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Ecological Modelling 299 (2015) 79–94
Contents lists available at ScienceDirect
Ecological Modelling
journa l h om epa ge: www.elsev ier .com/ locate /eco lmodel
athematical modelling of seasonal migration with applicationso climate change
ohn G. Donohue ∗, Petri T. Piiroinenchool of Mathematics, Statistics & Applied Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland
r t i c l e i n f o
rticle history:eceived 10 June 2014eceived in revised form 2 December 2014ccepted 3 December 2014
eywords:easonalityigration
limate changehenology
a b s t r a c t
The breeding success of many species depends on the alignment of the period of maximum demand ofoffspring with a period of food abundance. In this paper, we use piecewise-smooth differential equationsto model the variation in the size of a population that has a short interval each year during which successfulreproduction is possible. An increase in spring-time temperatures in recent decades has caused theseintervals to advance, leading to temporal mismatches between species on different trophic levels. Wefirst consider a one-species model which illustrates the dynamics of a population of specialist feedersover the course of a single breeding season and use it to examine how reproductive success depends onthe population’s distribution of breeding dates. We then introduce time-dependent switches to extendthe model to a broader class of species. Steady-state solutions are used to measure the extent to which a
onlinear systemsopulation modelling
migratory population can sustain itself over long time scales and repeated breeding events. The model’spredictions agree with the observed negative effect on population size associated with an advance in thefood season if not adequately compensated for by the population. Finally, we discuss how this model canbe extended in order to encompass a wider range of ecological systems and to highlight the mechanismsover which climate change may exert an influence.
© 2014 Elsevier B.V. All rights reserved.
. Introduction
Seasonal variation is one of the most fundamental driving forcesn ecology. Recurrent cycles in temperature create transient periodsf time at which critical activities such as flowering and breed-ng are possible. For many species, breeding success depends onynchronisation of the food requirements of their offspring with
period of food abundance (Visser et al., 2006). Migratory popu-ations can mitigate unfavourable aspects of seasonal variation byxploiting geographically distinct regions during different parts ofhe year, effectively defining an annual existence infrastructure thattherwise could not be employed. However, in a scenario of rapidlimate change, this strength can become a vulnerability. Migrantsypically rely on environmental cues and/or their own endoge-ous rhythms to determine the appropriate time to depart a region
Gauthreaux, 1980). This dependence may become maladaptive ifhese cues are no longer accurate indicators of conditions at theirestination (Carey, 2009).∗ Corresponding author. Tel.: +353 91492332.E-mail addresses: [email protected] (J.G. Donohue),
[email protected] (P.T. Piiroinen).
ttp://dx.doi.org/10.1016/j.ecolmodel.2014.12.003304-3800/© 2014 Elsevier B.V. All rights reserved.
Long-term studies have shown that climate change hasimpacted the timing of critical events for a variety of different taxa.While the impact on autumn phenology is either of a low mag-nitude or heterogenous among species (MacMynowski and Root,2007), trends in spring-time event shifts have been observed, par-ticularly in the timing of arrival to breeding quarters and the timingof breeding itself (Walther et al., 2002). These changes are typi-cally a response to a change in the phenology of the environment.An alteration in the timing or length of the period of maximumfood availability is of particular concern for a breeding population.For species who rely upon this brief window of food abundanceto achieve successful recruitment, even a slight mismatch in tim-ing can lead to significant population declines. The trend towardshigher spring temperatures in recent decades has caused such win-dows to advance (Walther et al., 2002). The efforts of affectedspecies to adaptively alleviate the resulting mismatches has beenthe focus of intense study. A common response, where possible, isfor a population to advance its reproductive schedule in an attemptto compensate for the earlier emergence of its prey (Visser and Both,
2005; Brown et al., 1999).In the case of migrating species, the earliest possible breedingdate depends on the date of their arrival to the breeding quar-ters. In general, long-distance migrants tend to be less flexible
8 ologic
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han short-distance migrants in their ability to adjust the timingf their arrival (Butler, 2003). One reason for this may be thathe cue for departure from the wintering area is independent ofemperature, meaning that their arrival time does not advance to
ake the earlier optimal breeding time feasible. Although there isome evidence of long-distance migrants altering their migrationues in order to circumvent this limitation (Jonzén et al., 2006),his appears to be the exception rather than the rule. The climate-hange-driven decoupling of phenological cues prevents manyong-distance migrants from arriving in their breeding quarters athe optimal time (Both and Visser, 2001), leading to pronouncedopulation declines (Both et al., 2006). Short-distance migrantsppear to be less restricted in their capacity to adjust the tim-ng of key events (Miller-Rushing et al., 2008; MacMynowski andoot, 2007). These species are able to arrive in the breeding quar-ers early enough to use temperature as a predictor of the optimalreeding time. However, climate change may affect different stagesf a season non-uniformly. In this case, the temperature at theeginning of the season may no longer provide a reliable fore-ast of the timing of the food season, causing a mismatch (Buset al., 1999). Such temporal inconsistencies have, for instance,een linked with reproductive failure in a population of great titsVisser et al., 1998). A dramatic rise in temperatures after layingas been initiated by a pair will speed up the development of theirrey. However, their own reproductive cycle may remain relativelyxed, with limited scope for bringing hatching forward after therst egg has been laid, other than via a reduction in the size ofhe clutch (Both, 2010). A second hypothesis proposed to explainhese reproductive failures is that pairs may simply be unable todvance their breeding schedule in step with the advancementn the food season if the comparatively low temperatures ear-ier in the spring impose an energy constraint on reproduction
hich is unaffected by climate change (Stevenson and Bryant,000).
In the first part of this paper, we describe a set of nonlinear dif-erential equations to represent the dynamics of a population ofpecialist feeders over the course of a single breeding season. Weefer to this set of equations as the breeding model and use it tonvestigate the relationship between a population’s reproductiveuccess and a temperature-dependent window of food abundance.
e focus on species that breed in regions characterised by a highlyrofitable but short-lived interval of food availability. Examples
nclude passerines in temperate forests, geese in arctic tundras andolden plovers in upland habitats. In such regions, an individualhat does not align the needs of its young with this interval willuffer reproductive failure for the season (Both, 2010). One appli-ation of models of this type is in determining an optimal breedingchedule for a population in a highly seasonal environment. By tak-ng a distribution viewpoint to this problem, we are able to gainnsight into the significance of the entire population’s timing deci-ions, instead of being restricted to those of an average individual.n keeping with the optimal-timing theory developed by Perrins1970), we consider constraints on an adult’s breeding time which
ay prevent the population from achieving its full reproductiveotential. For a given population, we can subsequently deduce thehape of the hatching distribution that is best equipped to exploithe environmental conditions, represented here by a short-livedood season.
Various mathematical models have been developed to helproaden understanding of the role of the migration process andow it may be affected by the aforementioned environmentalhanges (Taylor and Hall, 2012; Jonzén et al., 2007; Sutherland,
996). These models generally do not consider the annual cyclen its entirety, implicitly ruling out the possibility of seasonalnteractions influencing survival predictions. In the second partf this paper, we propose a population-based dynamical system
al Modelling 299 (2015) 79–94
framework in which measured data can be synthesised, and theinterplay of different seasonal factors, both within and betweenyears, can be analysed. As time is continuous, the size of thepopulation at any point in a cycle can be considered. The breed-ing and survival stages are considered as separate recurrentregimes in a time-dependent switching system. Systems of thistype fall into the category of piecewise-smooth dynamical sys-tems and have found application in engineering and biologicalsciences (Budd et al., 2008; di Bernardo et al., 2008). To datehowever, they have been less considered in an ecological con-text. We refer to this extended model as the switching model,where its name is derived from the sequence of switches thatpartitions each cycle into distinct stages. The model helps us todevelop an understanding of the relationship between increasingspring temperatures and the breeding success of an affected pop-ulation. We illustrate some of the effects of climate change andassess how well populations with different adaptation constraintscan compensate for them. By employing some basic behaviouralassumptions, we suggest possible adaptive mechanisms that mayunderpin ecological systems susceptible to changes in spring phen-ology.
We begin by describing the mechanisms which constitute ourmodels of seasonal breeding and migration in Section 2. Thesecomponent parts are then assembled to form the breeding model,as detailed in Section 3.1. We study typical solutions to this sys-tem in Section 3.2. These solutions are then used to addressan optimisation problem in Section 3.3. Section 4.1 contains amathematical description of the switching model. This is fol-lowed by an application of the model to a population whichhas been affected by heightened springtime temperatures inSection 4.2. We conclude with Section 5, a discussion on theassumptions underpinning our models and a variety of potentialextensions.
2. Mathematical modelling and model assumptions
While efforts have been made to ensure that the mathemat-ical framework we outline is as general as possible, we specifyinsectivorous birds as examples due to the wealth of studies ontheir seasonal behaviour. Therefore, in what follows, we considera population of birds in which at most one brood is produced byeach female in a given breeding season. The survival of this brooddepends on the timing and quantity of prey items procured by theparents. For simplicity, we further assume that each egg in thebrood is hatched synchronously (i.e. incubation by the parents doesnot begin until the last egg in the clutch has been laid). This meansthat for a given clutch size, there is a fixed lag between the timeat which an individual’s reproductive cycle begins (i.e. the time ofmating) and the time at which its brood is hatched. Difference equa-tions have traditionally been used to describe reproduction whichis seasonal (Fulford et al., 1995). The discrete step length is usuallychosen to be the time elapsed between the start dates of successivebreeding seasons which is typically one year. In the present case,fluctuations within annual cycles are inherent to the problem ofinterest, so differential equations employing continuous time aremore suitable.
Ultimately, the dynamics of any real populations are determinedby the relative influences of mortality and reproduction. The inter-play between these two forces determines the level of recruitmentin a breeding season i.e. the percentage of newborns that surviveto first breeding age. In this section, we will consider appropriate
methods of representing each of these mechanisms in turn. Thesemechanisms form the basis of population growth in both the breed-ing model and the switching model, presented in Sections 3 and 4,respectively.
