Population Dynamics - IITKhome.iitk.ac.in/~peeyush/mth426/l6.pdfFundamentals of Mathematical Ecology/Biology Provides an introduction to Classical and Modern Mathematical Models, Methods

Introduction Equilibria, Stability Single Species Dynamics Interacting Populations

Population Dynamics

Introduction 1 / 62


Introduction

Fundamentals of Mathematical Ecology/Biology

Provides an introduction to Classical and Modern MathematicalModels, Methods and issues in population dynamics.Devoted to simple models for the sake of tractability.Topics covered include single species models, Bifurcations,interacting populations that include predation.Suitable for Post graduate students and beginning researchers inMathematical Biology.

Introduction 2 / 62


Introduction



Introduction 2 / 62


Introduction



Introduction 2 / 62


Introduction



Introduction 2 / 62


Some Results

Some useful results from ODEA general first order initial value problem is given by

y ′ = f (t , y), y(0) = y0.

—a non-autonomous differential equation due to explicitinvolvement of the independent variable, t , in the right hand side.In the entire course we are going to deal with autonomous firstorder differential equations/systems.

Introduction 3 / 62


Some Results


y ′ = f (t , y), y(0) = y0.


Introduction 3 / 62


Some Results


y ′ = f (t , y), y(0) = y0.


Introduction 3 / 62


Some Results

Some useful results from ODEUnless and otherwise stated, we assume that all the differentialequations satisfy the Picard’s theorem. Hence every initial valueproblem admits a unique solution.

Theorem (Picard’s Theorem)

If g(t ,u) is a continuous function of t and u in a closed and boundedregion R containing a point (t0,u0) and satisfies the Lipschitz conditionin R then there exists a unique solution u(t) to the initial value problemu′ = g(t ,u),u(t0) = u0 defined on an interval J containing t0.

Introduction 4 / 62


Some Results

Some useful results from ODEUnless and otherwise stated, we assume that all the differentialequations satisfy the Picard’s theorem. Hence every initial valueproblem admits a unique solution.

Theorem (Picard’s Theorem)

If g(t ,u) is a continuous function of t and u in a closed and boundedregion R containing a point (t0,u0) and satisfies the Lipschitz conditionin R then there exists a unique solution u(t) to the initial value problemu′ = g(t ,u),u(t0) = u0 defined on an interval J containing t0.

Introduction 4 / 62


Concept of Equilibrium

Definition (Equilibrium Points)Let us consider the following first order autonomous differential equation(system) dN

dt = F (N). All the solutions of the equation F (N) = 0 are calledequilibrium solutions of the above equation.

Introduction 5 / 62





Introduction 5 / 62





RemarkThese solutions are also some times called as equilibrium points, criticalpoints, stationery points, rest points or fixed points.

Introduction 5 / 62



Example: x ′ = x2 − 3x + 2x∗ = 2 and x∗ = 1 are two critical points.

Introduction 6 / 62




Introduction 6 / 62




It is easy to verify that x∗ = 2 and x∗ = 1 satisfy the differentialequation x ′ = x2 − 3x + 2.

Introduction 6 / 62



Therefore, if N∗ is an equilibrium solution of the differentialequation N ′ = f (N) then N(t) = N∗ is the unique (constant)solution of the initial value problem (IVP) N ′ = f (N),N(t0) = N∗.Thus, note that the equilibrium solutions are special constantsolutions of the associated differential equation.

Introduction 7 / 62



Therefore, if N∗ is an equilibrium solution of the differentialequation N ′ = f (N) then N(t) = N∗ is the unique (constant)solution of the initial value problem (IVP) N ′ = f (N),N(t0) = N∗.Thus, note that the equilibrium solutions are special constantsolutions of the associated differential equation.

Introduction 7 / 62


Stability, Instability

Definition (Stability)An equilibrium solution N∗ is said to be Lyapunov stable, if for anygiven ε > 0 there exists a δ > 0 (depending on ε) such that, for all initialconditions N(t0) = N0 satisfying |N0 − N∗| < δ, we have|N(t)− N∗| < ε for all t > t0.

Alternatively, we say that an equilibrium solution is said to be stable ifsolutions starting close to equilibrium solution (in a δ neighborhood)remain in its ε neighborhood for all future times.

Introduction 8 / 62



Definition (Stability)An equilibrium solution N∗ is said to be Lyapunov stable, if for anygiven ε > 0 there exists a δ > 0 (depending on ε) such that, for all initialconditions N(t0) = N0 satisfying |N0 − N∗| < δ, we have|N(t)− N∗| < ε for all t > t0.

