Student

Vt 2011

Examensarbete, 15 hp

Statistik C, moment 3, 15 hp

A comparative study of Stochastic and Deterministic Population Models

Kristina Lundquist

Sammanfattning I denna uppsats kombineras teori från populationsbiologi med stokastiska differential-ekvationer. Syftet är att jämföra tidigare studerade deterministiska populationsmodeller med korresponderande stokastiska populationsmodeller. Som ett illustrativt exempel appliceras modellerna på ett klassiskt dataset (Gause, 1934). Första halvan av uppsatsen upptas av en kortfattad genomgång av den teoretiska bakgrund som behövs för förståelsen av den efterföljande analysen. Denna genomgång inleds med några matematiska begrepp, följt av populationsbiologi med tillhörande koncept och avslutas med statistisk teori, bl. a stokastiska processer och stokastiska differentialekvationer. Den andra halvan av uppsatsen består av resultat av analysen samt en efterföljande diskussion. I analysen diskuteras och jämförs de deterministiska modellerna med de stokastiska. Eftersom de stokastiska modellerna tar hänsyn till stokastiska fluktuationer i populationerna ger de en mer realistisk bild av verkligheten. Å andra sidan finns det fortfarande frågetecken kring hur de stokastiska modellerna på bästa sätt skall uttryckas. Vidare skulle det vara intressant att applicera de nya stokastiska modellerna på data från populationer som lever ute i naturen till skillnad från i laboratoriemiljö (som var fallet här).

Titel: En jämförande studie av stokastiska och deterministiska populationsmodeller

Abstract In this paper theory from population biology is combined with stochastic differential equations. The aim is to compare previously studied deterministic population models with corresponding stochastic population models. As an illustrative example the models are applied to a classical dataset (Gause, 1934). The first half of the paper gives a brief review of the theoretical background needed in order to understand the subsequent analysis. This review begins with some mathematical concepts, followed by concepts in population biology and ends with statistical theory, e.g. stochastic processes and stochastic differential equations. The second half of the paper deals with the results of the analysis and the subsequent discussion. In the analysis the deterministic models and the stochastic models are discussed and compared. Since the stochastic models capture stochastic fluctuations in populations they give a more accurate picture of reality. On the other hand, question marks still remain concerning the best way of expressing the stochastic models. Furthermore it would be interesting to apply the new stochastic models to data from wild life populations and not on data from laboratory populations (as was the case here).

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Contents Contents 1 I. Introduction 2 1. Differential equations, Initial value problems and Systems of differential equations 2 2. Approximation methods of finding solutions to differential equations 3 3. Population biology 4 4. Models for one species 5 5. Models for two species 6 6. Steady state and stability 7 7. Stochastic processes and the Wiener process 9 8. Stochastic differential equations 10 9. Euler-Maryuama approximation and Milstein approximation 11 10. Fokker-Planck and Stationarity 11 11. Gause’s experiment 12 II. Results 13 1. Deterministic analysis 13 1.1. The logistic growth model 13 1.2 The competition model 15 2. Stochastic analysis 18 2.1 The stochastic logistic growth model 18 2.2 The stochastic competition model 21 2.3 Alternative models 22 3. Comparison of deterministic and stochastic analysis of Gause’s data 24 III. Summary and Concluding remarks 25 1. Summary 25 2. Concluding remarks 25 IV. Acknowledgements 26 V. References 27 Appendix 1 28 Appendix 2 29 Appendix 3 31 Appendix 4 32 Appendix 5 34