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.1. Food supply and mortality
In what follows, we assume that adults and nestlings are subjecto density-dependent mortality at different rates. Furthermore, andult member of the population is assumed to be capable of surviv-ng equally well at any point in a season which means that the deathate for adults is constant. Conversely, nestlings rely on a plentifulupply of food in order to survive the initial period after hatching.
e consider an environment with a fleeting interval of food abun-ance, individuals born outside of which are unlikely to survive.e propose that the death rate for nestlings should vary with time
uring the breeding season and remain constant during the restf the year. The seasonality of the environment creates an intervaluring which food is abundant, resulting in a reduced likelihoodf a nestling starving to death for the duration. The death rate forewborns is therefore negatively related to food supply during the
ood season.We will denote the constant death rate for adults as c1, the
aseline death rate for newborns as c3 and the variable food sup-ly of the newborn population as g(t). We consider a season ofood abundance of duration �tF that commences at t = tF. As a firstpproximation, we have chosen to define a quadratic relationshipetween food supply and time during this interval. While g(t) coulde replaced by any function which fits observed behaviour, here we
et g(t) be a piecewise-smooth function given by
(t) :=
⎧⎨⎩4�(t − tF )(tF + �tF − t)
(�tF )2, tF < t < tF + �tF ,
0, Otherwise,
(2.1)
here 0 ≤ � ≤ 1 represents the magnitude of the variation. Weurther assume a linear relationship between food supply and mor-ality. The effect of this assumption, coupled with a food supplyhat is quadratic in time during the food season, is a death rate thatecreases to a minimum before returning toward its baseline levelf c3, where the minimum coincides with maximum food supply.he death rate for newborns at a particular time is then given by
3(1 − g(t)) =
{c3((�tF )2 − 4�(t − tF )(tF + �tF − t))
(�tF )2, tF < t < tF + �tF ,
c3, Otherwise.
(2.2)
Note that the mortality rate for newborns decreases from c3 to minimum value of c3(1 − �) at t = tF + �tF/2 when the food supplys at a maximum, as shown in Fig. 1(a).
.2. Reproduction
In order for a fixed breeding season to be defined in aontinuous-time setting, the population’s rate of reproductionust tend to zero at some point in the cycle and can therefore
e thought of as an explicit function of time. A constant ratef reproduction throughout the entire cycle, by contrast, wouldmply continuous rather than seasonal reproduction. We denotehe population’s rate of reproduction by r(t). This function isonzero during the breeding season and zero otherwise. Its precise
orm during the breeding season is determined by the aggregatedehaviour of the individuals in the population. The rate of repro-uction of the population at a particular instant during the breedingeason is influenced by the timing decisions of the individual mem-ers. While the shape of the function is given by the timing choicesf the population, the height is proportional to the number of off-pring produced by an average female over the course of a breedingeason.
We propose two distinct relationships between reproductionate and time. First, we consider a uniform distribution over thereeding season. This is given by the piecewise-constant functionU(t) and implies that the population as a whole is making the same
l Modelling 299 (2015) 79–94 81
concerted effort to reproduce at each point in the hatching intervalor that there is no correlation between the timing of breeding pairs.In such a case, we let
r(t) = rU(t) :={ r0
tE − tS
, tS < t < tE,
0, Otherwise,(2.3)
where tS and tE denote the start and end dates of the hatching inter-val, respectively, tE is the hatching time of the last successfullyrecruited nestling and the constant r0 > 0 is related to the typicalclutch size.
If, on the other hand, the population has a tendency to focusits reproductive efforts on a particular part of the breeding season,the rate at which the population reproduces cannot be thought ofas uniform. Therefore, as an alternative to rU(t), we propose thepiecewise-linear function rL(t), which increases to a local maximumat a peak hatching date tP. This could be thought of as the result ofthe population, acting on phenological cues, tending to hatch theireggs in an interval around t = tP. It can be described by the function
r(t) = rL(t) :=
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩2r0
tE − tS
t − tS
tP − tS, tS < t < tP, tP /= tS,
2r0
tE − tS
tE − t
tE − tP
, tP ≤ t < tE, tP /= tE,
0, Otherwise,
(2.4)
with the condition that tE /= tS . We note that if the peak in hatch-ing occurs at the first or last possible hatching date, then rL(t) is adiscontinuous function. In particular, if tP = tS, there is an instanta-neous switch from no activity to maximum activity at t = tS with alinear decrease to zero thereafter. On the other hand, if tP = tE , thefunction attains its maximum value at t = tE before switching backto zero.
If the hatching time of the last successfully recruited nestling isalso the latest time at which a hatching could theoretically occur(i.e. tE = tE), the area under the functions given by Eqs. (2.3) and(2.4) is equal to r0. This area can be interpreted as the total repro-ductive effort made by the population during the breeding season.An increase in r0 could be thought of as an increase in the typicalclutch size for the population and would therefore lead to greaterbiomass production over the course of the season. If tE < tE , onthe other hand, not all of this reproductive effort will be realisedand late contributions will be discarded. The relationships betweengrowth and time given by Eqs. (2.3) and (2.4) are illustrated inFig. 1(b). These piecewise-smooth functions allow us to model arange of different behaviours simply by varying the timing param-eters tS, tP, tE and tE .
2.3. Recruitment
While the population’s reproduction and the survival of theresulting newborns can, to an extent, be considered in isolation, it isonly by considering the interaction between these two factors thatwe can determine what proportion of the offspring from a breedingseason will survive to first breeding age. Nestlings that have beenprovided with sufficient food to survive the critical early stages oftheir lives have a strong likelihood of progressing to adulthood. Fol-lowing a period of fledging and the development of foraging skills,the young birds will be capable of surviving without parental care.In order to facilitate this in our model, we introduce a recruitmentmechanism that converts viable offspring into new members of the
adult population.It is important to note that the mechanism outlined here isdesigned specifically to fit the case of a population that relies uponan intense but short-lived season of food abundance. In this case,
82 J.G. Donohue, P.T. Piiroinen / Ecological Modelling 299 (2015) 79–94
F deatb
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ig. 1. Schematic diagrams of (a) the functional relationship between the newbornetween reproduction rate r(t) and time t given by Eqs. (2.3) and (2.4).
newborn individual must be in the nest during the food seasonf it is to be successfully recruited. We therefore choose the endf the food season to be the sole time of recruitment to the adultopulation during a given cycle. We further assume that the newlyecruited individuals will themselves reproduce during the nextycle and can therefore be assimilated directly into the breedingopulation, so that
breeding population �→ breeding population + newborn population,
newborn population �→ 0,
at t = tF + �tF . (2.5)
Implicit in (2.5) is the assumption that all surviving nestlings athe end of the food season, regardless of their age, will be just asikely to survive to the following year as their parents. As the endf the food season is assumed to signal the end of the period ofuccessful recruitment in our model, we explicitly cut the breedingeason short at t = tF + �tF. Therefore, we initially choose
E = tF + �tF . (2.6)
s the hatching time of the last successfully recruited nestling. Inrder to position the food season as a critical factor in determin-ng the breeding success of the population during the season, weave coupled the nestling mortality and reproduction functions viahe parameter tE and thereby truncate the reproduction rate func-ion at the point at which recruitment would become impossible.eproductive attempts at any time after t = tE will not gain any ben-fit from the season of food abundance and, in general, would needo be modelled independently. This is represented by the shadedegion in Fig. 1(b). In the case we are considering here, an adultreeding after the end of the food season cannot be successfulnd such attempts are simply discarded. Strictly speaking, to allowor the unlikely scenario of a food season taking place too late inhe breeding season for the population to avail of it, we could letE = min(tE, tF + �tF ) but, as we are primarily concerned with theffect of increasingly early food seasons, we omit this mechanismrom the models presented in this paper.
. The breeding model
.1. Model description
Here we will consider an adult population whose biomass isepresented by M(t) where t is the time elapsed (in days) since theeginning of the spring. The breeding of the individuals contained
h rate c3(1 − g(t)) and time t given by Eq. (2.2) and (b) the functional relationships
in M(t) generates a population of nestlings with biomass B(t). Wemodel the change in population size over a breeding season usinga set of time-dependent functions of the form discussed in Sections2.1–2.3. The contributions of these mechanisms are aggregated inthe system
dM(t)dt
= −c1M(t), (3.1)
dB(t)dt
= r(t)M(t)(
1 − B(t)K
)− c3(1 − g(t))B(t), (3.2)
with switches
M(t) �→ M(t) + B(t) at t = tF + �tF ,
B(t) �→ 0 at t = tF + �tF ,(3.3)
and initial conditions of M(0) = M0, B(0) = 0 where t = 0 is the firstday of spring. The parameters c1 and c3 are positive death rates asintroduced in Section 2.1. The parameter M0 is the biomass of theadult population at the beginning of spring and K is the maximumnestling population size that can be supported by the environmentat any point in time. The growth of the population is proportionalto the size of the breeding population and limited by the availabil-ity of territory. When the size of the adult population and/or thereproduction rate coefficient r0 is sufficiently high, competition forterritory becomes a crucial limiting factor and all individuals donot have the opportunity to reproduce. In this case, the
(1 − B
K
)term in Eq. (3.2) acts to slow down growth. The rate of increase inthe nestling population eventually declines due to the failure of asubset of the population to find territory in which to mate.