Alternatively, we say that an equilibrium solution is said to be stable ifsolutions starting close to equilibrium solution (in a δ neighborhood)remain in its ε neighborhood for all future times.

Introduction 8 / 62



Stability - Pictorial Representation

Introduction 9 / 62



Definition (Asymptotical Stability)An equilibrium solution N∗ is said to be asymptotically stable if

it is stableif there exists a ρ > 0 such that for all N0 such that|N0 − N∗| < ρ⇒ lim

t→∞|N(t)− N∗| = 0.

Alternatively, an equilibrium solution is said to be asymptotically stableif it is stable and in addition all solutions initiating in a ρ neighborhoodof the equilibrium solution approach the equilibrium solution eventually.

Introduction 10 / 62



Definition (Asymptotical Stability)An equilibrium solution N∗ is said to be asymptotically stable if

it is stableif there exists a ρ > 0 such that for all N0 such that|N0 − N∗| < ρ⇒ lim

t→∞|N(t)− N∗| = 0.

Alternatively, an equilibrium solution is said to be asymptotically stableif it is stable and in addition all solutions initiating in a ρ neighborhoodof the equilibrium solution approach the equilibrium solution eventually.




Asymptotical Stability - Pictorial Representation




Definition (Unstable Solution)A solution of the system

dNdt

= f (N).

is said to be unstable if it is not stable.




Theorem (A Useful Theorem in One-dimensional Space)Suppose that N∗ is an equilibrium point of the differential equationN ′ = f (N), where f (N) is assumed to be a continuously differentiablefunction with f ′(N∗) 6= 0. Then the equilibrium point N∗ is asymptoticallystable if f ′(N∗) < 0, and unstable if f ′(N∗) > 0.




Theorem (A Useful Theorem in One-dimensional Space)Suppose that N∗ is an equilibrium point of the differential equationN ′ = f (N), where f (N) is assumed to be a continuously differentiablefunction with f ′(N∗) 6= 0. Then the equilibrium point N∗ is asymptoticallystable if f ′(N∗) < 0, and unstable if f ′(N∗) > 0.

Example




Single Species Dynamics

B.S.R.V. Prasad (VIT University) Single Species Dynamics 14 / 62


Populations

Populations...Population will be a primitive concept for us.

It concerns groups of living organisms (plants, animals,micro-organisms..) which are composed of individuals with a similardynamical behavior.We postulate that every living organism has arisen from anotherone and populations reproduce.Note: we will study populations and not the individuals.

Populations change in size, they grow or decrease due to birth,death, migration.



Populations






Populations






Populations






Populations






The Basic Framework

The Basic FrameworkWe want to study laws that govern population changes in space andtime.We begin by restricting our study to how populations change in time.We call these changes dynamical. Our basic framework isFirst: a population is described by its number of individuals ( in somecases, however, by the biomass).

We will here study unstructured populations, but in reality structuredpopulations arises and are important. Here “structure” means classesof age, size, sex,...

Second: we need to describe the time variation of the population. Wewill use (ordinary) derivatives for this purpose. Alternatively, we couldalso work with stochastic processes or discrete-time formulations...Three: we need to know what causes these time variations. Whichbiological processes. Then we have to translate this into (convenient)mathematical language how these biological processes affect thetime-changes of the population.



The Basic Framework






The Basic Framework






The Basic Framework






The Basic Framework






The Basic Framework






Malthusian Model

Dynamics of a single species

N(t) represents total number or density of a population in anenvironment.dNdt stands for rate of change in the entire population.1N

dNdt represents per capita rate of change in the entire population.

(Change in the total population due to an individual.)To start with, we assume that the population changes due to birthsand deaths only.If b,d represent per capita birth and death rates then their differencerepresents per capita rate of change, i.e.,

1N

dNdt

= b − d .



Malthusian Model





1N

dNdt

= b − d .



Malthusian Model





1N

dNdt

= b − d .



Malthusian Model





1N

dNdt

= b − d .



Malthusian Model





1N

dNdt

= b − d .



Malthusian Model

Dynamics of a single species: 1N

dNdt = b − d .

Thus the governing dynamic equation for the population is

dNdt

= rN

where r = b − d called as intrinsic growth rate. This model iscalled exponential model or Malthusian model.If the initial population at time t0,N(t0) = N0, the solution of thisdifferential equation is given by

N(t) = N0ert .



