Additionally, when the value of the fecundity is close to that of
the threshold, the decrease in the proportion of vegetative propagation
is effective in protecting the plant from extinction. (b) The solid line
indicates the epidemic threshold at mS = 5. When the value of
parameters is below the threshold, the disease-free equilibrium is stable,
and when the value exceeds the threshold, the disease-free equilibrium
becomes unstable and epidemic equilibrium is stable. Thus, if invading
pathogens have low infectivity levels, the pathogens do not spread
within the plant population. Additionally, when the values of the
parameters are close to that of the threshold, the increase in the
proportion of vegetative propagation is effective in protecting the plant
from the epidemic. (c) The epidemic threshold shifts downward with an
increase in mS. Thus, a plant population with a high fecundity is likely
to lead to an epidemic, even when pathogens having a low infectivity
level invade the population. 71
0.2 0.4 0.6 0.8 1.0
0.02
0.04
0.06
0.08
0.2 0.4 0.6 0.8 1.0
0.05
0.10
0.15
0.2 0.4 0.6 0.8 1.0
0.05
0.10
0.15
0.20
0.2 0.4 0.6 0.8 1.0
0.05
0.10
0.15
0.20
0.25
Proportion of vegetative propagation (α)
Global
density
ofhealthyindividual(ρ
∗ S) (a) (b)
(c) (d)
Fig 18. The comparison of the equilibrium values of ρSbetween the pair approximation and Monte Carlo simulation.
The solid and dotted lines indicate the the pair approximation and
Monte Carlo simulation, respectively. The figure shows the ρ∗S in the
extinction and disease-free phases. (a-d) differ in the value of mS (=1.2,
1.3, 1.4 and 1.5, respectively). From the comparison between the pair
approximation and Monte Carlo simulation, the discrepancy increases
with an increase in α and a decrease in mS.
72
(ρ∗I = 0). The disease-free equilibrium became stable when the values of
the parameters were between the values of the extinction and epidemic
thresholds [Fig. 17 (a )and (b)]. The ρ∗S increased with a decrease in α
[Figs. 18 and 19] or an increase in µ [Fig. 20]. However, an increase in
the mS led to a lower value for the epidemic threshold [Fig. 17 (c)]. Thus,
the increase of mI led to a transition to the epidemic phase, even when
mS or α was low.
3.2.1.3 Epidemic phase
In this phase, pathogens could invade and spread within a plant pop-
ulation. The value of ρ∗S reached its maximum at α = 1 or 0, which
depended on µ. In particular, the increase in µ led to a shift in the value
of α from 1 to 0 [Fig. 21 (I.)]. However, the value of ρ∗I increased with
a decrease in α, regardless of µ (the ρ∗I reached its maximum value at
α = 0) [Fig. 21 (II.)]. In addition, the proportion of healthy individuals,
out of all of the individuals (ρ∗S/ (ρ∗S + ρ∗I )), also increased with a decrease
in α, regardless of µ [Fig. 21 (III.)]. The increase of µ led to a shift to the
disease-free phase and then a shift to the extinction phase [Fig. 17(a)].
In the model of a previous study [45], in which seed propagation was
73
0.2 0.4 0.6 0.8 1.0
0.788
0.790
0.792
0.794
0.796
0.798
0.2 0.4 0.6 0.8 1.0
0.780
0.785
0.790
0.795
0.2 0.4 0.6 0.8 1.0
0.76
0.77
0.78
0.79
0.2 0.4 0.6 0.8 1.0
0.824
0.826
0.828
0.830
0.832
0.834
0.2 0.4 0.6 0.8 1.0
0.80
0.81
0.82
0.83
0.2 0.4 0.6 0.8 1.0
0.76
0.78
0.80
0.82
0.2 0.4 0.6 0.8 1.0
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0.2 0.4 0.6 0.8 1.0
0.005
0.010
0.015
0.020
0.2 0.4 0.6 0.8 1.0
0.01
0.02
0.03
0.04
0.2 0.4 0.6 0.8 1.0
0.0005
0.0010
0.0015
0.0020
0.0025
0.2 0.4 0.6 0.8 1.0
0.01
0.02
0.03
0.04
0.2 0.4 0.6 0.8 1.0
0.02
0.04
0.06
0.08
Proportion of vegetative propagation(α)
Global
density
ofhealthyindividual(ρ
∗ S)
Global
density
ofinfected
individual(ρ
∗ I)
I.
II.
(a) (b) (c)
Epidemic Disease-free
Fig 19. The equilibrium values of ρS and ρI depend on α.
Groups I and II are plots of the values of ρ∗S and ρ∗I , respectively, in
disease-free and epidemic phases. The upper panels in each group show
the pair approximation, and the lower panels show the Monte Carlo
simulation. (a-c) differ in the value of µ (=3.6, 3.7 and 3.8, respectively)
and the appropriate value of fecundity (mS) was selected to show the
disease-free and epidemic phases (mS = 5 in the pair approximation and
mS = 6 in the Monte Carlo simulation). The ρ∗S increases with a
decrease in α during the disease-free phase and with an increase in α
during the epidemic phase.
74
0.2 0.4 0.6 0.8 1.0
0.45
0.50
0.55
0.60
0.65
0.70
0.2 0.4 0.6 0.8 1.0
0.4
0.5
0.6
0.7
0.8
0.2 0.4 0.6 0.8 1.0
0.5
0.6
0.7
0.8
0.2 0.4 0.6 0.8 1.0
0.4
0.5
0.6
0.7
0.8
0.9
Proportion of infectivity to fecundity(µ)
Global
density
ofhealthyindividual(ρ
∗ S) (a) (b)
Fig 20. The equilibrium values of ρS depend on µ in the
disease-free and epidemic phases. The upper panels show the pair
approximations, and the lower panels show the Monte Carlo
simulations. α = 0.8, and (a) mS = 4 and (b) mS = 5. The increase in
the mS leads to a lower epidemic threshold value.