3.2. The effects of different reproduction rates
We will now present solutions to the breeding model with theintention of highlighting the contributions of its constituent partsand the interplay between them. At this point, we wish to illus-trate typical dynamics of such a system over a single breedingseason, rather than make predictions about a particular populationand our parameter values are chosen accordingly. We first use theMATLAB® package ode45 to numerically solve Eqs. (3.1) and (3.2),without any consideration of a food season (i.e. � = 0) and with-out recruitment. This enables us to isolate how the dynamics of
the population depend on the choice of reproduction rate r(t). Thegrowth curves labelled U and L in Fig. 2(a) are solutions correspond-ing to a piecewise-constant and piecewise-linear reproduction rate,respectively, as discussed in Section 2.2. The parameter tE has no
J.G. Donohue, P.T. Piiroinen / Ecological Modelling 299 (2015) 79–94 83
0
1
2
t0 20 60 90
(a)
B(t)
M0
U
L
0
1
2
t0 20 35 55
(b)
B(t)
M0
U
L
Fig. 2. Panel (a) shows the change in size of a nestling population given by solving the system Eqs. (3.1) and (3.2) and normalising with respect to the initial size of the adultpopulation. Curves U and L correspond to a reproduction rate which is piecewise constant (r(t) = rU(t)) and piecewise linear (r(t) = rL(t)), respectively. There is no seasonalvariation in food supply (� = 0) and no recruitment mechanism which means that both curves eventually tend to zero. As there is no coupling between food supply and thereproduction rate, we must set a value for the last hatching time and we choose tE = tE so that all hatchings, regardless of timing, are included. Other parameter values areM s the cw 2.6) ant from
em
sthaswooteobdssgTbpw
atdmgFaocaahiflbaed
0 = 10, r0 = 1.5, K = 50, c1 = 0.002, c3 = 0.02, tS = 0, tP = 20 and tE = 60. Panel (b) showhich is composed of Eqs. (3.1)–(3.3). The time of last hatching tE is now set by Eq. (
he parameter values tF = 35, �tF = 20 and � = 1. All other parameters are unchanged
cological meaning in this case and we therefore set tE = tE . Thiseans that all hatchings, regardless of their timing, are included.When the reproduction rate is uniform throughout the breeding
eason (i.e. r(t) = rU(t), as outlined in Eq. (2.3)), we see in Fig. 2(a) thathe nestling population will grow monotonically because the likeli-ood of an adult producing offspring is equal at all times. However,s the growth rate of the nestling population is dependent on theize of the adult population, mortalities in the breeding populationill slow down growth. If we instead use the piecewise-linear form
f the reproduction rate (i.e. r(t) = rL(t), as outlined in Eq. (2.4)), webserve that curve L increases to a maximum at some point afterhe peak in the reproduction rate at t = tP. The reduced breedingffort of the population, a large proportion of which is now pre-ccupied with raising broods, manifests itself in a rapid decreaseeyond this point. Adult mortalities once again contribute towardsecreasing the rate of growth. As we assume that there is no foodeason and no recruitment of offspring, nestlings face a high risk ofuccumbing to starvation throughout the breeding season and therowth curves eventually decrease regardless of the form of r(t).he hatching of eggs is ultimately in vain because mortality will,eyond a certain point in time, overwhelm the efforts of the adultopulation to increase the number of nestlings. Nestlings die outithout having their numbers replenished.
Next we will solve the full breeding model which incorporates food season and recruitment and is given by Eqs. (3.1)–(3.3). Theime of last hatching tE was prescribed a value in the scenarioescribed above but with the inclusion of recruitment is deter-ined by the timing of the food season, as outlined in Eq. (2.6). The
rowth of the nestling population B(t) is presented graphically inig. 2(b) with U and L again corresponding to a piecewise-constantnd piecewise-linear reproduction rate, respectively. Prior to thenset of the food season (0 ≤ t < 35), the growth curves are identi-al to those presented in Fig. 2(a). Beyond this point, which appearss a kink in the growth curve, the increase in food supply inducesn increase in the survival rates of the nestlings. The offspringatched during this period are likely to be sufficiently well-fed dur-
ng the period they are most vulnerable and hence survive to theedgling stage. At the end of the food season, the size of the new-
orn population instantaneously switches to zero, as all offspringre assimilated into the adult population. Eggs hatched after thexpiration of this window are deemed less likely to survive and areisregarded in our model.hange in size of a nestling population B(t) given by solving the full breeding modeld nestlings are added to the adult population at the end of a food season described
panel (a).
3.3. Optimal timing of breeding
The breeding model can be used to locate the optimal time for apopulation to concentrate its hatching activity in an environmentcharacterised by a short food season. This theoretical exercise maybe undertaken for any family of non-uniform hatching distribu-tions. Here, we use the class of piecewise-linear functions definedby Eq. (2.4) on the hatching interval [tS, tE]. We can interpret thefunction r(t) = rL(t) as a hatching distribution from which all contrib-utions occurring too late to be viable are instantly discarded fromthe population. Recall from Section 2.2 that when the reproduc-tion rate is assumed to be a piecewise-linear function of time, thepopulation’s first hatching at t = tS is followed by hatching effortsreaching a peak at t = tP. The hatching season theoretically may con-tinue until t = tE , but will generally be cut short at the end of thefood season at t = tF + �tF as reproduction after this point is notexpected to lead to recruitment. We suppose that tS and tE aredetermined exogenously and cannot be altered by the population.We then measure the population gains associated with differentvalues of tP, allowing us to establish which distribution defined on[tS, tE] would maximise recruitment at the population level.
Theoretically, birds should time their reproduction in such a waythat the peak demand of their nestlings coincides with the peakin food supply. This synchronisation results in maximal reproduc-tive success for a pair (Lack, 1950, 1968). The rate at which thefood supply deteriorates following this peak differs significantlyamong habitats (Both, 2010) but our primary interest here is infood seasons that are of a much shorter time scale than the overallbreeding season. It has been observed in great tit populations thatfemales hatching eggs earlier or later than the optimal date willfledge fewer offspring and those fledged typically will have a loweraverage weight (van Noordwijk et al., 1995; Verboven et al., 2001).We would expect this to be a general characteristic of species forwhom the synchronisation of food season and breeding is the mainselection pressure on laying date. As it is unlikely that all femaleswill be capable of hatching their eggs exactly on the optimal date,a distribution which peaks on this date may be the best that can behoped for. Without any restriction on first or last hatching date, the
population may choose the hatching distribution which is globallyoptimal. In the context of the breeding model, this would corre-spond to simultaneously tuning all three timing parameters tS, tPand tE relative to a given food supply g(t).
8 ologic
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4 J.G. Donohue, P.T. Piiroinen / Ec
However, it is not necessarily the case that the time of peakatching tP, corresponding to this optimal distribution remainshe optimal value when the time of first hatching is inflexible.n some cases, this value of tP may not lie on the range of possi-le hatching dates (i.e. the interval [tS, tE] in our model). By fixinghe first hatching date tS, we are effectively imposing a constraintn the population’s ability to optimise breeding. In particular, annadequate food supply during the egg-laying or incubation stages
ay force individuals to breed after this date (as reviewed by vanoordwijk et al. (1995)) meaning that the population has to settle
or a locally optimal value of tP on the range of available hatch-ng dates. Most members of such populations therefore appear toehave sub-optimally by not advancing their laying dates to theheoretical (global) optimum. This means that the timing of breed-ng is such that the demands of nestlings are not synchronised withhe food peak. In this case, most of the population’s young will ben the nest after the food peak rather than during it. Our model isased upon the assumption that there is an inadequate food sup-ly for nestlings for a significant proportion of the season. For an
ndividual breeding in such conditions, having its young in the nestfter the food peak may result in a recruitment failure for the sea-on. In the context of the analysis to follow, this scenario couldnfold when the food season is early in relation to the start date ofhe hatching interval.
We now determine the locally optimal value of tP for a selectionf qualitatively distinct values of tF and otherwise use parameteralues consistent with the previous figures. By choosing a suffi-iently small adult population M0 = 10, relative to the maximumestling population size K = 50, we ensure that there is enough ter-itory for all members of the population to breed. This allows us tosolate the effect of choosing a particular breeding schedule with
ortality and mistiming of breeding the only obstacles to repro-uctive success for an individual. The time at which hatching effortseach a peak, tP, is bound between the extrema tS = 0 and tE = 60. Weormalise the recruited population at the end of the food season (i.e.(tF + �tF)) with respect to the initial size of the adult populationnd plot this quantity against tP. A value of 1, for example, implieshat the initial adult population has replicated itself once, a valuef 2 represents the generation of two new members of the popula-ion by each existing adult and so on. For reference, we also plot theood season g(t) corresponding to each of these recruitment curves,ith g(t) as given in Eq. (2.1). We begin by considering a newborneath rate c3 = 0.02, which corresponds to the solid lines in Fig. 3,nd consider three cases in turn.
First, we see in Fig. 3(a) that if the food season is unusually early0 < t < 20), beginning just as the eggs of the earliest breeders areeing hatched, then the optimal action is for the bulk of the popu-
ation to hatch their eggs as early as possible. In this case, there is decreasing relationship between tP and recruitment. If the popu-ation has predicted that the food season will be particularly early,elaying egg-laying any longer drastically reduces the likelihoodhat the resultant offspring will survive. Furthermore, hatching dis-ributions which peak too late result in the initial population noteing replicated (i.e. the recruitment curve crosses below the level). This may impose a strain on the population’s ability to withstandortality effects for the remainder of the year, with no opportu-
ity to enhance its numbers until the following year. The optimalatching distribution is discontinuous (i.e. tP = tS) with a jump from
nactivity to maximum activity at the earliest possible hatchingate. Breeding effort is steadily reduced throughout the hatching
nterval, if the optimal tP is chosen, as late-hatched nestlings facenfavourable conditions.
Second, when the food season is situated in the middle of theatching period (20 < t < 40) as in Fig. 3(b), the dependency of repro-uctive success on tP is less straightforward. The optimal time tooncentrate breeding efforts is such that most hatchings occur prior
al Modelling 299 (2015) 79–94
to the peak in food supply but not so early that nestlings cannotsurvive long enough to benefit from it. The population’s span ofhatching dates [tS, tE] is well-chosen relative to the food season,providing a greater robustness in the specific choice of hatching dis-tribution on this interval. There is an interval of possible tP choiceson the solid curve which span roughly one third of the overallinterval and are associated with a relatively constant recruitmentrate. Hatching too many eggs before or after this time leads toreduced chick survival over the course of the season but all dis-tributions on [tS, tE] which reach a peak during this interval willyield comparable levels of recruitment. The recruitment curve is,in this scenario, bounded below by 1, with all possible hatching dis-tributions leading to the initial population being replicated at leastonce.
Third, if the food season occurs at the end of the hatching period(40 < t < 60) as in Fig. 3(c), an increasing relationship between tP andrecruitment success is observed. If pairs breed too early, the resul-tant offspring will struggle to survive to the beginning of the foodseason and hence recruitment. An optimal hatching distribution inthis scenario is one which steadily increases to a maximum, locatedat the very end of the hatching interval. Some pairs will breed wellin advance of the food season but the majority discern from tem-perature cues that their young would be better equipped to surviveif breeding is delayed. All choices of hatching distribution again leadto the population replicating itself at least once.
Some care needs to be taken in making direct comparisonsbetween Fig. 3(a) and 3(c) as each of the left vertical axes denotesa measure of recruitment taken at a different stage of the overallseason and therefore, the typical time at which the population’snestlings fledge will also vary. This adds another layer of compli-cation as there is evidence of a decline in fledging success over thecourse of the breeding seasons of many single-brooded bird popu-lations (Daan et al., 1989). However, if such effects are negligiblefor a particular population, it is possible to make some meaningfulinferences from the relative position as well as the relative variationof the different recruitment curves. In each case, we note the differ-ence between the maximum and minimum possible reproductiveoutput, which we denote by D. We notice that this value decreasesfrom Fig. 3(a) to (c) the food season becomes progressively laterrelative to the population’s breeding schedule. Therefore, for a pop-ulation which is capable of hatching its eggs well in advance of thefood season, the precise distribution of hatching dates is of lessimportance.