Malthusian Model

Dynamics of a single species: 1N

dNdt = b − d .

Thus the governing dynamic equation for the population is

dNdt

= rN

where r = b − d called as intrinsic growth rate. This model iscalled exponential model or Malthusian model.If the initial population at time t0,N(t0) = N0, the solution of thisdifferential equation is given by

N(t) = N0ert .



Malthusian Model

Thomas Robert Malthus



Malthusian Model

Malthusian’s Model, dNdt = rN,N(t0) = N0

Per capita growth rate, 1N

dNdt , is always constant (b − d).

The growth rate of the population is always increasing(decreasing) if r > 0(r < 0).The population grows (decays) exponentially from the initial valueN0 since N(t) = N0ert . The population will remain constant onlywhen births and deaths balance each other, i.e., b = d or r = 0.



Malthusian Model







Malthusian Model







Malthusian Model




Malthusian Model

Malthusian’s model, dNdt = rN,N(t0) = N0

From the model we observe that the population either blows up toinfinity or decays to zero exponentially which is not realistic.This calls for a modification in the model. The present modelassumes that the per capita growth rate is independent of thepopulation.It is more realistic to assume that the per capita growth rate to bea function of total population in view of the fact that the populationalways has to share the limited food resources which naturallylimits their growth.Before going into this, some examples:



Malthusian Model





Malthusian Model





Malthusian Model





Malthusian Model

Exponential Growth

Figure: The population of India. Until 1951-1961, the growth is wellapproximated by an exponential curve.



Malthusian Model

Exponential Growth

Examples

Figura : The population of USA . Until 1920, the growth is well approximated by anexponential.

Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 32

Figure: The population of USA. Until 1920, the growth is well approximatedby an exponential curve.



Logistic Model

Modified ModelHence we assume that the per capita growth rate of the populationis linearly decreasing function of the total population,given by

1N

dNdt

= r(

1− NK

)where K is called carrying capacity which represents the totalpopulation the environment can support.Observe that the per capita growth rate continuously reduces fromr as the population N increases from zero and it becomes zerowhen the population reaches K . This seems reasonable asresources are always limited and the population are controlled bythese resources.



Logistic Model

Modified ModelHence we assume that the per capita growth rate of the populationis linearly decreasing function of the total population,given by

1N

dNdt

= r(

1− NK

)where K is called carrying capacity which represents the totalpopulation the environment can support.Observe that the per capita growth rate continuously reduces fromr as the population N increases from zero and it becomes zerowhen the population reaches K . This seems reasonable asresources are always limited and the population are controlled bythese resources.



Logistic Model

Logistic ModelThe modified model representing growth in a species is given by

dNdt

= rN(

1− NK

),N(t0) = N0

which is called as Logistic Model or Verhulst Model.



Logistic Model

Logistic ModelLogistic equation

Figura : Pierre-François Verhust, first introduced the logistic em 1838: “’Notice sur la loi quela population pursuit dans son accroissement”. On the right side, , Raymond Pearl, who"rediscovered"Verhust’s work.

Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 10 / 32

Figure: P-F. Verhulst, first introduced the logistic equation in 1838. On theright side, Raymond Pearl, who “rediscovered” Verhulst’s work.



Logistic Model

Parabolic Population Growth: f (N) = rN(

1− NK

)Let us analyze the Logistic model in the light of the theorem donepreviously.We have

f (N) = rN(

1− NK

)



Logistic Model


1− NK


f (N) = rN(

1− NK

)



Logistic Model


1− NK


f (N) = rN(

1− NK

)



Logistic Model

Analysis of Logistic Equation dNdt = rN

(1− N

K

),N(t0) = N0

f (N) = 0⇒ N = 0 or K .

Logistic equation admits two equilibrium points given byN1 = 0,N2 = K

f ′(N) = r(

1− 2NK

), f ′(N1) = r > 0, f ′(N2) = −r < 0

The trivial equilibrium, N1 = 0 is unstable and N2 = K isasymptotically stable.



Logistic Model


(1− N

K

),N(t0) = N0

f (N) = 0⇒ N = 0 or K .


f ′(N) = r(

1− 2NK

), f ′(N1) = r > 0, f ′(N2) = −r < 0




Logistic Model


(1− N

K

),N(t0) = N0

f (N) = 0⇒ N = 0 or K .


f ′(N) = r(

1− 2NK

), f ′(N1) = r > 0, f ′(N2) = −r < 0




Logistic Model


(1− N

K

),N(t0) = N0

f (N) = 0⇒ N = 0 or K .


f ′(N) = r(

1− 2NK

), f ′(N1) = r > 0, f ′(N2) = −r < 0




Logistic Model


(1− N

K

),N(t0) = N0

If N0 is close to 0 then N2

K will be much smaller, hence can beignored.Thus the solution behaves as per dN

dt ≈ rN leaving theneighborhood of 0. This illustrates the instability of zeroequilibrium solution.