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I. Introduction In population biology deterministic models have been rigorously studied; in contrast stochastic models have not been studied that much. Although stochastic analyses have been performed, few of these involved stochastic differential equations. Since population models often consist of differential equations, it is surprising that the use of stochastic differential equations have been so limited. However, the development of biological models with stochastic differential equations is increasing, see for example Arató (2003) and Khasminskii and Klebaner (2001). The aim of this paper is to combine the theory of population biology with that of stochastic differential equations. More specifically I will analyse two deterministic population models, “convert” them into corresponding stochastic population models and analyse these stochastic models. To illustrate, the dataset from a classical experiment (Gause, 1934) is analysed. The deterministic and stochastic models will be compared and evaluated by how well they predict the data from Gause’s experiment. This paper consists of three major parts; Introduction (I), Results (II) and Summary and Discussion (III). In the introduction I give some theoretical background necessary for understanding the subsequent analysis. First some mathematical concepts (section 1 and 2), then population dynamics (section 3-6), followed by a short description of the statistical theory needed (section 7-10). The introduction closes with a description of Gause’s experiment (section 11). The results (II) are presented in two different sections: first the deterministic analysis (section 1) is presented and then the stochastic analysis (section 2), followed by a comparison of them (section 3). In the last major part I summarize and discuss the findings. 1. Differential equations, Initial value problems and Systems of differential equations A differential equation connects an unknown function, x, with its derivatives. The differential equation is called an ordinary differential equation (ODE) if x is a function of only one variable, e.g. )t(xx = , and is called a partial differential equation (PDE) if x is a function of more than one variable, e.g. )v,u(xx = . The differential equation is called a differential equation of order n if it contains derivatives up to order n. To uncover the unknown function the differential equation must be solved. Unfortunately not all differential equations have explicit solutions. When this is the case, we must content ourselves with finding an approximation to the solution (see the next section). Before starting the procedure of finding an approximate solution, it is imperative to investigate if a solution even exists and if it is unique. A simple example of a differential equation is the model of radioactive decay (Nagle et al. 2004). Here one assumes that the rate of decay is proportional to the amount of radioactive substance present. This yields the differential equation

)t(kxdt

dx −= ,

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where x(t) is the amount of radioactive substance present at time t and k is the proportionality constant. This equation can be solved by integration, which yields the solution

ktCe)t(x −= , where the constant C should be the initial amount of the radioactive substance, )(xC 0= . Thus the radioactive substance decays exponentially. An example of a simple PDE is the Laplace equation

,02

2

2

2

=∂∂+

∂∂

v

x

u

x

where ∂ denotes the partial derivative. An initial value problem (IVP) contains a differential equation and the initial value of the function. Let’s return to our example of radioactive decay. The IVP is stated as

=

−=

,)0(

),(

0xx

tkxdt

dx

and the solution is ktex)t(x −= 0 .

An example of a simple system of differential equations is

+=

+=

)t(xa)t(xadt

dx

)t(xa)t(xadt

dx

2221212

2121111

, (aij constants)

which can be written in matrix form as

xAdt

xd = ,

where

=

2

1

x

xx and

=

2221

1211

aa

aaA .

2. Approximation methods of finding solutions to differential equations Consider finding the solution to the IVP for a first order ODE

≤≤=

=

.0

,)0(

),,(

0

Tt

xx

xtfdt

dx

Further assume that we have investigated and found out that there indeed exists a unique solution; denote this solution by xu(t). Now we want to approximate xu(t) over the interval

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[0,T]. The simplest approximation method is the Euler method (Nagle et al. 2004). First the interval [0,T] is discretized into

Tt...t...tt Tn =<<<<<= 100

and the step size, hn, is defined as

nnn tth −= +1 .

The Euler method is based on the following recursive formula

)x,t(f*hxx nnnnn +=+1 , where nx is an approximation to ).( ntx

The accuracy of the method can be increased by decreasing the step size, hn. As hn tends to zero the approximate solution approaches the unique solution, xu(t). For fixed t and step size h denote the approximate solution by xa(t,h), then

)t(x)h,t(xlim uah

=→0

and xa(t,h) is said to converge to xu(t) at t. The limit can be rewritten as

00

=−→

)t(x)h,t(xlim uah

,

where )t(x)h,t(x ua − is the error of the approximation.