75
0.0 0.2 0.4 0.6 0.8 1.0
0.197
0.198
0.199
0.200
0.201
0.2 0.4 0.6 0.8 1.0
0.1835
0.1840
0.1845
0.1850
0.2 0.4 0.6 0.8 1.0
0.172
0.173
0.174
0.175
0.0 0.2 0.4 0.6 0.8 1.0
0.16
0.17
0.18
0.19
0.20
0.2 0.4 0.6 0.8 1.0
0.25
0.26
0.27
0.28
0.2 0.4 0.6 0.8 1.0
0.130
0.135
0.140
0.145
0.150
0.2 0.4 0.6 0.8 1.0
0.20
0.21
0.22
0.23
0.24
0.2 0.4 0.6 0.8 1.0
0.19
0.20
0.21
0.22
0.23
0.24
0.2 0.4 0.6 0.8 1.0
0.19
0.20
0.21
0.22
0.23
0.2 0.4 0.6 0.8 1.0
0.26
0.28
0.30
0.32
0.34
0.36
0.2 0.4 0.6 0.8 1.0
0.20
0.25
0.30
0.2 0.4 0.6 0.8 1.0
0.25
0.30
0.2 0.4 0.6 0.8 1.0
0.46
0.48
0.50
0.52
0.2 0.4 0.6 0.8 1.0
0.46
0.48
0.50
0.0 0.2 0.4 0.6 0.8 1.0
0.44
0.46
0.48
0.50
0.2 0.4 0.6 0.8 1.0
0.45
0.50
0.55
0.2 0.4 0.6 0.8 1.0
0.34
0.36
0.38
0.40
0.42
0.44
0.2 0.4 0.6 0.8 1.0
0.32
0.34
0.36
0.38
0.40
0.42
Proportion of vegetative propagation(α)
Global
density
ofhealthyindividual(ρ
∗ S)
Global
density
ofinfected
individual(ρ
∗ I)
Proportion
ofhealthy
individual
inallindividual
I.
II.
III.
(a) (b) (c)
Fig 21. The values of ρ∗S (I.), ρ∗I (II.) and ρ∗S/ρ∗S + ρ∗I (III.)
depend on α in the epidemic phase (mS = 4). The upper panels
show the pair approximation, and the lower panels show the Monte
Carlo simulation. The panels (a-c) differ in the value of µ (=2.1, 2.3 and
2.5, respectively). From the panels in I, the value of α, which maximizes
the value of ρ∗S, depends on the value of µ. When µ is low, the ρ∗Sreaches its maximum at α = 0. However, when µ is high, the value of α
that maximizes the ρ∗S is equal to 1. Panels II and III show that the
value of ρ∗I and the proportion of healthy individuals out of all of the
individuals, respectively, increase with a decrease in α, regardless of µ.
76
not assumed, the epidemic equilibrium shifted to unstable and the oscilla-
tory solution (limit cycle) was observed at a high mS and mI , indicating
that the Hopf bifurcation occurred. The oscillatory solution indicated
that competition between plants and pathogens occurred, and the com-
petition cost was the maintenance of the plant population. However, in
the present model, the epidemic equilibrium was always stable if the val-
ues of the parameters exceed the epidemic threshold, and the bifurcation
was not observed. Thus, the plants could evade competition through seed
propagation.
3.2.2 Optimal proportion of vegetative propagation (Mixed
population)
To explore the defensive behavior through the breeding systems against
pathogen spread by applying the above model (single population), it was
assumed that the mixed plant population consisted of 11 types of plants
having different proportions of vegetative propagation (α = 0, 0.1, 0.2, · · · , 1).
The system was analyzed in two cases, uninfected and infected popula-
tions, using only the MCS.
77
3.2.2.1 Uninfected population
In this case, pathogens did not invade the plant population. Thus, the
initial density of the infected individuals was equal to 0 (ρI (0) = 0). The
plant type with the lower α became dominant when the mS was low, and
the α of the dominant type increased with mS [Fig. 22]. Additionally,
when the mS was low enough, plants became extinct. Therefore, when
plants had a high fecundity, vegetative propagation was advantageous
over seed propagation.
3.2.2.2 Infected population
In this case, plants were infected by pathogens (pathogens invaded the
plant population). The system had three equilibrium phases, extinction,
disease-free and epidemic, and the stable phase switched depending on
the values of the parameters, similar to in a single population. Here,
the pathogen type with the lower α became dominant when mS was low,
regardless of µ, and when mS was high, the α of the dominant type shifted
from an intermediate to lower value with an increase in µ (mI) [Fig. 23].
Thus, seed propagation was advantageous over vegetative propagation
even when the plants had a high fecundity, which was different from in
78
0 5 10 15 20 25 30
0.2
0.4
0.6
0.8
1.0
Fecundity of plant (mS)
Proportion
ofvegetative
propagationof
dom
inan
ttype(α
)
Fig 22. The transition of the dominant type of plant depends
on mS in mixed population. This shows that pathogens did not
invade the plant population. The initial density of the infected
individual is set equal to 0, and the value of α for the dominant type is
plotted. The type with the lower α becomes dominant when the mS is
low, and the α changes to a higher value with an increase of mS.
Additionally, when the mS is low enough, plants become extinct.
79
an uninfected population.
0 1 2 3 4 5
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5
0.2
0.4
0.6
0.8
1.0
Proportion of infectivity to fecundity(µ)
Proportion
ofvegetative
propagationofdominanttype(α
)
(a) (b)
Fig 23. The transition of the dominant type of plant depends
on µ in a mixed population. This shows the case when plants are
infected by pathogens. (a) mS = 30 and (b) mS = 15, and the value of α
of the dominant type is plotted. From panel (a), the α of the dominant
type shifts from an intermediate to lower value with an increase in µ.