If late-spring effects are not significant, we can consider the ver-tical axes in Fig. 3(b) and (c) to represent a consistent measure ofreproductive output for the season and can overlay the two plots.The solid recruitment curves will then coincide at a certain valueof tP (not shown). This implies that there is a hatching distributionwhich enables the population to achieve an identical reproductiveoutput irrespective of which of the two distinct climatic scenar-ios occurs. Though the size of the recruited population is invariant,the traits of the individuals who successfully reproduce dependson which of the two food seasons occurred. In this example, par-ticularly late breeders will miss the food season entirely if tF = 20whereas the offspring of the earliest breeders may not be able tosurvive long enough to avail of the food season if tF = 40.
A comparison of the maxima attained by the solid curves inFig. 3(b) and (c) suggests that the longer period of hatching associ-ated with the later food season has only a limited benefit. When thefood season is later, a higher proportion of the area under r(t) = rL(t)(see Eq. (2.4)) can be exploited, resulting in a greater number ofhatchings prior to the food season. However, the high constant mor-
tality rate experienced by the nestlings diminishes these gains. Theearliest breeders in this slightly extreme case may not be able tosustain their broods long enough for them to benefit from the foodseason.
J.G. Donohue, P.T. Piiroinen / Ecological Modelling 299 (2015) 79–94 85
Fig. 3. Reproductive outputs are plotted against tP , the time at which the population’s hatching effort reaches a maximum, and the associated food supplies g(t) are plottedagainst time t. In each panel, the left vertical axis represents reproductive output which is calculated at the close of the food season and is normalised with respect to the sizeof the adult population at the beginning of spring. The right vertical axes represents the magnitude of the food seasons. The parameter values tS = 0 and tE = 60 define theearliest and latest hatching dates for the population. Panels (a), (b) and (c) correspond to food seasons which, relative to this interval, are early (tF = 0), well-timed (tF = 20)a c3 = 0.m .02. T
craeoPeTIiaiaiitstatrt(geisi
4
gtiWtrmtstTIt
nd late (tF = 40), respectively. A solid curve corresponds to a newborn death rateaximum and minimum reproductive output possible in each scenario when c3 = 0
While the solid recruitment curves in Fig. 3(b) and (c) attainomparable maxima, the solid recruitment curve in Fig. 3(a), cor-esponding to an early food season, lies below both of them forll choices of tP. In other words, for any given hatching strat-gy, a late food season may be less problematic than an earlyne when c3 = 0.02. The reason for this may be, as suggested byerrins (1991), that there will be newborns in the nests to ben-fit from the sudden increase in food supply when it does occur.hese will most likely be the offspring of the later-breeding birds.f a deterioration in environmental conditions brings about anncrease in c3 however, there are more severe consequences for
population which tends to breed early. To highlight this, wencrease the baseline death rate for newborns to c3 = 0.06 andgain plot reproductive output against the peak time in hatch-ng. This is represented by the dashed curves in Fig. 3. We seen Fig. 3(b) and (c) that there is a pronounced downward shift inhe recruitment curves corresponding to well-timed and late foodeasons, with a large portion of the curve in Fig. 3(c) now belowhe level-1 self-replication boundary. The recruitment curve forn early food season in Fig. 3(a) is comparatively less affected. Inhese less hospitable surroundings, the earliest broods experienceeduced survival because they may now, for example, be exposedo much lower prey densities before the onset of the food seasonPearce-Higgins et al., 2005; Visser et al., 2006). There is now areater urgency to time breeding correctly as hatching eggs tooarly becomes just as detrimental to recruitment success as hatch-ng too late. This is further demonstrated by the formation of aharper maximum in the relationship between tP and recruitmentn Fig. 3(b).
. The switching model
With a structure in place to assess the outcome of a sin-le breeding event, as was described in Section 3.1, we proceedo fit the event into the annual cycle of the population, allow-ng us to analyse the interplay between different components.
e separate each breeding season by an interval during whichhe population is concerned with other activities such brood-earing, molting and foraging. We will assume, in developing thisodel, that the species is migratory which means that part of this
ime will be spent in a separate non-breeding range. So-calledeasonal interactions (Knudsen et al., 2011) allow the effects of fac-
ors in one stage of the annual cycle to carry over into another.his is of particular significance for species that are migratory.n this case, the cycle can be thought of as a sequence of dis-inct regimes governed by a switching mechanism. This means02 and a dashed curve to c3 = 0.06. We also include the difference D between thehe parameter values used are � = 1, �tF = 20, M0 = 10, r0 = 3, K = 50 and c1 = 0.002.
that we can define independently the forces acting on a populationduring each part of the cycle. In accounting for the primary sea-sonal features of a cycle, these switches are complemented by abreeding-support function that defines an interval on which repro-duction is possible. This is a simple formulation of the idea thatthe range of possible breeding times for a population is con-strained, either by its current environment or by its own pastactions.
Once we are able to represent the variation in population sizeduring a single annual cycle, we can then study the extent towhich the population can sustain itself over long time scales andrecurrent breeding events. This is achieved using a piecewise-smooth dynamical system which depends on the annual phase,rather than time per se. A disturbance in one year will influencethe initial conditions of the next and so on. This more generalmodel is periodic in time and will be referred to as a switchingmodel in the remainder of the text. The long-term behaviour ofthe system can be thought of as the outcome of a succession ofswitches as the years pass. In general, for a fixed set of parame-ter values, the system will evolve towards a certain configurationfrom which it has no tendency to change, unless influenced bysome external forces. This configuration is a unique limit cycleof the dynamical system and represents the evolution of the sys-tem over one complete cycle. It is subsequently straightforwardto determine how the system will react to permanent changesin its parameters, merely by finding the corresponding steadystate.
4.1. Model description
We now extend the breeding model described in Section 3.1 toconsider a migrating population which we split into two subgroups,M(t) and S(t). The biomass of the population inhabiting a breedingrange at time t is given by M(t) while S(t) is the biomass of thepopulation on a non-breeding range, where the mating is carriedout by the individuals contained in M(t) only. This population isassumed to be single-brooded and results in a nestling populationwith biomass B(t). The variable t ≥ 0 is the time elapsed (in days)since the date of the last arrival to the breeding quarters on a partic-ular year. That is, we select the date of last arrival to the breedingquarters as the beginning of the migratory cycle. The initial con-ditions at t = 0, therefore, are the sizes of the populations in eachsubgroup at the beginning of the first migratory cycle considered.
Seasonal migration is represented by a periodic transfer of biomassbetween M(t) and S(t). We impose periodicity on the system bydefining a parameter T > 0 which represents the length of a migra-tory cycle, typically 365 days. The phase � records the number of
8 ological Modelling 299 (2015) 79–94
db
�
aTsmdiicptbqBttaWtmiwes
f
artsi
wtws3sutfnwttttstbovrmmuTpnb
Breeding
Migration
Migration
FirstSpring Arrival
FirstAutumn Arrival
LastSpring Arrival
LastAutumn Arrival
Survival
t0 = 0
t1
t2
t3 time
Fig. 4. A schematic diagram of the migratory cycle as represented in the switchingmodel. The cycle is split into four separate parts using the sequence of event timest0, t1, t2 and t3. Each of these markers corresponds to either the earliest or latestarrival time for the population in spring or autumn. The beginning of a cycle isdesignated as t0 = 0, coinciding with the date of the last spring arrival. The shaded
6 J.G. Donohue, P.T. Piiroinen / Ec
ays elapsed since the beginning of the current cycle and is definedy
:=t mod T (4.1)
nd thus 0 ≤ � < T with � = 0 representing the beginning of a cycle.his definition ensures that � = 0 when t = 0 which means that thetarting point for the system coincides with the beginning of aigratory cycle. In order to sub-divide the cycle into four parts, we
efine an increasing sequence of parameters, denoted by ti where = 0, 1, 2, 3, 4. Each of these values represents the end of one stagen a cycle and the beginning of the next. In particular, each markerorresponds to the beginning or end of a migratory period for theopulation. We let the parameter t0 be the time of the last arrivalo the breeding quarters, t1 be the time of first arrival to the non-reeding quarters, t2 be the time of last arrival to the non-breedinguarters and t3 be the time of first arrival to the breeding quarters.y enforcing t4 = t0 + T, we ensure that t0 ≡ t4modT which meanshat t4 is the time of the last arrival to the breeding quarters duringhe next cycle. Eq. (4.1) implies that the phase � will successivelyssume the values on the interval [0, T] over the course of one cycle.ithout loss of generality, we designate the date of the last arrival
o the population’s breeding quarters as the beginning of a newigratory cycle, so that t0 = 0 and hence, t4 = T. When � = T, the phase
s reset to � = 0, signifying the beginning of a new cycle. The systemill therefore periodically revisit each of these markers as time t
volves. This parameter sequence is then used to model distincttages in the cycle via the switching functions
i(�) ={
1, ti−1 ≤ � < ti,
0, Otherwise,i = 1, 2, 3, 4. (4.2)
A schematic of a migratory cycle as it is represented in our modelppears in Fig. 4. Note that while the population’s journeys betweenegions do not appear explicitly in this model, their position rela-ive to the four regimes which compose a cycle is indicated by thehaded region of Fig. 4. We will now consider each of these regimesn turn.