The convergence property is an important trait of the Euler method. Another important property is the rate of convergence, i.e. how fast the approximate solution converges to the unique solution. This rate is often reported in terms of a power of h. If the error tends to zero proportional to hp, then the approximation method is said to be of order p. The larger the p the faster the approximation method converges (1<h ) (Nagle et al. 2004). The Euler method is of order 1, which is a fairly slow convergence. Hence other methods have been proposed. One such method is the Taylor method of order p (Nagle et al. 2004). It is derived from the Taylor polynomial about the point tn, yielding the recursive formula

)x,t(f!p

h...)x,t(f

!

h)x,t(f*hxx nnp

p

nnnnnnn ++++=+ 2

2

1 2

where fi is the i:th derivative of x (e.g. )t(''xf =2 ). It is now apparent that the Euler method is a Taylor method of order 1. 3. Population biology In population biology the dynamics and changes of populations are studied. The dynamics include the growth and decline of populations and the interaction of species. To model the dynamics, different mathematical population models have been developed, many in the form of differential equations. The study of mathematical population models was initiated in the 19th and early 20th centuries by the pioneering work of Malthus (1798), Verhulst (1838) and Lotka and Volterra (1920s and 1930s), to name a few. The models they formulated are still

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being taught today; they may, however, be considered too simple and lacking in certain biological features, such as environmental effects, change random events and spatial heterogeneity (Edelstein-Keshet, 1988). However, according to Edelstein-Keshet (1988) “the importance of these models stems not from realism or the accuracy of their predictions but rather from the simple and fundamental principles that they set forth; the propensity of predator-prey systems to oscillate, the tendency of competing species to exclude one another, the threshold dependency of epidemics on population size are examples”. Below (sections 4 and 5) I list only a few of these models, classified according to the number of interacting species. Models for more than two interacting species also exist, but are not discussed here. 4. Models for one species The general form to model the growth of one population is

)N(fdt

dN = .

Where N is equal to the number of individuals at time t, i.e. the population size. Many different forms of f(N) exist. Two simple examples are described below. The most simple form is due to Malthus (1798). He assumed that individuals give birth with rate b and die with rate d, thus

=

−=

.)0(

,)(

0NN

Ndbdt

dN

By solving you get t)db(eN)t(N −= 0 . Thus, depending on the value of (b-d), the population

size will increase or decrease exponentially or stay constant, which can be seen in figure 1.

Figure 1. The solution to the Malthus model, when 0>− db (red curve), when 0<− db (blue curve) and when

0=− db (black line) The model of an exponential growth is, however, not that realistic. For example, the resources run out at some point. One of the first to model this limitation was Verhulst (1836) in introducing the logistic growth model

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=

−=

.)0(

,1

0NNK

NrN

dt

dN (1)

Here dbr −= is a measure of the growth rate, and the carrying capacity, K, limits the growth of the population. The constant r is (often) assumed to be positive. Upon solving (1) you obtain

( ).1)(

0

0

−+= rt

rt

eNK

KeNtN (2)

This solution is sketched in figure 2, showing the behaviour of the solution with different values of the constants r and K. In the figure the red and blue curves correspond to a small value of r and the green and magenta curves to a bigger value of r, the red and magenta curves correspond to a small value of K and the blue and green curves to a bigger value of K.