From panel (b), the individual with lower α becomes dominant when
mS is low, regardless of µ.
3.3 Discussion
Adjusting the breeding system’s balance is effective in defending against
the spread of systemic pathogens, which transmit to close individuals,
within a plant population. According to field studies, the vegetative
propagules have lower mortality rates than seedlings because of resource
sharing through interconnected ramets. However, the ramets also trans-
port viruses along with the resources through their vascular systems, and
80
provide space for the hyphal growth of fungal pathogens in their vascu-
lar vessels. Additionally, the pathogens also spread through air and a
dense population of plants assists the pathogen’s spread. Thus, vegeta-
tive propagation has a disadvantage over seed propagation when systemic
pathogens invade a plant population because the breeding system repro-
duces offspring close to the parents.
There are many approaches that use the lattice model to analyze the
breeding dynamics of clonal plants [37, 38, 39, 44], as well as pathogen
transmission dynamics [45, 46]. Among them, studies of breeding dy-
namics examined the effects of spatial structures on the reproductive
strategy, represented in plants by the competition between vegetative
and seed propagation. Harada and Iwasa (1994), Harada et al. (1997)
and Harada (1999) considered two types of plants. One type reproduces
through both seed and vegetative propagation (mixed strategy), and the
other one reproduces through only vegetative propagation (pure strat-
egy). They analyzed the competition dynamics between the two types
by adopting the PA. Additionally, they assumed that the proportion of
both breeding propagation systems in the mixed strategy depends on the
81
distance from the parents, and they examined the ESS of the balance
between vegetative and seed propagation using computer simulations.
Ikegami et al. (2012) considered the effects of plant density and mor-
tality on the adoption of breeding systems. They assumed that each
individual switches between seed and vegetative propagation depending
on the local density of the individuals and that the switching threshold
of the reproductive pattern is affected by mortality. They analyzed, by
computer simulation, the optimal switching strategy based on local den-
sity and mortality. These studies did not consider the effects of pathogens
on the reproductive strategy.
However, studies of pathogen transmission dynamics examined the
transition threshold of the equilibrium phase (mainly extinction, disease-
free and epidemic phases). Sato et al. (1994) analyzed in particular the
phase transition in the TCP using the PA. Haraguchi and Sasaki (2000)
assumed that multiple pathogens have different virulence levels. They ex-
amined the ESS of the infection rate using a numerical simulation. Their
simulation suggested that pathogens evolve to an intermediate infection
rate. However, their models could not express the seed propagation pro-
82
cess because the models were constructed based on the basic TCP. Thus,
it is necessary to modify their models to describe both the plant reproduc-
tion process, including seed propagation, and the pathogen propagation
process.
The new model, as presented here, has three (positive) equilibrium
phases, extinction, disease-free and epidemic, using the PA and the spa-
tial structure affects the system when the fecundity is low, similar to
supreinfectin model (in Section.Chapter 2). The stability condition of
the extinction phase requires that the values of the parameters are be-
low the extinction threshold, which does not depend on the virulence of
the pathogens, as determined by an analysis using the PA. Thus, the
plants become extinct due to low fecundity, and with the death of the
host, the pathogens also become extinct. A stable equilibrium shifts from
extinction equilibrium to disease-free equilibrium when the values of the
parameters exceed the extinction threshold. In the disease-free phase,
the increase in fecundity leads to a large plant population size. Addi-
tionally, the population size increases with the proportion of vegetative
propagation. Thus, seed propagation has an advantage over vegetative
83
propagation in a homogeneous environment. However, high fecundity or
a low proportion of vegetative propagation leads to the downward shift
of the epidemic threshold [Figs. 17(c)]. Therefore, when plant fecundity
or the proportion of vegetative propagation is high, pathogens spread
within the plant population, even if the infectivity of the pathogen is
weak. Thus, the decrease in the production of offspring or aggressive
vegetative propagation (even when plants produce a number of offspring)
makes it difficult to prevent an epidemic because plants assist the spread
of the pathogen by reproducing susceptible individuals.
A stable equilibrium shifts from disease-free to epidemic equilibrium
when the values of the parameters exceed the epidemic threshold. In this
phase, the relative merit of both breeding propagation systems changes
depending on the relative scale to plant fecundity of pathogen infectivity.
Vegetative propagation has an advantage when the relative scale is low,
and seed propagation has an advantage when the relative scale is high. If
the infectivity of the pathogen is weak, then the breeding destinations of
new vegetative propagules become farther and farther away from infected
individuals with time because of a faster reproduction rate relative to the
84
pathogen transmission rate. The breeding destination of seedlings is se-
lected randomly in the entire area, and the seedlings may be reproduced
close to the infected individual (assisting the spread of the pathogen),
even when the habitat of the parents is far from the infected individ-
ual. Thus, plants inhibit pathogen spread by increasing the proportion
of vegetative propagation if the pathogen is weakly infective. If the infec-
tivity of the invading pathogen is strong (pathogen transmission rate is
faster than the plant reproduction rate), then the production of offspring
close to the parents in vegetative propagation can assist the pathogen’s
spread, and the probability of an epidemic within the plant population is
high. Plants can reproduce their offspring far from infected individuals
by seed propagation, even though the pathogen is widely spread. Thus,
plants block the spread of pathogen by increasing the proportion of seed
propagation when the pathogen is strongly infective. The increase in the
reproductive proportion of seed propagation leads to an increase in the
probability of having an infected individual within a plant population,
and this increases the possibility of an epidemic. Seed propagation is not
effective in inhibiting an epidemic in a single population, and plants can
85
avoid competition with pathogens through seed propagation, as seen in
previous studies [45, 47]. It follows that plants should adjust the balance
of the breeding propagation systems when the disease becomes epidemic
within a plant population.