Logistic Model


(1− N

K

),N(t0) = N0

If N0 is close to 0 then N2

K will be much smaller, hence can beignored.Thus the solution behaves as per dN

dt ≈ rN leaving theneighborhood of 0. This illustrates the instability of zeroequilibrium solution.



Logistic Model


(1− N

K

),N(t0) = N0

Let us study the nature of the equilibrium N = K . Let us definex = N − K and substitute in Logistic equation.

We obtain dxdt = −rx − rx2

K . If N0 is closer to K then x2 will besmaller and can be neglected. Hence dx

dt ≈ −rxThus x(t) = N(t)− K decays to zero exponentially. This illustratesasymptotic stability of K .



Logistic Model


(1− N

K

),N(t0) = N0







Logistic Model


(1− N

K

),N(t0) = N0







Logistic Model

Solution of Logistic equation: dNdt = rN

(1− N

K

),N(0) = N0

dNdt = rN

(1− N

K

).(

1N + 1

K−N

)dN = rdt .

integrating either side and using the initial condition N(0) = N0 weobtain

ln(

NK − N

K − N0

N0

)= rt ⇒ N

K − N=

N0

K − N0ert .

N(t) =KN0ert

K − N0 + N0ert =KN0

(K − N0)e−rt + N0.



Logistic Model


(1− N

K

),N(0) = N0

dNdt = rN

(1− N

K

).(

1N + 1

K−N

)dN = rdt .


ln(

NK − N

K − N0

N0

)= rt ⇒ N

K − N=

N0

K − N0ert .

N(t) =KN0ert


(K − N0)e−rt + N0.



Logistic Model


(1− N

K

),N(0) = N0

dNdt = rN

(1− N

K

).(

1N + 1

K−N

)dN = rdt .


ln(

NK − N

K − N0

N0

)= rt ⇒ N

K − N=

N0

K − N0ert .

N(t) =KN0ert


(K − N0)e−rt + N0.



Logistic Model


(1− N

K

),N(0) = N0

dNdt = rN

(1− N

K

).(

1N + 1

K−N

)dN = rdt .


ln(

NK − N

K − N0

N0

)= rt ⇒ N

K − N=

N0

K − N0ert .

N(t) =KN0ert


(K − N0)e−rt + N0.



Logistic Model

Logistic Growth



Simulation

Logistic Growth

Simulation


Simulation_1_1.wp2


Glory and Misery of Logistic Model

GlorySimple and its solvable.Allows us to introduce the concept of carrying capacity.A good approximation in several cases.

MiseryToo simple.Does not model more complex biological facts.

Why one should like the logistic equation?Minimal model using which we can build morecomplex/sophisticated ones.














































Further modification to the Logistic equationLogistic equation assumes that Per capita growth large when thepopulation is small.




Further modification to the Logistic equation ...If the population is small there may not be any interaction at allamong the population.Hence it is reasonable to assume that an environment requires aminimum number of population to enable growth in them.Per-capita growth rate of the population requires modification.











Allee Effect

Per-capita Growth With Allee Effect



Allee Effect

Equation with Allee Effect

dNdt

= rN(

NK0− 1)(

1− NK

)N1 = 0,N2 = K0,N3 = K are the three equilibrium points.



Allee Effect


dNdt

= rN(

NK0− 1)(

1− NK




Allee Effect


dNdt

= rN(

NK0− 1)(

1− NK




Simulation

SimulationSimulation


Simulation_1_2.wp2


Simulation

Interacting Populations

B.S.R.V. Prasad (VIT University) Interacting Populations 41 / 62


Predator-Prey Models

Interacting populationsTill now we have discussed about the dynamics of a singlespecies.Present discussion is devoted to study the dynamics of twointeracting population.In particular Predator-Prey interactions.












Predator-Prey Model developmentWe assume coexistence of predators and prey in an environment.N(t) - number or density of prey.P(t) - number or density of predator.The food is abundant for the prey and they grow as per theequation dN

dt = rN in the absence of the predator.



