Figure 2. The solution to logistic growth model (2). Red curve: small r and K. Blue curve: small r and big K. Magenta curve: big r and small K. Green curve: big r and K. 5. Models for two species When describing how two populations affect each other, a system of two coupled differential equations is obtained. The general form is

=

=

).,(

),,(

212

211

NNgdt

dN

NNfdt

dN

As in the case of one population there exist many forms of the two functions f(N1,N2) and g(N1,N2). The most famous model is the Lotka-Volterra predator-prey model (1921). Here

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−=

−=

,

,

2212

2111

NNNdt

dN

NNNdt

dN

δγ

βα

where N1 is the number of prey and N2 the number of predators. The constant α describes the growth rate of the prey without any predation, and δ describes the rate of loss of predators without prey (starvation). The constant β describes the rate of loss due to predation and γ describes the growth rate due to predation (Britton 2005). Another model proposed by Lotka and Volterra, and later studied by Gause (1934), is the Lotka-Volterra competition model. Here

−−=

−−=

.1

,1

2

121

2

222

2

1

212

1

111

1

K

Nb

K

NNr

dt

dN

K

Nb

K

NNr

dt

dN

(3)

Note that the first two terms in the equations are just the logistic growth (1). The constants b12

and b21 are the competition constants and describe the amount of competition between the species, or in other words how much one species influences the other (Britton 2005). 6. Steady state and stability Many of the models of population biology cannot be solved explicitly (exceptions are the two models mentioned in section 3). To understand the behaviour and to learn more from the differential models some concepts have been formulated (Edelstein-Keshet 1988, Britton 2005). One such concept is steady state, which stems from the idea of absence of change in the system. Definition 1 [One population]: The steady states of

)N(fdt

dN =

are the constants N* such that 0=*)N(f . Thus upon reaching a steady state, N*, the population size will not change. The definition for a steady state in two populations is analogous. Definition 2 [Two populations]: The steady state of the system

=

=

)N,N(gdt

dN

)N,N(fdt

dN

212

211

is the vector (N1*,N2*) such that 021 =*)N*,N(f and 021 =*)N*,N(g .

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Definition 3 [Two populations]: The nullclines of f, respectively g, are the curves defined by 021 =*)N*,N(f , respectively 021 =*)N*,N(g . The steady states are the points where the nullclines cross. Another concept, connected to steady state, is stability. A steady state can either be stable or unstable. It is denoted as stable if neighbouring states are attracted to it and unstable if neighbouring states are not attracted to it. In other words, the steady state is stable if the system, upon slightly moving away from the steady state, returns to it and unstable if the system doesn’t return. As an illustration, consider a ball resting on top of a hill or in a pit. If the ball is slightly disturbed it will in the case of the hill roll down and not return, but in the case of the pit the ball will roll up a bit and then return to the pit. The hill is an example of an unstable steady state and the pit an example of a stable steady state. Definition 4 [One population]: A steady state, N*, of

)N(fdt

dN = , is

(i) stable if 0<== *NNdt

dN*)N('f

(ii) unstable if 0>== *NNdt

dN*)N('f

(iii) neutrally stable if 0=== *NNdt

dN*)N('f

The definition of stability in two populations relies on the eigenvalues of the Jacobian matrix. The Jacobian matrix, J, of a system of functions is a matrix of the partial derivative of those functions. For example, let the functions )y,x(ff = and )y,x(gg = depend on the two variables, x and y; then the Jacobian is

.

∂∂

∂∂

∂∂

∂∂

=

y

g

x

gy

f

x

f

J

Definition 5 [Two populations]: Let λ1 and λ2 be the eigenvalues of the Jacobian matrix to the system

=

=

),,(

),,(

212

211

NNgdt

dN

NNfdt

dN

evaluated at the steady state (N1*, N2*). Then the steady state is (i) an unstable node if λ1, λ2 > 0 (ii) a stable node if λ1, λ2 < 0 (iii) a saddle point if λ2 < 0 < λ1 (iv) an unstable/stable spiral or a centre if λ1 and λ2 are complex

The different kinds of stability are sketched in figure 3.