In the mixed population, multiple plant types have different propor-
tions of vegetative propagation (α), and two cases, uninfected and in-
fected populations, were analyzed. In the uninfected population, plants
are not infected by pathogens, and competition among the different types
of plants (competition for breeding destinations) occurs. Consequently,
the plant type with lowest proportion of vegetative propagation becomes
dominant when the fecundity is low, and the proportion of vegetative
propagation in the dominant type changes to a higher value with an in-
crease in fecundity, even though seed propagation has an advantage over
vegetative propagation in homogeneous environments [Fig. 22]. Thus,
when plants are highly fecund, vegetative propagation becomes effective
with increasing spatial competition among different plant types (because
of a decrease in the open area). In the infected population, if plants have
high fecundity, then the optimal proportion of vegetative propagation
86
decreases slightly relative to the uninfected population, and it does not
depend on pathogen infectivity. Thus, in a plant population with high fe-
cundity, the pathogen infection has less of an effect on the optimal balance
than in a plant population with low. By contrast, when the fecundity is
high enough, the dominant type’s proportion of vegetative propagation is
lower than in the uninfected population. Thus, plants should increase the
proportion of seed propagation to escape from pathogens when systemic
pathogens invade a population.
In conclusion, seed propagation is an effective defensive behavior against
systemic pathogens in: (i) single populations: The plants increase their
population by increasing the proportion of seed propagation when the
epidemic pathogen is highly infective. However, the plants cannot reduce
the epidemic using this strategy; and in (ii) mixed populations: When
plants are not infected by the pathogen, and the fecundity is high, plants
increase their reproduction through vegetative propagation. When the
plants are infected by the pathogen, the high fecundity leads to a de-
crease in the effects of the pathogen infection on the optimal balance of
the breeding systems. However, if the fecundity is low, then increasing
87
the proportion of seed propagation is the optimal breeding strategy to
defend against the spread of a systemic pathogen.
88
Chapter 4 Conclusion
In this thesis, simple models, which considered spatial structures, were
constructed to express the relationship between plant reproduction and
pathogen propagation. Additionally, the effects of two characteristic
phenomena, superinfection by pathogens and seed propagation in clonal
plants, were studied. A simple case was analyzed to examine the basic re-
lationships between plants and pathogens, especially the plant fecundity
and pathogen infectivity. Then, the evolution of plants and pathogens
caused by the competition among the multiple types of individuals (of
pathogens in Chapter 2 and plants in Chapter 3) using computer simu-
lations were discussed.
In the superinfection model (in Chapter 2), the superinfection event is
an important factor in the evolution of virulence through the maintenance
of genetic diversity. A lower superinfection rate (but non-zero) leads to
an easing of the coexistence of multiple strains of pathogens. Addition-
ally, the plant reproductive dynamics is also an important factor in the
selection of virulence level. According to the previous studies (assuming
constant supply of the host), the strain of pathogen with intermediate
89
virulence becomes dominant (i.e. the virulence will evolve toward an
intermediate value). However, a high fecundity of the plant led to occu-
pation by the strain with the higher virulence level (or lower infectivity)
in the superinfection model. Thus, considering the both superinfection
and plant reproductive dynamics are necessary to examine the evolution
of pathogens more thoroughly.
In the seed propagation model (in Chapter 3), seed propagation is an
effective defensive behavior against systemic pathogens. Generally, when
the pathogen infectivity is low relative to plant fecundity, plants can es-
cape from infected individuals through the vegetative propagation, and
the feature is expressed in the seed propagation model. However, in analy-
sis of the model, the effect of pathogen abilities (infectivity and virulence)
on the optimal balance of breeding systems decreases with the increase in
the fecundity of pathogen. Thus, the adjustment of the breeding systems
has an important role to block the pathogen transmission when the plants
have low fecundity.
In summation, the intrinsic fecundity of the plant as well as superin-
fectin makes a major impact on the evolution of pathogen within plant
90
population and on the optimal balance of breeding systems in the clonal
plant. If the fecundity of plant is high enough, then the superinfection
event affects the size of plant population rather than the evolution of
the balance of breeding systems in infected clonal plants. The plants
can increase the size of their population when the parameter range of
plant fecundity remains in a place of coexistence of multiple strains of
pathogen, which increase due to lower superinfectin rate. On the con-
trary, the decrease in the size of plants population through the death
from disease is minimized with the increase in the fecundity, because the
pathogen population is occupied by strain with lowest infectivity through
the superinfection.
If the pathogens are not capable of superinfection, then the defensive
behavior through seed propagation is effective. When the pathogens with
high infectivity invade the plant population, plants increase the propor-
tion of seed propagation to escape from the pathogen infection, and then
it is expected that the strain with lower fecundity becomes dominant.
Because, the increase in seed propagation leads to the decrease in the
healthy individual from the neighborhood of the infected individual, and
91
it is expected that the increase in seed propagation yields similar result
to the decrease in the fecundity in the superinfection model. Thus, the
invading pathogen can not spread within the plant population (the fitness
of the pathogen decreases). Contrastingly, if the pathogens are capable
of superinfection, then the pathogens can spread widely within the plant
population in spite that plants escape by seed propagation. Because,
when the plant has low fecundity, the strain which has high infectivity
becomes dominant due to superinfection event. Thus, the superinfection
is an important ability for pathogens to increase in their fitness and plants
can not block the pathogen spread even though plants adjust the balance
of breeding systems due to the superinfection event.