Prey EquationThere are encounters between prey and predator.They result in consumption of prey by the predator.Number of encounters is proportional their densities.In presence of predators the prey dynamic equation gets modifiedto

dNdt

= rN − cNP, r , c > 0





dNdt

= rN − cNP, r , c > 0





dNdt

= rN − cNP, r , c > 0





dNdt

= rN − cNP, r , c > 0




Predator EquationIn the absence of prey the predators die at an exponential rate.In the presence of prey they increase at a rate proportional to thenumber of encounters.Thus the predator dynamic equation is given by

dPdt

= bNP −mP, b,m > 0.

c > b.





dPdt

= bNP −mP, b,m > 0.

c > b.





dPdt

= bNP −mP, b,m > 0.

c > b.





dPdt

= bNP −mP, b,m > 0.

c > b.



Lokta-Volterra Equations

Predator Prey dynamics: dNdt = rN − cNP, dP

dt = bNP −mP.

r is intrinsic growth rate of the prey population.c consumption rate of the predator per prey.b conversion factorm is death rate of the predator.These equations are called Lotka - Volterra predator-preyequations.





dt = bNP −mP.






dt = bNP −mP.






dt = bNP −mP.






dt = bNP −mP.





Lotka-Volterra

Vito Volterra (1860-1940), an Italian mathematician,

proposed the equation now known as the Lotka-

Volterra one to understand a problem proposed by his

future son-in-law, Umberto d’Ancona, who tried to ex-

plain oscillations in the quantity of predator fishes cap-

tured at the certain ports of the Adriatic sea.

Alfred Lotka (1880-1949), was an USA mathemati-

cian and chemist, born in Ukraine, who tried to trans-

pose the principles of physical-chemistry to biology.

He published his results in a book called “Elements

of Physical Biology”. His results are independent from

the work of Volterra.




Lotka - Volterra equations: N ′ = rN − cNP,P ′ = bNP −mp.We wish to study this nonlinear two dimensional coupleddifferential system.Find if there are any equilibrium points for the system andinvestigate their nature.Obtain information about the qualitative behaviour of its solutions(N(t), P(t)).











Equilibrium Solutions

A few definitionsConsider an autonomous system x ′ = F (x , y), y ′ = G(x , y)Equilibrium points of this system are the points satisfyingF (x , y) = 0 = G(x , y).F (x , y) = 0 is called x - isocline andG(x , y) = 0 is called y - isocline.Equilibrium points are nothing but the intersection points of x andy isoclines.
















Isoclines




Equilibrium Points of Lotka-Volterra Equations

Isoclines: (r − cP)N = 0, (bN −m)P = 0Prey isocline: N = 0, r − cP = 0.Predator isocline: P = 0, bN −m = 0⇒ (N1,P1) = (0,0), (N2,P2) = (m

b ,rc )






b ,rc )






b ,rc )






b ,rc )






b ,rc )

Isocline Figure



Trivial Equilibrium Analysis

N ′ = rN − cNP, P ′ = bNP −mP: Analysis near (0,0)

We study the stability/instability nature of these critical points.In the vicinity of (0,0) we can neglect the terms involving NP.Hence we have dN

dt ≈ rN, dPdt ≈ −mP.

If the initial population is (N0,P0), the solution is (N0ert ,P0e−mt).Near (0,0) the prey grows exponentially and the predatorsdecrease exponentially.Thus the solutions go away from (0,0) indicating instability of(0,0).







































Nature of (0,0)



Interior Equilibrium Analysis

N ′ = rN − cNP, P ′ = bNP −mP: Analysis near (mb ,

rc )

Define u = N − N2, v = P − P2.This transforms the original system tou′ = −mc

b v − cuv , v ′ = rbc u + buv .

Let (N0,P0) be in the vicinity of (N2,P2).Since u, v are small we can neglect their product terms and hencewe obtain dv

du ≈ −rb2

mc2uv

mc2vdv + rb2udu = 0rb2u2 + mc2v2 = c2 ⇒ rb2(N − N2)

2 + mc2(P − P2)2 = C2,

represent ellipses around (N2,P2).





rc )




du ≈ −rb2

mc2uv


2 + mc2(P − P2)2 = C2,






rc )




du ≈ −rb2

mc2uv


2 + mc2(P − P2)2 = C2,






rc )




du ≈ −rb2

mc2uv


2 + mc2(P − P2)2 = C2,






rc )




du ≈ −rb2

mc2uv


2 + mc2(P − P2)2 = C2,






rc )




du ≈ −rb2

mc2uv


2 + mc2(P − P2)2 = C2,





Nature of (N2,P2)



Global Dynamics

Global behaviour of N ′ = rN − cNP, P ′ = bNP −mPEliminating t and rearranging the system we obtain(r−cP)dP

P = (bN−m)dNN .