During the first stage, when t0 ≤ � < t1, the population, as ahole, is located in the breeding quarters. Assuming migration is
otal, the entire adult population is contained in the M(t) subgrouphile S(t) = 0. The reproduction of M(t) during a given breeding sea-
on is represented as in the breeding model outlined in Section.1. However, in the remainder of the paper, we will only con-ider the proposed piecewise-linear reproduction rate, allowings to model the tendency of a population to focus its reproduc-ive efforts on a particular moment in the breeding season. Theunctions g(t) (see Eq. (2.1)) and rL(t) (see Eq. (2.4)) representingewborn mortality and the population’s reproduction respectively,ill also be used in the switching model. They are now applied
o the phase � instead of time t but are otherwise unchanged. Ashe timing of arrival of an individual will partially determine theiming of its breeding, the hatching distribution is influenced byhe arrival distribution. This imposes certain restrictions on thehape of the hatching distribution r(�) as well as its position inhe migratory cycle. A feasible hatching distribution may simplye, for example, a phase-shift of the arrival distribution. The extentf this shift would be the sum of the typical time taken for an indi-idual to find territory, mate, commence laying and incubate theesultant eggs. Variation among individuals within the populationay distort this distributional symmetry. In some cases, individualsay complete the reproductive process more quickly (slowly) than
sual, skewing the bulk of hatchings towards earlier (later) dates.
his idea will be addressed in more detail in Appendix A: S1 whenarameter values are assigned. For now, recall that the number ofestlings cannot exceed K at any point during a breeding seasonecause of the scarcity of territory. At the end of each food season,areas of the diagram correspond to the migratory journeys of the population andtherefore contain the dates of the first and last arrival in the destination region.
nestlings are recruited and the reproductive window is effectivelycut short. To enable us to impose restrictions on the original layingdates of the population, we introduce a new parameter tS whichdenotes the time of the first possible hatching in the population.The interval [tS, tE] is then the set of all possible hatching timesduring a given cycle and we can define a breeding-support functionp(�) by
p(�) ={
1, tS < � < tE,
0, Otherwise.(4.3)
The values taken by tS and tE may depend on other systemparameters, as we will discuss in detail in Section 4.2. At this point,we can summarise the basic structure of a breeding season in ourmodel using the schematic in Fig. 5. We consider three interrelatedtime-lines which contain key events relating to migration, breed-ing and food supply. Hatching cannot begin until the populationhas migrated to the breeding quarters, while the success of thesereproductive attempts depend on the timing of the food food sea-son. The coupling represented by Eq. (2.6) is indicated by a dottedline between the food season end time tF + �tF and the hatchingtime tE of the last successfully recruited nestling.
The second stage of the cycle, when t1 ≤ � < t2 is concerned withthe population’s arrivals to the non-breeding quarters. As a firstapproximation, we assume that the rate of movement out of eachregion is positively related to the size of the population currentlyinhabiting the region. This gives rise to an exponentially decay-ing departure distribution and generates an arrival distribution
which is skewed towards early arrivals. In practise, these density-dependent terms could be replaced by an appropriately scaledarrival distribution derived from observation.
J.G. Donohue, P.T. Piiroinen / Ecological Modelling 299 (2015) 79–94 87
Fig. 5. A schematic diagram of the timing of events during a breeding season with respect to the phase �. The three time-lines represent, from top to bottom, the times ofevents associated with migration, breeding and food supply, respectively. Migratory timing is described by the parameters t3, t0 and t1, which represent the time of the firstspring arrival, last spring arrival and first autumn arrival. The first and last possible times of hatching are denoted by tS and tE . Hatching begins at � = tS and reaches a peak at� foodn .6). Thw
scisncv
tfdis
w
wtpdterrttbrcp
= tP . The start and end times of the food season are tF and tF + �tF . We connect theestling with a dotted line to indicate that these times are equal, as given by Eq. (2ill not survive.
There is no migration between the two ranges during the thirdtage, when t2 ≤ � < t3. If total migration occurred during the pre-eding stage, we now have the entire adult population containedn S(t) while M(t) = 0. As in the breeding quarters, we assume a con-tant death rate for the adult population throughout its stay in theon-breeding region, in the absence of detailed information to theontrary. These death rates could also be interpreted as the meanalues of minor fluctuations throughout the season.
The fourth stage, when t3 ≤ � < t4, sees the return of members ofhe population to the breeding quarters. The departure distributionrom the non-breeding quarters is again assumed to exponentiallyecay. This stage completes the cycle, with a new cycle commenc-
ng at � = T. We can now incorporate all of the components of ourwitching model by means of the hybrid dynamical system
dM(t)dt
= ˛vkvf4(�)S(t) − kvf2(�)M(t) − c1(f1(�) + f4(�))M(t), (4.4)
dS(t)dt
= ˛vkvf2(�)M(t) − kvf4(�)S(t) − c2(f2(�) + f3(�))S(t), (4.5)
dB(t)dt
= r(�)p(�)f1(�)M(t)(
1 − B(t)K
)− c3(1 − g(�))B(t), (4.6)
d�(t)dt
= 1, (4.7)
ith switches
M(t) �→ M(t) + B(t) at � = tF + �tF ,
B(t) �→ 0 at � = tF + �tF ,
� �→ 0 at � = T,
(4.8)
here c1, c2, c3, K, kv and ˛v are positive constants and the ini-ial conditions for the system are M(0) = M0, S(0) = S0, �(0) = 0. Thearameters c1 and c3 are, as in Section 3.1, the breeding-quarters’eath rates for adults and newborns, respectively. The death rate onhe non-breeding range is given by c2. The death rates c1 and c2 onlyxert an influence on the population during the periods that the cor-esponding regions are inhabited. Similarly, movement betweenegions will only occur between the specified first and last arrivalimes. The rate constant kv determines the extent of the migra-ion while ˛v is the survival rate. While, the population’s journeys
etween regions is not explicitly modelled, mortality effects en-oute are captured by the survival parameter ˛v. A value of ˛v = 1orresponds to total survival while ˛v = 0 would mean that theopulation is wiped out over the course of its journey. If ˛v = 1, theseason end time tF + �tF and the hatching time tE of the last successfully recruitedis represents our assumption that nestlings born after the end of the food season
effect of the ±kvf2(�)M(t) pair of terms in Eqs. (4.4) and (4.5) is, ast→ ∞, to move the entire biomass of M(t) into S(t). However, thesystem will only spend a finite time in this regime. A sufficientlylarge value of kv can approximate the same effect over a given finitetime interval, ensuring that the migration of the population is com-plete. In practise, there may sometimes be an overlap between thearrival and ‘residency’ regimes with, for example, the first hatchingfor some pairs occurring before the entire population has arrivedin the breeding quarters. For the sake of brevity, we ignore thesecases here.
While we are primarily concerned with describing possi-ble mechanisms governing the variation in a population that isimpacted by climate change, a reference set of parameter values isnecessary for comparative purposes. Throughout the paper, the fol-lowing values are employed: c1 = c2 = 0.002, c3 = 0.02, r0 = 5, K = 50,kv = 0.25, � = 1, �tF = 20, T = 365, t0 = 0, t1 = 152, t2 = 182, t3 = 335,tS = tS = 5, tF = 10, tP = 15, tE = 115. A discussion on how these val-ues were estimated appears in Appendix A: S1 and a detailed studyof the corresponding steady-state solution follows in Appendix A:S2. Finally, we note that t0 = 0 corresponds to a calendar date of May15 (compare with Figs. 4 and 5) which means that the phase vari-able � records the number of days elapsed since the most recentoccurrence of May 15.
4.2. Effect of advancing food season
There is a vast body of literature indicating that climate changein recent decades has resulted in the advancement of the period offood abundance (i.e. movement to an earlier set of dates) (Waltheret al., 2002). Long-term studies of the intervening period suggestthat this may lead to species mistiming their reproduction withrecruitment negatively affected. In this section, we examine indepth the effect of selecting progressively earlier seasons of foodabundance for a fixed reproduction window and migration sched-ule. We will then show how a population with the ability to advanceits range of hatching dates can compensate for such effects. Thereare many possible factors which may hinder a population’s abilityto synchronise breeding with the newly timed food season. In thelatter part of this section, we employ two different hypotheses and
test how they affect population sizes in our model. We will employa steady-state based approach which demonstrates the outcome ofannually repeated instances of a particular phenological configu-ration of breeding population and environment. In each case, we
8 ologic
wfm
anb
t
s(
g
taflft
t
cb
btrimiplaitynp22bptprma
t
r)
�
tP(
r)
t
tE(0.
8 J.G. Donohue, P.T. Piiroinen / Ec
ill focus on the size of the adult population immediately after theood season ends as this seems to be the most direct and intuitive
ethod of measuring breeding success.We begin by defining a time-shift ıF ≥ 0 which represents an
dvancement in the timing of the food season. We then define theew food season start time tF (ıF ) as a function of the time-shift ıF
y
F (ıF ) := tF − ıF . (4.9)
The timing parameter tF corresponds to the start date of a foodeason which is unaffected by climate change. We replace it in Eq.2.1) by the more general form tF (ıF ), so that
(�) :=
{4�(t − tF (ıF ))(tF (ıF ) + �tF − �)
(�tF )2, tF (ıF ) < � < tF (ıF ) + �tF ,
0, Otherwise.
(4.10)
In the scenario without climate change, where ıF = 0, we getF (0) = tF . The function defined in Eq. (2.1) can thus be interpreteds a special case, corresponding to an environment which is unaf-ected by climate change. Additionally, we note that the time of theast successful hatching is assumed to coincide with the end of theood season and consequently will advance with the phenology ofhe environment. We therefore generalise Eq. (2.6) to
E(ıF ) = tF (ıF ) + �tF . (4.11)
This means that the hatching distribution will now be trun-ated at � = tF (ıF ) + �tF , irrespective of the population’s choice ofreeding schedule.
For a migratory population, it may only be possible to advancereeding if accompanied by an equal advancement in its migra-ion schedule. The simplest adaptation is to leave the winteringange at an earlier date and arrive in the breeding range early,n order to react to the earlier food season. This adjustment
ay be borne out of plasticity in individual migratory tim-ngs (Cotton, 2003) or by new recruits arriving earlier than inrevious years (Gill et al., 2014). In either case, at the popu-
ation level, we arrive at the conclusion that the times of firstnd last arrival to the breeding quarters are flexible. By bring-ng these dates forward at the same rate, we are reducing theime spent in the non-breeding quarters over the course of theear and increasing the stay in the breeding quarters. This phe-omenon has been observed in many short-distance migrantopulations (Miller-Rushing et al., 2008; MacMynowski and Root,007) and occasionally in long-distance migrants (Jonzén et al.,006). Recall that the times of first and last arrival to thereeding quarters were designated t3 and t0, respectively. Theopulation’s hatching distribution is described by the parame-ers tS, tP and tE . Now, in order to bring the population’s springhenology forward, we define a second time-shift ır ≥ 0 whichepresents a homogeneous advancement in reproduction andigration and define the population’s new timing parameters
r(�) = rL(�) :=
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩2r0
tE(ır) − tS(ı
2r0
tE(ır) − tS(ı
s
3(ır) := t3 − ır, t0(ır) := t0 − ır, tS(ır) := tS − ır,
tP(ır) := tP − ır, tE(ır) := tE − ır. (4.12)
al Modelling 299 (2015) 79–94
To allow us to alter the timing of the breeding season, we rede-fine the function f4(�), originally given by Eq. (4.2), as
f4(�) ={
1, t3(ır) ≤ � < t0(ır),
0, Otherwise.(4.13)
Similarly, in order for breeding to advance in step with springmigration, we update the hatching distribution definition given byEq. (2.4), so that
− tS(ır)ır) − tS(ır)
, tS(ır) < � < tP(ır), tP(ır) /= tS(ır),
E(ır) − �
ır) − tP(ır), tP(ır) ≤ � < tE(ıF ), tP(ır) /= tE(ır),
Otherwise.