Additionally, both the balance adjustment of breeding systems and
the selection of virulence level through superinfection have influence on
each other. Specifically, it is expected that the increase in proportion
of seed propagation leads to the ease of coexistence of multiple strain
of pathogen. Thus, the seed propagation can assist the maintenance
of genetic diversity of pathogen although it is effective defense behavior
against pathogen spread. Then, the optimal balance of breeding systems
92
might depend on the virulence of dominant strain or virulence and density
of coexisting strains. Therefore, the model including both superinfection
event and seed propagation leads to different result with respect to the
optimal balance of breeding system. Furthermore, the analysis of the
model is necessary to examine the effect of seed propagation in defensive
behavior against systemic pathogen minutely.
A comparison of the three methods, MA, PA and MCS, indicated that
the spatial structure impacts the dynamics of a plant population infected
by a systemic pathogen. In the analysis using the MA (neglecting the
spatial structure), the oscillatory solution (Chapter 2) and the effects of
the balance of the breeding systems on the plant population dynamics
(especially, equilibrium and threshold) (Chapter 3) were not observed,
unlike in the analyses using the PA or MCS. In the superinfection model,
the oscillatory solution indicates that the competition among plants and
pathogen strains occurs and that plants and pathogens do not main-
tain stable populations. Therefore, the competition leads to the further
evolution of the plants and pathogens to stabilize the population. Addi-
tionally, the spatial structure has an important role in the maintenance
93
of the pathogen’s genetic diversity, as indicated by the PA, resulting in a
wider pathogen range in the coexistence phase than the MA. Thus, the
pathogens’ dynamics within a plant population is greatly affected by the
plant’s spatial factors, such as the configuration of the ramets and the size
of the genets. In the seed propagation model, using the MA, there was no
dependence on the proportion of vegetative propagation because the dif-
ference between vegetative and seed propagation was the distance of the
breeding destination in the present model. Thus, the difference between
the breeding systems is nothing. In the escape strategy, the optimal bal-
ance of the breeding systems is profoundly affected by spatial structures
when the systemic pathogen invades the plant population. In particu-
lar, when the plant’s growth rate is slower than the pathogen’s infection
rate, then the influence of the spatial structures increases as indicated by
the increasing quantitative discrepancy between the PA and MCS. How-
ever, the PA can effectively analyze the qualitative characteristics of the
dynamics.
As is usual with CP analyses, the system examined here is very com-
plicated. The PA is very useful to approximately close and solve the
94
system. Fortunately, several analytical solutions for the equilibrium and
phase transition thresholds were obtained. However, the equilibrium and
thresholds were not analytically derived and approximations do not al-
ways work well, compared with numerical calculations, leading to dis-
crepancies. In the future, more details will have to be analyzed, such
as the bifurcation condition, the values of the equilibrium, and the sta-
bility of the equilibrium state. More complicated models will have to
be constructed and analyzed to express the dynamics of real plants and
pathogens. For instance, a model that integrates the superinfection and
seed propagation models, and realistic assumptions, such as those for the
mortality cost and the proportion of vegetative propagation, should be
configured. The mortality cost has a more complex relationship with the
infection rate and virulence level, and the vegetative proportion would be
adjusted depending on several factors (such as the distance from an in-
fected individual). Additionally, the parameter values, notably differences
in morality costs among strains (∆m), should be estimated by compar-
isons among the available quantitative data. However, a more complex
model increases the difficulty of analysis and the discrepancies from nu-
95
merical calculations. Developing the new and convenient approximation
method, such as a higher-order analysis, will be indispensable in the fu-
ture to examine complicated biological systems.
96
Acknowledgements
The author is deeply grateful to Takenori Takada for his practicable
comments and advice in the accomplishment of this study. The au-
thor gratly appreciates Takashi Kohyama, Toshihiko Hara, Masaharu Na-
gayama, Tomonori Sato, and Kazunori Sato for their variable suggestions.
The author thanks Ryo oizumi, Akiko Satake, Motohide Seki and Yuya
Tachiki for checking this study and making it better. The author thanks
our laboratory and Sato’s group members for support and encourage.
97
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Appendix
A. Simplification of the master equation
The set of Eq. (4) was simplified by the following process, and the
variables have the following properties from their definitions:
Pσiσj= Pσjσi
, (7)
∑σj∈S
Pσiσj= ρσi
(for any σi) , (8)
∑σi∈S
ρσi= 1, (9)
∑σj∈S
qσj/σi= 1 (for anyσi) . (10)
A differential equation for each variable was obtained from Eqs. (3) and
(8),
109
ρσi=∑σj∈S
Pσiσj, (11)
Pσiσj= ρσi
qσj/σi+ ρσi
qσj/σi,
qσj/σi=
Pσiσj− ρσi
qσj/σi
ρσi
. (12)
In addition, the following variables were replaced using the remaining
variables:(ρS, ρI, q0/S, q0/I, qS/I
)from Eqs. (3), (7), (9) and (10).
ρ0 = 1− ρS − ρI, q0/0 = 1−ρSq0/S
1− ρS − ρI
− qI/0,
qS/0 =ρSq0/S
1− ρS − ρI
, qS/S = 1− q0/S −ρIqS/IρS
,
qI/S =ρIqS/IρS
, qI/0 =ρIq0/I
(1− ρS − ρI),
qI/I = 1−(1− ρS − ρI) qI/0
ρI
− qS/I.