Upon integration, we obtain Pr e−cP = KN−mebN .represents ovals about (N2,P2) in anti clockwise direction.



Global Dynamics


P = (bN−m)dNN .




Global Dynamics


P = (bN−m)dNN .




Global Dynamics


P = (bN−m)dNN .


Thus theconsideredsystem admitsperiodic solutions.



Species Average

Periodic SolutionsLet T be the period of a solutions.Consider the equation dN

dt = rN − cNPSeparating the variables and integrating over a time period T we

obtaint0+T∫t0

dNN =

t0+T∫t0

(r − cP)dt

⇒ ln[

N(t0+T )N(t0)

]= rT − c

t0+T∫t0

Pdt

Thus rT − ct0+T∫t0

Pdt = 0

⇒ 1T

t0+T∫t0

Pdt = rc



Species Average



obtaint0+T∫t0

dNN =

t0+T∫t0

(r − cP)dt

⇒ ln[

N(t0+T )N(t0)

]= rT − c

t0+T∫t0

Pdt


Pdt = 0

⇒ 1T

t0+T∫t0

Pdt = rc



Species Average



obtaint0+T∫t0

dNN =

t0+T∫t0

(r − cP)dt

⇒ ln[

N(t0+T )N(t0)

]= rT − c

t0+T∫t0

Pdt


Pdt = 0

⇒ 1T

t0+T∫t0

Pdt = rc



Species Average



obtaint0+T∫t0

dNN =

t0+T∫t0

(r − cP)dt

⇒ ln[

N(t0+T )N(t0)

]= rT − c

t0+T∫t0

Pdt


Pdt = 0

⇒ 1T

t0+T∫t0

Pdt = rc



Species Average



obtaint0+T∫t0

dNN =

t0+T∫t0

(r − cP)dt

⇒ ln[

N(t0+T )N(t0)

]= rT − c

t0+T∫t0

Pdt


Pdt = 0

⇒ 1T

t0+T∫t0

Pdt = rc



Species Average



obtaint0+T∫t0

dNN =

t0+T∫t0

(r − cP)dt

⇒ ln[

N(t0+T )N(t0)

]= rT − c

t0+T∫t0

Pdt


Pdt = 0

⇒ 1T

t0+T∫t0

Pdt = rc



Species Average

1T

t0+T∫t0

Pdt = rc

Note that the LHS is nothing but the average of the predatordensity over a cycle i.e., Paverage = r

c .

Following similar lines we can show that Naverage = mb .



Species Average

1T

t0+T∫t0

Pdt = rc

Note that the LHS is nothing but the average of the predatordensity over a cycle i.e., Paverage = r

c .

Following similar lines we can show that Naverage = mb .



Simulation

SimulationSimulation


Simulation_3_1.wp2


Glory and Misery of Lotka-Volterra Model

GloryAlthough it is an oversimplified model for predator-prey system, itcaptures an important feature that this kind of systems exhibitsoscillations − which are inherent to dynamics.

MiseryAs Lotka-Volterra model exhibits orbital stability, once you are on acertain orbit in the phase space, it has certain amplitude and period.If we perturb this orbit, the system will stay on a new orbit, withdifferent amplitude and period.But the real systems are under perturbations all the time, they wouldjump between trajectories but may not always be periodic. Moreover,the periodicity if exits is not effectively periodic.In fact, real predator-prey oscillations better described by limit cyclebehaviour.
























Some ReferencesElements of Mathematical Ecology, M. Kot ( Cambridge Univ.Press, 2001).Mathematical Biology I, J.D. Murray ( Springer, Berlin, 2002).Complex Population Dynamics, P. Turchin (Princeton Univ. Press,2003).Theoretical Ecology - Principles and Applications, R. May and A.R. McLean (Oxford, 2007).A Primer of Ecology, N.J. Gotelli ( Sinauer, 2001).Ecology of Populations, E. Ranta, P. Lundberg, V. Kaitala(Cambridge Univ. Press, 2006).




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Documents

Population Dynamics - IITKhome.iitk.ac.in/~peeyush/mth426/l6.pdfFundamentals of Mathematical Ecology/Biology Provides an introduction to Classical and Modern Mathematical Models, Methods