(4.14)
We observe that, when the time-shift ır = 0, the functionsdefined in Eq. (4.12) return the event times specified in Section4.1, with
t3(0) = t3, t0(0) = t0, tS(0) = tS, tP(0) = tP, tE(0) = tE.
(4.15)
In this case, Eq. (4.13) reduces to the definition contained inEq. (4.2), giving us the original timing of the breeding season. Ifthe phenology of both the environment and the population remainunchanged, ıF = 0 and ır = 0 and Eq. (4.14) reduces to Eq. (2.4).This hatching distribution corresponds to the timing decisions ofthe population in the absence of any climate change effects. Anadvancement of the hatching distribution, achieved here by set-ting ır > 0, is equivalent to shifting the piecewise-linear functionrL(�) forward in time. This may only be possible if the first possibletime of hatching is also free to advance. Consider a fixed length oftime W > 0 which represents the waiting time which must elapsebefore a female arrival is ready to hatch its eggs. In the absence
of any other restrictions, the earliest possible hatching date tS(ır)could then be expressed as
tS(ır) = t3(ır) + W. (4.16)
However, if conditions in the region prevent a hatching at thispoint, then the earliest possible hatching date will necessarily belater in the season. In general, we can write
tS(ır) := max(t3(ır) + W, t∗), (4.17)
where t* is the earliest time at which conditions allow hatching tooccur and marks the beginning of a steady improvement in theseconditions. Recall that the function r(�) is multiplied by the discon-tinuous function p(�) in Eq. (4.6), with the product p(�)r(�) being
non-zero on the interval [tS(ır), tE(ır)] only. Thus, if tS(ır) falls out-side of this range of dates, there will be a time interval duringwhich no hatching will occur. While we will refer imprecisely toa hatching failure at a particular moment, we are really concernedwith a planned hatching which could not materialise due to somedeficiency at an earlier stage of the reproductive process. In fact,it is generally at the laying or incubation stage that reproductionwill falter as this is where the highest energetic costs are incurred(Carey, 2009). As a starting point, we suppose that the breedingquarters imposes no environmental constraint on the breedingdecisions of the breeding population. Thus, an advancement of the
migration schedule brings the earliest possible hatching date for-ward with it and Eq. (4.17) simplifies to Eq. (4.16), which meansthat there is nothing preventing the population from adjusting tothe higher spring temperatures prompted by climate change.
J.G. Donohue, P.T. Piiroinen / Ecological Modelling 299 (2015) 79–94 89
Fig. 6. A schematic diagram highlighting shifted event times with respect to the phase �. Such shifts are induced by higher spring temperatures (in the case of food supply)or the population’s attempt to adjust to them (in the case of migration and breeding). Here, the food season start date moves forward from its original position at � = tF to� = tF (ıF ). The population attempts to compensate by migrating into the breeding quarters ır days earlier than in a scenario without climate change. This leads to a stretchingof the original migratory time-line, shown in Fig. 5, by ır . This allows the hatching distribution to move forward by ır days also. The dotted line between the breeding andfood time-lines indicates that the time of the last successful hatching depends on the timing of the food season via Eq. (4.11). The bottom time-line includes the earliest time
t* at which environmental conditions allow a hatching to occur. It is notable that t* is not coupled with tS(ır ) in this case, as we have assumed that there is sufficient food for
a . The
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To summarise the revisions made to our framework so far, theood season will now begin at � = tF (ıF ) and end at � = tF (ıF ) + �tF .he times of first and last arrival to the breeding quarters are
3(ır) and t0(ır), respectively, while the hatching distribution is
escribed by tS(ır), tP(ır) and tE(ır). The hatching distribution willow be truncated at � = tF (ıF ) + �tF and heterogeneous shifts inhe phenology of breeding and the phenology of the environmenti.e. ıF /= ır) will affect the level of population growth. The adjust-
ent in the timing of events in our model is illustrated in the revisedchematic diagram shown in Fig. 6.
The parameters t3, t0, tS, tP, tE and tF will remain fixed at thealues they were assigned in Section 4.1 for the remainder of theaper. The quantities (ıF, ır) give us a natural coordinate system inhich to consider the effect of bringing forward the critical events
n our model, relative to these reference points, and therefore it ishe values of ıF and ır we will vary. In the numerical example thatollows, we set the time W between the first arrival and the firstossible time of hatching to be 35 days. This interval is made upf 10 days of pre-laying preparation (Jonzén et al., 2007), 12 daysf laying and 13 days of incubation (Perrins, 1991). Fig. 7(a) showshe dependency of the steady-state size of the adult population athe end of the food season (i.e. M(tF (ıF ) + �tF )) on ıF and ır. Weonsider a starting point of (ıF, ır) = (0, 0) which corresponds to theystem without any influence from climate change. This positionan be thought of as an ‘equilibrium’ state and it corresponds tohe peak in the adult population at the time of the last successfulatching. We see in Fig. 7(a) that advancing migration and repro-uction for this fixed food season by 10 days (i.e. moving from (ıF,r) = (0, 0) to (ıF, ır) = (0, 10)) allows the population to align theemands of its nestlings with the timing of the food season moreuccessfully, increasing recruitment (following curve I in Fig. 7(a)).erhaps of more interest here is the effect that moving from fromıF, ır) = (0, 0) to (ıF, ır) = (10, 0) has on the steady-state populationize (following curve II in Fig. 7(a)). The outcome of a ten-day shiftn the food season without an accompanying shift in the reproduc-ive schedule is a total collapse in population size. The food seasons already underway when the first eggs are being hatched which
eans that relatively few nestlings will avail of it. The cumulative
ffect of these annual mismatches is the population size tendingo zero. If, on the other hand, the population reacts to this changen its environment by advancing migration and reproduction, wenstead move from (ıF, ır) = (0, 0) to (ıF, ır) = (10, 10) (followingdotted line connecting the first possible time of hatching tS(ır ) and the first time of
curve III in Fig. 7(a)). The sizes of the population at this point and atthe final point are equal, the population having matched the shift inthe phenology of its environment with an identical shift in its own.Note also that the population size is constant along the line ır = ıF
(curve III in Fig. 7(a)). Such (ıF, ır) pairs correspond to a populationthat has perfect information about optimal migration times and therequisite level of energy to breed at the optimal time when it arrivesin the breeding quarters. Furthermore, if the difference between thetwo time-shifts ır and ıF is fixed, the steady-state population sizeagain remains constant. This is reflected in the fact that the con-tours of the surface are parallel to curve III when projected in the(ıF, ır) plane (see Fig. 7(a)). As the population becomes more adeptat tracking climate change, it is able to move from low-level con-tours to high, thereby returning towards its original steady-statesize.
However, the ecological reality is rarely so straightforward.There is often some impediment to achieving a resynchronisationof reproduction with the food season. In this context, we inter-pret tS(ır), in a slightly more general way, as the first attempted
hatching. So far, we have ensured that tS(ır) < tS(ır) and so therewas no need to distinguish between the first attempted hatchingand the first successful. We now look at the case where tS(ır) fallsoutside of the range of viable dates with any planned attempts tohatch eggs between then and the first possible date unsuccessful.We assume that individuals that are unsuccessful would then takethe next best alternative and hatch as close as possible to this date(Perrins, 1965). Adjustments in the timing of hatching at an indi-vidual level may be represented by an adjustment in the shape ofthe population’s hatching distribution. This process is formulatedmathematically in Appendix A: S3. The total reproductive effortmade by the population during a breeding season is conserved at avalue of r0 for all possible values of the first hatching date tS(ır).This means that failed laying attempts by an individual female,made early in the breeding season do not preclude further attemptslater. We can nominally choose any first hatching date, therefore,but if it is not contained within the interval of possible hatchingdates, there will be no growth at the times which are invalid andthe original area under the growth function will be compressed
into a narrower interval of time. This introduces a discontinu-ity in the hatching distribution at the beginning of the hatchinginterval as a significant proportion of the population choose tohatch at the earliest available opportunity. We now consider two
90 J.G. Donohue, P.T. Piiroinen / Ecological Modelling 299 (2015) 79–94
Fig. 7. Panel (a) shows the steady-state size of the adult population at the end of the food season (M(tF (ıF ) + �tF )) corresponding to different hatching schedules ır andfood season times ıF . Curve I represents an advancement in breeding without an advancement in the food season, curve II represents an advancement in the food seasonwithout an advancement in breeding and curve III matches an advancement in the food season with an equal advancement in breeding. We also plot a selection of contoursof the surface, each of which are parallel to curve III and correspond to a fixed difference between the time-shifts ır and ıF . Panel (b) shows the steady-state size of theadult population at the end of the (advanced) food season plotted against the population’s attempted advancement in hatching ır when there is a constraint on breedingtime. The attempted advance in the hatching distribution is a response to an advancement in the food season start date from � = tF (0) to � = tF (10). The dashed curve Urepresents the outcome of a theoretical shift in the hatching distribution if there were no constraint on breeding time. It is a slice of the surface in panel (a), taken at ıF = 10.