(13)
Thus, a set of equations with five variables was obtained from Eqs. (4),
(8) and (12), using the PA,
110
ρS = P0S + PSS + PIS
= ρS
(βSq0/S −mIqI/S
), (14)
ρI = P0I + PSI + PII
= ρI
(mIqS/I − 1
), (15)
˙qI/0 =PI0 − ρ0qI/0
ρ0
=ρI
ρ0
(qI/I − qI/0
)+
βSqI/0 + (z − 1)mIqI/Sz
qS/0 − qI/0, (16)
˙q0/S =P0S − ρSq0/S
ρS
= qI/S
(mIq0/Sz
+ 1)− βSq0/S
(q0/S +
1− (z − 1)(q0/0 − qS/0
)z
), (17)
˙qS/I =PSI − ρIqS/I
ρI
= qI/S
[(z − 1)mI
(qS/S − qI/S
)− 1
z−mIqS/I
]+
(z − 1) βSqS/0z
. (18)
Thus, the set of simplified equations was obtained by substituting (13)
in Eq. (14)-(18).
111
B. Analysis in superinfection model
B.1 Mean-field approximation
The Jacobian at Eqs. (5) is:
J =
−1− βSρS −βSρ0 − 1
(βS +mI) ρS βSρ0 +mIρS −mI (1− ρ0 − ρS)
.
B.1.1 Extinction region
In the case of the extinction region, the Jacobian is:
JM ≡ J(EM) =
−1 −βS − 1
0 βS
,
thus,
Tr(JM) = βS − 1, Det(JM) = −βS
The stability condition (Tr(JsiM) < 0 andDet(JM) > 0) is 0 > βS. There-
fore, this equilibrium is always unstable under the given assumptions..
B.1.2 Disease-free region
112
In the case of the disease-free region, the Jacobian is:
JM ≡ J(EM) =
−1− βS −1
(βS +mI) mI
,
thus,
Tr(JM) = mI − βS − 1, Det(JM) = βS (1−mI)
The stability condition is:
0 < βS, mI < 1.
Therefore, the stability condition at the equilibrium is mI < 1.
B.1.3 Epidemic region
In the case of the epidemic region, the Jacobian is:
JM ≡ J(EM) =
−βS+mI
βS−mI(1+βS)
βS+mI
βS+mI
βS1
,
thus,
113
Tr(JM) = − βS
mI
, Det(JM) =βS (mI − 1)
mI
The stability condition is:
0 < βS, 1 < mI
Therefore, the stability condition at the equilibrium is 0 < βS and mI > 1.
B.2 Pair approximation
To analyze the local stability at each equilibrium state, we used the
Routh–Hurwitz stability criterion. Let the characteristic polynomial of
the Jacobian of n degrees at equilibrium state be
a0λn + a1λ
n−1 + a2λn−2 + · · ·+ an−1λ+ an,
and the Hurwitz determinant be ∆n.
B.2.1 Extinction region
In the stability analysis of the extinction equilibrium, the same five
114
variables were selected as in Appendix.A. as were the simplified Eqs. (4).
ρS =ρS
(mIqI/S − βSq0/S
),
ρI =mIρSqI/S − ρI,
qI/0 =1
(1− ρS − ρI) z
(ρSq0/S
[βSqI0 + (z − 1)mIqI/S
]+[ρI
(1 + qI/0
)− ρSqI/S − 2 + (d− 2) ρS qI/0
]z),
q0/S =(1 +
mIq0/Sz
)qI/S
− βSq0/S
[q0/S +
1
z− z − 1
z
((1− ρS − ρI)
(1− qI/0
)− 2ρSq0/S
1− ρS − ρI
)],
qI/S =1
z
[βSq0/S
((z − 1) qI/0 − qI/Sz
)+mIqI/S
((z − 2)
(1− qI/S
)− (z − 1) q0/S
)]− qI/S.
(19)
115
The Jacobian of Eqs. (19) at EP (ρ∗0 = 0, ρ∗S = 0, q∗I/0 = 0) is:
JP ≡ J(EP),∣∣∣λI − JP
∣∣∣ = (λ+ 2) (λ+ 1)(λ+mIq
∗I/S − βSq
∗0/S
)∣∣∣∣∣∣∣∣∣λ+
2βS−mIq∗I/S
z+−βS(1− 2q∗0/S) −1−
mIq∗0/S
z
((βS+mI)z−mI)q∗I/S
zλ+ 1 + βS −
m[(z−2)(1−2q∗
I/S)−(z−1)q∗0/S
]z
∣∣∣∣∣∣∣∣∣ .(20)
Two remaining variables have the following equilibrium values:
q∗0/S =mI (z − 2)− z
mI (z − 1), q∗I/S =
βS (mI (z − 2)− z2) (mI (z − 2)− z)
m2I (z − 1) (z − 2) (mI + z)
.
Here,
mIq∗I/S − βSq
∗0/S
=− 2βSz (mI (z − 2)− z)
mIz (z − 2) (mI + z). (21)
From q∗0/S > 0, the Eq. (21) is negative, thus this equilibrium is always
unstable.
116
B.2.2 Disease-free region
In the stability analysis of the disease-free equilibrium, five other vari-
ables(ρS, ρI, q0/0, q0/S, andqI/S
)were selected, as were the simplified Eqs. (4).
ρS =βSρSq0/S −mIρIqS/I,
ρI =ρI
(mIqS/I − 1
),
q0/0 =− 1
z (1− ρS − ρI)
[(z − 2) βSρSq0/0q0/S
+z(2ρS
(1 + q0/Sq0/0
)− ρIq0/0 − 2
(1− ρI − q0/0
))],
q0/S =(1 +
mIq0/Sz
)qI/S − βSq0/S
[q0/S +
1
z− (z − 1)
z
(q0/0 −
ρSq0/S1− ρS − ρI
)],
qI/S =βSq0/Sz
[(z − 1)
(1− q0/0
)− zqI/S −
(z − 1) ρSq0/S1− ρS − ρI
]+
mIqI/Sz
[(z − 2)
(1− qI/S
)− (z − 1) q0/S
]− qI/S.