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Figure 3. The different kinds of stability described in definition 5. (a) unstable node (b) stable node (c) saddle point (d) unstable spiral (e) stable spiral (f) centre 7. Stochastic processes and the Wiener process A stochastic process is a family of random variables that depend on time. The time can either be discrete, ,...,,t 210= , or continuous, [ )∞∈ ,t 0 . Thus the process can either be a “discrete-time” process or a “continuous-time” process (Grimmett 2006); an example of the former is random walk and of the latter is Wiener process or Brownian motion. Brownian motion is named after the botanist R. Brown (1773-1858), who in 1827 studied the motion of tiny particles suspended in water. He observed that the particles moved in an erratic random fashion (Grimmett 2006). Random walk can also be thought of as the motion of a particle. Here the particle moves or jumps “one step at the time”. While, in the Wiener process the particle moves continuously, with no jumps. Definition 6: The Wiener process, W(t), 0≥t , satisfies the following properties. (i) [Independence of increments] )s(W)t(W − , ts< , is independent of the past. (ii) [Normal increments] )s(W)t(W − is normally distributed with mean 0=µ

and variance .2 st −=σ (iii) [Continuity of paths] W(t) is a continuous function of t. (iv) 00 =)(W .

An implication of (ii) and (iv) is that ( )t,N~)t(W 0 , since )t(W)(W)t(W =− 0 . Figure 4 shows simulation of five sample paths of the Wiener process.

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Figure 4. Five sample paths of the Wiener process.

8. Stochastic differential equations By introducing noise into an ordinary differential equation (ODE) a stochastic differential equation (SDE) is obtained. Let X(t) be an unknown stochastic process, W(t) a Wiener process, µ(x,t) and σ(x,t) known functions, then

( ) ( ) )t(dWt),t(Xdtt),t(X)t(dX σµ += is an SDE driven by a Wiener process. The function µ(x,t) is called the drift coefficient and σ(x,t) is called the diffusion coefficient. An IVP for SDE is

( ) ( )

=+=

.)0(

),(),(),()(

0XX

tdWttXdtttXtdX σµ (4)

In many applications SDEs are the result of incorporating random fluctuations, noise, in deterministic models. The noise is called additive if the diffusion coefficient, σ, doesn’t depend on the stochastic process, X(t), and is called multiplicative if it does depend on X(t). SDEs in applications can arise by incorporating additive or multiplicative noise in an ODE (Kloeden and Platen 1999). SDEs can also arise when the coefficients of an ODE are perturbed by white noise (this is done for the logistic growth model (1) in Appendix 5). The white noise process, ξ(t), is defined as the (generalised) derivative of the Wiener process (Klebaner 2007).

dt

)t(dW)t( =ξ

As in the case of deterministic differential equations, SDEs may not be explicitly solvable. When this is the case, numerical methods can be used to obtain an approximation of the solution (see section 9) – if a solution exists. It is thus imperative to first ascertain if a solution exists. When the three conditions stated in Theorem 1, Appendix 2, are satisfied then there exists a unique strong solution.

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9. Euler-Maryuama approximation and Milstein approximation Consider finding the solution to (4), Tt ≤≤0 . Assume that a unique solution exists, i.e. the conditions in Theorem 1 are satisfied. The simplest method of obtaining an approximate solution is the Euler-Maryuama method (Kloeden and Platen 1999). As in the deterministic case (see section 2) the interval [0,T] is first discretized into

Tt...t...tt Tn =<<<<<= 100 ,

and the step size, hn, is defined as .1 nnn tth −= +

The Euler-Maryuama method is then based on the following recursive formula

nnnnnnnn W*)t,X(h*)t,X(XX ∆σµ ++=+1 ,

where nX is an approximation to )( ntX , and ).()( 1 nnn tWtWW −=∆ +

It can be shown that the Euler-Maryuama method is of convergence order ½ (Kloeden and Platen 1999). To improve the order of convergence, G. Milstein proposed an approximation method based on the following recursive formula

( )( ),*),('*),(*21

*),(*),(

2

1

nnnnnn

nnnnnnnn

hWtXtX

WtXhtXXX

−∆+

∆++=+

σσ

σµ

where ( )