The curve CA results from the migration timing parameters t3(ır ) and t0(ır ) and the hatching timing parameters tS(ır ), tP (ır ) and tE(ır ) advancing simultaneously by ır , while
t curve
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cologically distinct obstacles to adaptation. Each of these obsta-les imposes a constraint on the population’s ability to track climatehange. We examine each obstacle separately. In both cases, we willssume that the food season has advanced by ten days (i.e. ıF = 10)nd compare the response curves to the unconstrained responsehown in Fig. 7(a). The population size we consider in our anal-sis, again measured at the end of the food season, is therefore(tF (10) + �tF ).Constraint A: Response constrained by food limitations It has been
oted by Stevenson and Bryant (2000) that food shortages at theeginning of spring may constrain a population’s first laying date.hile climate change may heighten late spring temperatures, the
attern of temperature change will not necessarily be consistenthroughout the season. This means that the source of nutrientsequired for a female to reproduce at its preferred date may notdvance to the same extent as the source of nutrients for nestlings.e model this phenomenon by assuming that the time at which
emales have sufficient energy to begin laying is independent ofhe start time tF (ıF ) of the food season upon which nestlings willepend. An insufficiency in the food supply for females at the timef laying imposes a restriction on the first possible time of hatch-ng. In particular, for a given clutch size, the first possible time ofatching will be the first possible time of laying plus the time takeno lay and incubate the eggs. The earliest time t* at which femalesre ready to hatch is subsequently independent of tF (ıF ) also.
We again assume that the population can freely adjust its migra-ion schedule. The population is aware that the optimal responseould be to advance its reproductive schedule and migrates to the
reeding quarters earlier in order to do so. However, the short-ge in food supply acts as an energetic bottleneck and attemptso begin laying at this earlier date are unsuccessful. This meanshat the population’s timing parameters are shifted forward by ır
s before but the earliest possible hatching date tS(ır) will only
dvance as far as environmental conditions allow. Without knowl-dge of the precise time at which a female dwelling in a habitatould have sufficient energy to lay an egg, we are unable to accu-ately prescribe a value for t* but here, we we will assume that t* = 0.
CB results from the hatching timing parameters tS(ır ), tP (ır ) and tE(ır ) advancing
ue to a constraint on migration times.
In real terms, this would mean that May 15 is the earliest date atwhich this environment would allow a hatching to take place (seeSection 4.1). In the scenario with no climate change, there is noadvancement in the phenology of the population (ır = 0) and thefirst arrival date to the breeding quarters occurs at � = t3(0) = t3.In this case, Eq. (4.17) reduces to Eq. (4.16) as t3(ır) + W > t∗. Thismeans that, as we increase ır away from zero, the first possible
hatching date tS(ır) initially varies with the first arrival date. How-ever, as we continue to advance migration, we decrease t3(ır) untilthe relationship between the first possible hatching date and thefirst arrival date breaks down. The food supply for laying femalesbecomes a limiting factor as t∗ > t3(ır) + W . This decoupling of the
first time of arrival t3 and the first possible time of hatching tS(ır)reduces Eq. (4.17) to
tS(ır) = t∗, (4.18)
when migration has advanced sufficiently. The value of tS(ır) isdetermined solely by environmental factors, regardless of howmuch earlier the population migrates.
Fig. 7(b) shows a plot of the steady-state population size atthe end of the (advanced) food season, M(tF (10) + �tF ), againstthe population’s attempted advancement in hatching ır. The solidcurve CA corresponds to the size of a population whose attempt toadvance its breeding is opposed by food limitations. The dashedcurve U, included for comparative purposes, corresponds to a sliceof the surface shown in Fig. 7(a) and represents a population’sunconstrained adjustment to a ten day shift in the timing of thefood season (i.e. ıF = 10). If, as before, the food season shifts forwardby ten days from its reference value of tS(0) = tS , the populationis able to simultaneously advance migration and breeding by fivedays. We see in Fig. 7(b) that the population’s response follows thesame path as the unconstrained system from ır = 0 to ır = 5, as the
curves CA and U coincide for these values. The population has fiveunexploited breeding days which, with an earlier arrival time, cannow be used to mitigate the effects of the earlier food season fornestlings. Beyond this point, the population can migrate earlier in
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n attempt to respond to this change but is unable to advance itsrst hatching date any further because of an inadequate food supplyarly in the spring. At ır = 5, the underlying hatching distributionecomes discontinuous and the curves CA and U diverge for valuesf ır larger than 5. The population as a whole is now able to manip-late its hatching distribution, only by increasing the proportion ofairs which hatch at the earliest possible date. We observe that CAontinues to increase with ır and therefore, this response recovers
higher population size than if no action were taken. However,he rate of recovery of population size is lessened after ır = 5, withA associated with lower population sizes than U for these valuesf ır. This means that, despite the population’s tendency towardsarlier breeding, the steady-state population size is less than whatt would be if the population were able to move the first hatchingate back further. It should be noted, in addition, that establishing
discontinuity in the hatching distribution at the beginning of theatching interval will only make sense if the increase in food sup-ly at � = t* is rapid. If this is not the case, there may not be enoughood available at this point for a majority of the population to beginaying and a truncation of the curve CA at ır = 5 may be the best thatan be hoped for.
At the individual level, a female can respond to a ten daydvancement in its offspring’s food supply with a ten day advance-ent in its arrival to the breeding quarters. If there was a sufficient
ood supply of food, the individual would be ready to lay eggs tenays earlier than in previous years. However, it is forced to wait foronditions in the breeding ground to improve before it can beginaying. In this case, as the earliest arriving female does not reducehe size of its clutch, the waiting time W remains fixed. This indi-idual’s eggs will then hatch at a time when the prey item uponhich its nestlings’ survival relies will be approaching a peak in
bundance. The offspring of later arrivals will receive little, if any,enefit from this food source and population declines result.
Constraint B: Response constrained by a photoperiodic migrationue Even in cases where adults have a plentiful supply of food, theiming of their arrival to the breeding quarters may act as a bar-ier to earlier breeding (Both and Visser, 2001). For some migrants,articularly those that travel long distances, the cue to depart theintering range is independent of temperature and therefore, cli-ate change will not alter its timing (Bradley et al., 1999). The
opulation does not realise that its interval of hatching dates shoulde brought forward until it has arrived in the breeding quarters, athich point it is too late. This means that the times of first arrival
nd last arrival to the breeding quarters will not be adjusted as reaction to rising temperatures. We fix them at t0(ır) = t0 and3(ır) = t3, so that the time-shift ır ≥ 0 now applies solely to theiming of breeding. To isolate the effect of this constraint on theopulation, we will assume that the increase in temperature is con-istent throughout the spring, so that there is an ample supply ofood for the breeding population. As in the unconstrained case, thismplies that the earliest possible hatching date depends only on therst arrival date. A fixed time of first arrival t3(ır) then implies that
he earliest possible hatching date tS(ır) is fixed also. It’s value is
et to tS(ır) = t3 + W . The population’s late arrival, relative to theeak in food supply, gives us another energetic bottleneck at theeginning of the breeding season.
The solid curve CB in Fig. 7(b) shows the relationship betweenhe population size M(tF (10) + �tF ) and the extent of the popu-ation’s attempts to advance breeding, contained in ır. Recall thathe dashed curve U represents the idealised case with no constraintn advancement. With a restriction imposed on arrival time, the
opulation’s attempt to advance its breeding schedule results in aiscontinuous hatching distribution for all ır > 0, because there areo unexploited hatching dates. As a consequence, CB lies strictlyelow U for all positive values of ır. As with Constraint A above, anl Modelling 299 (2015) 79–94 91
increase in the proportion of early breeders relative to the fixed firsthatching date achieves a lower steady-state population size thanan advancement of this first hatching date. In this case, attemptsto advance reproduction serve to erode the interval between thetime of first hatching tS and peak time of hatching tP (see Section4.1). The removal of this interval is completed when ır = 10, leav-ing a monotonically decreasing hatching distribution. As the arrivaldistribution in this case is also a decreasing function of time, thehatching distribution is analogous to a phase-shift of the arrival dis-tribution, with all individuals breeding as quickly as possible upontheir arrival in the breeding quarters. Even with the benefit of thisadaptation, the population size on the curve CB at ır = 10 is roughlyhalf of the corresponding value on U, suggesting that an inflexibilityin arrival time may limit the size of migratory populations in thelong-term.
5. Discussion
In this work, we have developed a framework in which thetemporal variations in the size of a population over the course ofits annual cycle can be incorporated. The representation of sea-sonal breeding in a dynamical systems context is a foundationupon which a new class of time-dependent population models canbe developed. Although theoretical models of population growthover a single breeding season already exist, this work is, to ourknowledge, the first that includes the entire annual cycle andrecurrent breeding seasons. In recent years, periodic forcing hasbeen introduced to classical ecological (Rinaldi et al., 1993) andepidemiological (Bacaër, 2012) models in order to illustrate theimportance of including seasonal variation. In population models,this has often taken the form of a sinusoidal forcing in the growthrate (Taylor et al., 2012), mortality rate (Bessho and Iwasa, 2010) orresources (Scheffer et al., 1997). However, for a species that alter-nates between distinct habitats and/or has just a short window eachyear during which successful reproduction is possible, a sinusoidalvariation may not accurately capture the temporal properties ofthe system. When a sudden onset of activity is typical in a system,piecewise-smooth forcing may be a more faithful representationof the biological reality. We used a piecewise-smooth dynamicalsystem approach to model the variation in a single-brooded popula-tion of migratory birds over a single breeding season (the breedingmodel) and then over long time scales (the switching model).
Using the breeding model, we considered how the growth of apopulation could be maximised in a single breeding season whenfood is abundant for a short time only. A sudden increase in tem-perature at the beginning of spring can lead to the food seasonoccurring too early for a large part of the population to fully profitfrom it. In the case of a specialist predator, this is associated withreproductive failure for all but the earliest breeders. For the sake ofcomparison, we also considered somewhat idealised cases in whichthe population has ample time to prepare for an upcoming foodseason and found optimal hatching distributions for the popula-tion under these conditions. By varying the level of risk inherentto the environment, encapsulated by the baseline mortality ratefor newborns, we were able to highlight two distinct scenarios. Thefirst of these is characterised by an asymmetry in the consequencesof population-wide mistiming, with late breeding of much graverconsequence than early. When the environment prior to the foodseason is more inhospitable, this asymmetry dissipates and breed-ing too early leads to a higher proportion of newborns starving todeath before the food season begins. An analysis of this type whichincorporates prey-switching behaviour would be a natural progres-
sion and allow one to consider species that do not suffer as a resultof their young being in the nest after one particular prey item is nolonger available to them.
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We showed that when the reproduction rate explicitly dependsn time and there is assimilation of surviving offspring into thedult population at the end of each breeding season, the typicalteady-state solution to the switching model is a limit cycle with aeriod of one year. This is equivalent to a stable non-zero fixed point
n a classical seasonal breeding model with self-limiting growth andiscrete time. We note that the assumption that new recruits areeady to breed after one year is appropriate for passerines and otherelatively short-lived birds (Perrins et al., 1987). Otherwise, a modelhich is structured by year of birth may be more appropriate.