(22)
117
The Jacobian of Eqs. (22) at EP is:
JP ≡J(EP),∣∣∣λI − JP
∣∣∣ =(λ+ 2 +2 (z − 1) βS
z
)∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
λ+ 1 0 0 −mI
0 λ −βS −1
(z−1)βS
z(z−1)βS
zλ+mI
(2− 1
z
)−1
(z−1)βS
z(z−1)βS
z(z−1)βS
zλ+ 1− (z−2)mI
z
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣. (23)
Thus, the coefficients of the characteristic polynomials and the Hurwitz
118
determinants are
a4 =2β2
S (z − 1) (z − (z − 1)mI)
z2,
a3 =βS
z2[z (3z − 2 + 3βS (z − 1))−mI
(z2 (a+ 3)− 3z (a+ 2) + 2 (a+ 1)
)],
a2 =z2 (1 + βS (βS + 5− 2mI)−mI)− z (βS (βS + 3− 5mI)− 2mI)− 2βSmI
z2,
a1 =2 (βS + 1)− βS + (z − 2)mI
z,
a0 =1,
∆2 =1
z3[z (z − 1) (2z − 1) β3
S + β2S
z (3 + z (9z − 10))− (z − 1) (2z − 1)2mI
+ βS
z2 (9z − 5) + (z − 2)2 m2
I − 2zmI (5 + z (4z − 11))
+zmI ((z − 2)mI − 2z) ((z − 2)mI − z)] ,
∆3 =βS
z5[2mI (βS + 1) + z (2 + βS (βS + 3)−mI (5βS + 6))− z2 (2βS + 3) (1 + βS −mI)
][β2
S (z − 1)2 ((z − 2)mI − 3z)− βS (z − 1)((z − 2)2 m2
I − 2zmI (2z − 5) + 5z2)
−z ((z − 2)mI − 2z) ((z − 2)mI − z)] .
Here, from a4 > 0,
mI <z
z − 1, (24)
In addition, we confirmed that all of the coefficients and determinants
119
are positive under these conditions. Thus, the stability condition of the
disease-free equilibrium is Eq. (24).
B.2.3 Epidemic region
In the stability analysis of the epidemic equilibrium, five other variables
(ρ0, ρS, q0/S, q0/I, qS/I
)were selected, as were the simplified Eqs. (4).
ρ0 =1− ρ0 − ρS
(1 + βSq0/S
), (25)
ρS =βSρSq0/S −mI (1− ρS − ρI) qS/I, (26)
q0/S =1
ρ0ρS
[mIρ0 (1− ρ0 − ρS) q0/SqS/I + z (1− ρ0 − ρS) ρ0qS/I
+βSρSq0/Sρ0
(z(1− q0/S
)+ (z − 1) q0/I − 2
)− (z − 1)
(2ρSq0/S + (1− ρS) q0/I
)],
(27)
q0/I =1 +
((z − 1)mIq0/S
ρS
− 1
)qS/I −
q0/I(zρ0
(1 +mIqS/I
)+ (z − 1) βSρSq0/S
)zρ0
,
(28)
qS/I =1
ρ0ρS
[(z − 1) βSρ
2Sq0/Sq0/I
+ρ0qS/ImI
(ρS
[(z − 2)
(1 + qS/I
)− (z − 1) q0/S
]− 2 (z − 1) (1− ρ0) qS/I
)].
(29)
From the above equations (Eqs. (25)-(29)), the following equilibrium
120
value was derived, and all of the other variables were derived by Eq. (13).
ρ∗S =z (mI − 1) + (z − 1)mI − 2zmIq
∗0/I[
z (mI − 1) + (z − 1)mIq∗0/S
](1 + βS)−
[(z − 1) βSq∗0/S + 2z
]mIq∗0/I
,
ρ∗I =βSρ∗S q
∗0/S,
q∗0/S =zmI
(1− q∗0/I
)− (z +mI)
(z − 1) βS
,
q∗S/I =1
mI
,
q∗0/I =1
2zβSmI (βS +mI)
(β2
S (mI − z) + βSm2I (z − 1)− zmI (3βS +mI)
+√
4zβS (βS +mI) [(z − 1)m3I + zmI (βSmI − βS −mI)]
+ (zmI (3βS +mI)− (z − 1) βSm2I − (mI − z) β2
S )2).
Here, the Jacobian at equilibrium was abbreviated as EP, because the
coefficients of the characteristics polynomials and the Hurwitz determi-
nants are too long to write in this paper. In addition, it is too difficult
to derive the stability condition analytically. Thus, the values of the co-
efficients and determinants were confirmed numerically. As a result, all
of the coefficients of ∆2 and ∆3 are always positive, and the sign of ∆4
varies depending on the parameter values. Thus, the Hopf bifurcation
121
occurs to exceed the parameter values at the threshold.
C. Analysis in seed propagation model
C.1 Mena-field approximaation
Using the MA, the set of Eq. (6) was rewritten as follows:
ρ0 = ρS (1−mSρ0) + ρI, (30)
ρS = ρS (mSρ0 −mIρI − 1) , (31)
ρI = ρI (mIρS − 1) . (32)
Thus, the following three equilibria were obtained:
E ≡ (ρ∗0 , ρ∗S , ρ
∗I ) ,
E1 = (1, 0, 0) : extinction,
E2 =(
1mS
, 1− 1mS
, 0)
: disease-free,
E3 =(
mI
mS+mI, 1mI, mSmI−mS−mI
mI(mS+mI)
): epidemic.
These equilibria do not depend on the proportion of vegetative propa-
gation, α. Thus, the effects of seed propagation on the system were not
122
examined using the MA.