.,

),('x

txtx

∂∂= σσ

The Milstein approximation method is of convergence order 1 (Kloeden and Platen 1999). In section 2 it was shown that the (deterministic) Euler method is derived from Taylor polynomials. Correspondingly, the Euler-Maryuama and Milstein methods are derived from stochastic Taylor polynomials (discussed in Kloeden and Platen 1999). 10. Fokker-Planck and Stationarity As a stochastic process X evolves with the passage of time, so does its probability density function (PDF). The Fokker-Planck equation tells us how the density evolves (Roberts 2009). Let ( )t),t(Xpp = be the PDF of the random variable X(t); then the Fokker-Planck equation is

( )

∂∂+−

∂∂=

∂∂

2

2

2

2 p

xp

xt

p σµ . (5)

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This equation can be generalised to higher dimensions. Let ( ))t(X),...,t(X),t(X)t(X d21=

be a vector of one-dimensional stochastic processes and let ( )t),t(Xpp = be the joint PDF of X1(t),..., Xd(t), then the Fokker-Planck equation is

( ) ∑∑∑= ==

∂∂∂+−

∂∂=

∂∂ d

i

d

j

ij

ji

d

ii

i

p

xxp

xt

p

0 1

22

1 2

σµ . (6)

As time progresses the PDF may become stationary (Roberts 2009).

Definition 7. The PDF ( )t),t(Xpp = is stationary when 0=∂∂

t

p.

11. Gause’s experiment In 1934 the Russian biologist Georgy F. Gause (1910-1986) published the groundbreaking book “Struggle for existence”. In it he discusses the many facets of the struggle for existence, also including the mathematicians’ point of view. Gause performed several studies of competing species, some which included the unicellular organisms Paramecium aurelia and Paramecium caudatum. In these experiments the two species either lived in isolation or were allowed to compete for resources (Leslie 1957, Pascual and Karieva 1996). The data used here comes from an experiment performed during a 25 day period. Daily counts of individuals per 0.5 cm3 were performed for both of the species in isolation and in competition (de Vries et al. 2006). The data is attached in Appendix 1 and plotted in figure 5.

Figure 5. Data of Gause’s experiment; where P.aurelia and P.caudatum were grown in isolation (blue and green crosses respectively) and in competition (red and cyan crosses respectively). Over the whole period of the experiment, the food supply was held constant. The species reproduce continuously by simple division. Hence the growth mechanism of the species is simple and complications, such as time lags or age structure, can be ignored (Pascual and Karieva 1996). Below I will refer to P.aurelia as species 1 and P.caudatum as species 2.

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II. Results The aim of this study was to compare a deterministic analysis with a corresponding stochastic analysis on the data obtained from Gause’s experiment (section 11). The comparison is evaluated by a “comparison criteria”, defined as how well the model predict the data from Gause’s experiment. For reasons of comparison, even though this is a paper in Statistics, I will briefly discuss the deterministic analysis. To model the growth of the two species in isolation the logistic growth model (1) is applied and to model the growth of the two species in competition the competition model (3) is applied. For the stochastic analysis the differential equations (1) and (3) are converted into SDEs. The outline is as follows: First the deterministic analysis is performed (section 1) – divided into the logistic growth model (1.1) and the competition model (1.2). This is followed by the stochastic analysis (2), which also is divided into the stochastic logistic growth model (2.1) and the stochastic competition model (2.2). An alternative way of converting (1) and (3) into SDEs is presented in section 2.3. In section 3 I compare the deterministic models with the corresponding stochastic models, based on the “comparison criteria”. 1. Deterministic analysis Below the values of the parameters, r1, r2, K1, K2, b12 and b21, are taken from de Vries et al. (2006). The parameters are estimated with a kind of iterative least square method. In this method the first step is to obtain a good initial guess of the parameter value by trial and error, and then update the guess by minimising the “residual” sum of squares. First, the parameters of the logistic growth model are estimated from the data of the species in isolation. Then 12b

and 21b are estimated from the data of the species in competition. The estimates are