We then used a rudimentary model of this type to show howhenomena associated with global warming impact upon the sur-ival of a population whose reproductive success relies uponynchrony between breeding and an annual period of food abun-ance. There has been a trend towards earlier occurrences of suchood seasons in recent decades. With the assumption of no recruit-
ent after the expiration of the food season in place, our modelualitatively matches the expected population declines associatedith an advancement in the food season. Furthermore, it pre-icts that, if this trend persists, then population collapses couldesult. This forecasted decline in average population size is ofourse only relevant for populations which obey the underlyingssumptions of our model. A population which is multi-brooded,or example, would not necessarily face a decline in recruitmentuccess throughout the breeding season (Daan et al., 1989) or selec-ion pressure from synchronisation of food season and breedingVerboven et al., 2001). An advancement of the food season there-ore may not hold the same significance for such species.
A population may respond to an advancement in the timing of food supply by advancing its breeding schedule. For a migratoryopulation, this may necessitate an advancement in the timing of
ts spring migration. We assumed that the reaction of a migrantopulation to an advancement in the time of the food season woulde to shift the entire spring migration period (defined as the lengthf time between the first and last arrival) forward, thereby increas-ng the total time spent in the breeding quarters each year athe expense of the time spent in the survival quarters. We werehen able to show that an advancement in migration and hatch-ng matching the original advancement in the food season restoreshe population to its original steady-state size. An alternative pos-ibility is that the migration period is actually extended ratherhan being shifted (Buskirk et al., 2009; MacMynowski and Root,007; Sparks et al., 2005). Additionally, there is some evidence ofopulations moving both of their yearly migrations forward so thathe time spent in each range is constant (Cotton, 2003). It woulde relatively straightforward to implement such behaviour in thewitching model and compare the resultant effects on populationize.
Constraints on this adaptation process may prevent the popula-ion from fully compensating for the advancement in the phenologyf its environment. We assumed that some individuals would maken unsuccessful attempt to realise the gains associated with earlyreeding and subsequently wait until the first available opportunityo make a second attempt. This behavioural assumption manifeststself in a re-shaping of the hatching distribution when it cannote shifted forward any further. We showed that even if individualso behave in this ostensibly rational way, the increased emphasisn earlier breeding is not enough to compensate for the inabil-ty to profit from the beginning of the food season, resulting in aecline in population size. When the constraint on breeding timeas the result of an inflexible migration schedule, the adjustedatching distribution was a monotonically decreasing function,
hich we interpreted as a phase-shift of the exponentially decayingrrival distribution. Many forms of arrival distribution have beenbserved in studies of real populations, both for the spring andhe autumn migrations. In particular, there is evidence to support
al Modelling 299 (2015) 79–94
both unimodal (Wilson, 2013) and multi-peaked (MacMynowskiand Root, 2007) spring arrival distributions. As discussed in Section4.1, the shape of this distribution in a given year will condition theshape of the subsequent hatching distribution, although the exactnature of this coupling is difficult to predict. While we have cou-pled the times of the first arrival event and the first hatching eventin the preceding analysis, there is no explicit mechanism linkingthe shapes of the two distributions. We take care to note, there-fore, that the decreasing hatching distribution which evolved inour analysis may be compatible with a given arrival distributiononly if later arrivals are able to reduce the interval between arriv-ing and hatching. This may be achieved by reducing the layingpreparation time, spending less time incubating eggs or, perhapsmost commonly, by reducing clutch size (Both, 2010). In the con-text of the model, this would suggest that the reproduction ratecoefficient r0 may depend on the timing of the food season. Onthe other hand, if late arrivals are unable to reduce the intervalbetween arrival and hatching, then the insight gleaned via the gen-eration of this decreasing hatching distribution may represent a‘best-case’ scenario which is unattainable for many forms of arrivaldistribution. A framework which couples the shapes of the arrivaland hatching distributions may then reveal that the population’sresilience to shifts in the food season depends on its spring-arrivaldistribution.
A key advantage of our approach is the ease with which it canbe generalised. The model presented here can be expanded upon ina myriad of ways, simply by replacing the relevant components inour equations with functions of density and/or time that have beeninferred from observation of real populations. For example, whenwe considered the case of climate change acting inconsistentlywithin a season, we assumed that tS , the time at which a femalehas sufficient food to commence laying, and tF, the beginning ofthe season of food abundance, are uncoupled. This is perhaps unre-alistic, suggesting as it does that there is no correlation betweentemperatures at the beginning of spring and temperatures later inthe season. While the exact nature of the relationship between cli-matic variation and time may be difficult to ascertain, it has beensuggested that chaotic behaviour at an atmospheric level gives riseto inter-annual variation in temperature and therefore the timingof events (Lorenz, 1967). The timing and length of the food seasonas well as the time of first laying are also linked to climatic factorsand will therefore vary from year to year. Studies on the migratorypatterns of various species suggest that there is also likely to besignificant inter-annual variation in arrival distributions for a par-ticular population (Morton and Pereyra, 1994; Sparks et al., 2005).In principle, this variation could be incorporated, either determin-istically or via stochastic terms, and the statistical properties of theresulting system studied.
The differential equations presented in this paper could alsobe used to explore the significance of the food season length �tF
for a breeding population. It has been suggested that if the foodseason is too narrow, reproduction will inevitably falter after off-spring are born, as parents are unable to provide a sustained sourceof nutrients for their newborns, leading to starvation and dimin-ished recruitment rates (Visser and Both, 2005). One of the effectsof higher spring temperatures may be to lessen the duration of thiswindow of high food availability (Both et al., 2006). The interdepen-dencies between the length, timing and intensity of the food seasonmay prove to be critical. It has been noted in Buse et al. (1999)that particularly high spring temperatures lead to a narrowing andamplification of the food season, as well as an advancement in itsphenology. Therefore, an early food season is likely to be narrower
than a late one but also more profitable for the individuals whoavail of it. In the limiting case of a food season shrinking to zerolength, however, the logical arguments underpinning our mod-elling choices may no longer be credible, with the nature of the
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ystem fundamentally altered by the severity of the climatic shift. different approach would therefore need to be considered.
We note that the growth terms used in both the breeding andwitching models are more likely to depend on M(t − d) than M(t),here M(t) represents the biomass of an adult population at time
, and d is the typical length of a reproductive cycle (i.e. the timelapsed between mating and hatching). This approach yields a delayifferential equation. Omitting the delay d and using the M(t)-ependent form found in the given equations was a convenienttarting point. Incorporating a delay would allow us to examineow certain migration patterns, expressed via arrival distributions,nd their interplay with other time-dependent functions containedn our model affect the reproductive success of the population. Thiss, therefore, a secondary means through which the shape of therrival distribution could become a determinant of the species’ sur-ival. As well as conditioning the shape of the hatching distribution,t would control the rate of growth in a delay-driven model. In a
odel of this type, an arrival distribution which is skewed towardsate arrivals would directly limit the population’s growth early inhe breeding season.
It is important to note that our model, by design, places criticalmportance on the period up to and including the food season as
determinant of recruitment success. It is, therefore, fundamen-ally biased against pairs who hatch their eggs after this windowas passed. The exclusion of any individuals that hatch after the
ood season has ended is perhaps too blunt a mechanism, even for species that preys on a narrow range of organism types. A possibleethod of including the contributions of later breeding would be
o transfer fledglings to a non-breeding compartment at t = tF + �tF.he breeding season would then continue with more fledglingsecruited after a second sub-season of breeding has been com-leted. Such fledglings would necessarily be fed with a more variediet with no one particular prey item being abundant. Extendinghe model would therefore require detailed and species-dependentnowledge of the pattern of prey growth throughout the season asell as predatory preferences in order to accurately model the vari-
tion in food supply g(t). Additional functional relationships maylso need to be considered in these cases. For example, individualshat lay eggs at the height of food abundance may have sufficientnergy to increase the size of their clutches. Under the parame-er regime specified in Section 4.1, the resultant offspring wouldatch after the food season has expired and would therefore bexcluded from our model. In a more general setting, the popula-ion’s typical clutch size r0 could be a function of the food supply at
certain point in time. However, it should be noted that a decreasen recruitment success for single-brooded species of birds is to bexpected over the course of a breeding season (Daan et al., 1989),hether explicitly due to seasonal variation (the date hypothesis) orue to quality differences between early and late breeding pairs (theuality hypothesis) (Verhulst and Nilsson, 2008). While this may bettributed partly to an increase in the newborn mortality rate afterhe food supply has peaked, there is evidence that nest-desertionnd other post-hatching difficulties become more frequent later inhe breeding season (Newton and Marquiss, 1984). A reduced ratef conversion from nestlings to adults therefore may be appropriateor later-hatched nestlings. Allowing clutch size to vary with foodupply would then introduce an additional trade-off for the popu-ation as late breeders could hatch a larger brood but the chancesf each individual surviving may be lessened.
In summary, a rigorous study of the coupling within theeal ecosystem of interest is required in order to generate morensightful predictions. Our model, in its current guise, is a first
pproximation to what is a richly complex chain of interrelatedvents. The framework we have outlined, however, is well-suited toeal with both seasonal interactions and the propagation of recruit-ent failures from one year to the next. The breeding model servesl Modelling 299 (2015) 79–94 93
two important functions. Firstly, it represents seasonal breeding ata population level and places it into a dynamical systems frame-work, allowing us to avail of the rich body of theory underpinningit. Secondly, it gives us a means to examine how the presence of atransient period of favourable conditions in the environment influ-ences population recruitment, given the assumptions of our model.The development of the switching model enables us to predict howthe system will react to transient changes in its components and/orhow long it will take to recover. It also enables us to assess theimpact of persistent temporal mismatches brought about by chang-ing climatic conditions. In some cases, the effect of total recruitmentbeing below its potential each year will be a collapse in populationsize. This approach may therefore be most useful as a complementto the ongoing experimental and theoretical work towards a clearerunderstanding of the possible consequences of continued climatechange on ecological systems.
Acknowledgements
J.G. Donohue was funded by an EMBARK Scholarship from theIrish Research Council. The authors gratefully acknowledge theanonymous reviewer whose critical reading of an earlier drafthelped to improve the manuscript.
Appendix A. Supplementary Data
Supplementary data associated with this article can be found,in the online version, at http://dx.doi.org/10.1016/j.ecolmodel.2014.12.003.
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