C.2 Extinction phase in pair approximation
In this section, The set of Eqs. (6) were simplified and rewritten using
qσ/σ′ and their definitions (Appendix. A.), as follows:
ρS =ρS
(αmSq0/S +mS (1− α) (1− ρS − ρI)−mIqI/S − 1
), (33)
ρI =mIρSqI/S − ρI, (34)
˙q0/0 =αmSρSq0/0 + z(q0/0 (mSρS − 1) + 2
)−
(z − 2)αmSρSq0/0q0/S1− ρS − ρI
, (35)
˙q0/S =1 +mIq0/SqI/S
z− q0/S −mS (1− α)
[(1− ρI) q0/S − (1− ρS − ρI) q0/0
]− αmSq0/siS
[1
z
1− (z − 1)
(q0/0 −
ρSq0/S1− ρS − ρI
)], (36)
˙qI/S =mS (1− α)[(1− ρS − ρI)
(1− q0/0 − qI/S
)− ρSq0/S
]+
1
z
[αmSq0/S
(z − 1)
(1− q0/0 −
ρSq0/S1− ρS − ρI
)− zqI/S
]+
1
z
[mIqI/S
(z − 2)
(1− qI/S
)− (z − 1) q0/S
]. (37)
From Eq.(33), the following two equilibria of ρS were obtained:
ρ∗S = 0,mS
[(1− α) (1− ρ∗I ) + αq∗0/S
]−mIq
∗I/S − 1
(1− α)mS
.
123
The value ρ∗S = 0 was chosen to analyze the extinction equilibrium. Then,
the positive equilibria of the other variables from Eqs.(34)-(37) were de-
rived.
E ≡(ρ∗S , ρ
∗I , q
∗0/0, q
∗0/S, q
∗I/S
)=
0, 0, 0,2αmS (z − 1)− z (mS + 1) +
√(2αmS + z (mS + 1))2 − 8zα2m2
S
2zαmS
, 0
(38)
To analyze the local stability, the eigenvalues of the Jacobian were
124
obtained at the equilibrium. The Jacobian J is:
J =
mS
(1− α
(1− q∗0/S
))− 1 0 0
0 −1 0
−mS
((z−2)αq∗0/S
z+ (1− α)
)− 1 −1 −2
−mS
((z−1)α
(q∗0/S
)2
z+ (1− α)
)−mS (1− α)
(1− q∗0/S
)−mS
((z−1)αq∗0/S
z− (1− α)
)−
mSq∗0/S
(z(1−α)+(z−1)αq∗0/S
)z
0 −mS
((z−1)αq∗0/S
z+ (1− α)
)0 0
0 0
0 0
−mS
((z−1)αq∗0/S
z− (1− α)
)mIq
∗0/S
z
0 −1−mS
(1− α
(1− q∗0/S
))+
mI
((z−2)−(z−1)q∗0/S
)z
.
125
The eigenvalues of the Jacobian are:
λ1 = −1,
λ2 = −2,
λ3 = −mS
(1− α
(1− q∗0/S
))− 1, (39)
λ4 = −mS
((z − 1)αq∗0/S
z− (1− α)
), (40)
λ5 = −1−mS
(1− α
(1− q∗0/S
))+
mI
((z − 2)− (z − 1) q∗0/S
)z
. (41)
Substituting q∗0/S in Eq. (38) into Eqs. (39) and (40),
λ3 =z (mS − 3)− 2αmS +
√(z (mS + 1) + 2αmS)
2 − 8zα2m2S
z,
λ4 = −
√(z (mS + 1) + 2αmS)
2 − 8zα2m2S
z.
The condition of λ3 < 0 is:
mS <− (z − α) +
√(z + α)2 − 4zα2
2α (1− α):= f(α). (42)
The right-hand side of the inequality was set to f(α). The λ4 is negative
because q0/S is a real number.
126
Then, it was shown that the Eq. (41) is negative. The second term on
the right-hand side of Eq. (41) is negative because mS > 0 and α, q∗0/S ∈
[0, 1]. Thus, if the third term is negative, then Eq. (41) is negative. When
the equilibrium q∗0/S is greater than (z − 2)/(z − 1), Eq. (41) is negative.
The value of the equilibrium decreases with an increase in mS and α from
following analysis:
∂q∗0/S∂mS
= −z(mS + 1) + 2αmS −
√(z (mS + 1) + 2αmS)
2 − 8zα2m2S
2αm2S
√(z (mS + 1) + 2αmS)
2 − 8zα2m2S
< 0,
(43)
∂q∗0/S∂α
= −(mS + 1)
(2αmS + z (mS + 1)−
√(z (mS + 1) + 2αmS)
2 − 8zα2m2S
)2α2mS
√(z (mS + 1) + 2αmS)
2 − 8zα2m2S
< 0.
(44)
Additionally, the f(α) reaches its maximum at α = 1 because the f(α)
is a monotonically increasing function of α. Therefore, the equilibrium is
at its minimum value at α = 1, and the maximum value of mS is within
the parameter range of Eq. (42).
127
Let f(α) = g (α) /h (α),
limα→1
g (α) = limα→1
− (z − α) +
√(z + α)2 − 4zα2 = 0,
limα→1
h (α) = limα→1
2α (1− α) = 0.
Using l’Hopital’s rule,
limα→1
g′(α)
h′(α)= lim
α→1
1 + 2α+2z−8αz
2√
(z+α)2−4α2z
2(1− α)− 2α=
z
z − 1.
Thus,the Eq. (42) at α = 1 is
mS < limα→1
f(α) =z
z − 1,
Therefore, the q∗0/S under the parameter range is
q∗0/S >z − 1
z
(>
z − 2
z − 1
)
Therefore, Eq. (41) is always negative.
In conclusion, the extinction equilibrium became stable under the con-
ditions of Eq. (42).
128