7901 .r = , 6602 .r = , 15431 .K = , 62022 .K = , 17.212 =b and 36.021 =b . Below, if not otherwise specified, the time variable is set to lie in the interval [0,25] since this is the time span of Gause’s experiment. 1.1 The logistic growth model The logistic growth model (1) is applied to the data of Gause’s experiment. In figure 6 the left hand side of equation (1) is plotted against the right hand side (red curve for species 1 and blue curve for species 2), thus illustrating the behaviour of dN/dt, as a function of the number of individuals, N.

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Figure 6. Phase plot of the logistic growth model (1), for species 1 (red) and species 2 (blue).

As can be seen in the figure, the change in growth first increases then decreases and finally is zero at the carrying capacity, K. The solution of the logistic growth model (1) is sketched in figure 7 (red line for species 1 and blue line for species 2), which shows how the number of individuals in the populations changes with time. From the figure it is clear that at first the populations grows exponentially and thereafter settle down at a population of K individuals.

Figure 7. The solution to the logistic growth model (2), for species 1 (red) and species 2 (blue).

The steady states of the logistic growth model (1) are 0=*N and K*N = , as can be derived from Definition 1, or seen in figure 6. Thus, the population is in a steady state when no individuals are present, or when the number of individuals have reached the carrying capacity. In order to derive the stability of the steady states, the derivative

K

N*rr)N('f

2−=

is calculated and evaluated at the steady states. According to Definition 4, the steady state 0=*N is unstable, since 00 >= r)('f , and the steady state K*N = is stable, since

0<−= r)K('f . Thus, upon reaching a population size of K individuals the population will not change (which is visible in figure 7).

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1.2 The competition model The competition model (3) is applied to the data of Gause’s experiment. This model cannot be solved explicitly. Hence it is approximated with the Euler method. The approximate solution is shown in figure 8.

Figure 8. Euler approximation of equation (3), for species 1 (red) and species 2 (blue).

An analysis of the steady states and the stability of them are done with the aid of nullclines. The steady states are the points where the nullclines cross. The nullclines for species 1 are the trivial 01 =N and the non-trivial 21211 NbKN −= and the corresponding nullclines for species

2 are 02 =N and 12122 NbKN −= . The nullclines are sketched in figure 9. The solid lines represent the nullclines of species 1 and the dotted lines the nullclines of species 2.

Figure 9. Nullclines for species 1 (solid lines) and species 2 (dotted lines). Four different scenarios exist

depending on the values of K1, K2, 12

1

b

K and21

2

b

K .

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Four different scenarios exist depending on the values of K1, K2, 12

1

b

K and

21

2

b

K. In scenario 1

(left top) and 2 (right top) the steady states are

( ) ( )00211 ,N,Ns == , ( ) ( )01212 ,KN,Ns == . An additional steady state exists in scenario 3 (left bottom) and 4 (right bottom).

( ) ( ),,0, 2213 KNNs == ( )

−−

−−==

11 2112

2121

2112

1212214 bb

KKb,

bb

KKbN,Ns .

In the first steady state, s1, both populations have gone extinct. In the second and third steady states, s2 and s3, one of the populations have gone extinct and the other has reached its carrying capacity. In the fourth steady state, s4, the populations coexist. The stability of the four steady states has been extensively discussed elsewhere (see e.g. Britton 2005, de Vries et al 2006). I will therefore not go into any details, rather I will only interpret the result. The stability of the steady states depends on the value of the parameters, K1, K2, b12 and b21, and subsequently on the different scenarios. This dependence is sketched in table 1. Figure 10 shows the stability properties of the steady states in the different scenarios.

Table 1. The stability of the four different steady states in the four different scenarios. scenario steady state

1 2 3 4

s1 unstable node unstable node unstable node unstable node s2 stable node saddle point stable node saddle point s3 saddle point stable node saddle point stable node s4 - - unstable node stable node

Figure 10. The stability properties in the four different scenarios.

17

From table 1 and figure 10 one can conclude that in scenario 1 species 1 thrives and species 2 goes extinct. The opposite occurs in scenario 2. In scenario 3, both s2 and s3 are stable indicating that (depending on the initial value) one of the species will go extinct while the other will survive (this is called bi-stability). In scenario 4 the fourth steady state is stable, thus the species coexist. To summarize, in most cases only one of the species survives, and only in one case out of four the species coexist. Coexistence occurs when the competition within a species is greater than the competition between species (Britton 2005). As can be seen in the data from Gause’s data (see figure 5) the species seem to coexist. With parameter values estimated from de Vries et al. (2006) this is also what the competition model (3) will predict, as can be seen in figure 11. Figure 11 also shows the approximate solution to (3) in the (N1,N2)-plane.

Figure 11. Non-trivial nullclines for species 1 (red) and species 2 (blue) and the approximate solution to (3) (black). The coexistence steady state is marked with a black star (*). The figure shows that the steady state of coexistence lies just below the carrying capacity of species 1, 1.5431 =K , but much below the carrying capacity for species 2, 6.2022 =K . The figure also shows that the slopes of the nontrivial nullclines for the two species are almost the same. This indicates that if the parameter values are slightly altered, species 1 will thrive but

species 2 will go extinct, i.e. scenario 1 (this happens if21

21 b

KK > ; see figure 9).

To elucidate what happens with the populations over a longer period of time than that of Gause’s experiment, the time variable is set to lie in the interval [0,100] and the approximate solution to (3) (Euler method) is sketched in figure 12. Here a note of caution is required. The approximate solutions (the curves in the figure) depend on the parameters estimated from Gause’s data and hence extrapolating outside the time interval of Gause’s experiment gives unreliable information.

18

Figure 12. Euler approximation for equation (3), species 1 (red) and species 2 (blue), for the time interval [0,100]. 2. Stochastic analysis The ODEs (1) and (3) were converted into SDEs by incorporating the linear multiplicative noise, cX(t); see equations (7) and (9) respectively. 2.1 The stochastic logistic growth model

)t(dW)t(cXdtK

)t(X)t(rX)t(dX +

−= 1 . (7)

First we need to check if a unique strong solution to (7) exists (see Appendix 2); which indeed is the case. The solution can be shown to be (see Kloeden and Platen 1999, p. 125).

∫+

−

+

−

+

=t )s(cWs

cr

)t(cWtc

r

dserXK

KeX)t(X

0

20

20

2

2

. (8)

If there is no noise in the system, i.e. when 0=c , (8) reduces to (2). The integral in the denominator is not readily obtainable. However, the expected value of it can be calculated (see Appendix 3), and is found to be

r

e))t(I(E

rt 1−= , where ∫+

−

=t )s(cWs

cr

dse)t(I0

2

2

.

Since equation (8) is a quotient of two dependent random variables its expected value is not readily obtainable. However, applying the Delta method the expected value can be approximated by the expected value of the numerator divided by the expected value of the denominator (see Appendix 3). This yields

19

( ) ( )10

0

−+≈ rt

rt

eXK

KeX)t(XE ,

which is (2). Thus, the expected value of the solution to the stochastic logistic growth model can be approximated with the solution of the deterministic logistic growth model. The Euler-Maryuama approximation was used to approximate the solution of (7). The values of the constants of the drift coefficient, r and K, were set in accordance with the parameter estimation produced in de Vries et al. (2006) (see section 1.3). Different values of the constant in the diffusion coefficient, c, were applied – see figure 13.

Figure 13. Two different simulations of the Euler-Maryuama approximation to (7) for species 1. The red curve is the deterministic solution, the blue and black curves are the approximations, with 080.c = (blue) and 80.c = (black). In figure 13 the red curve is the solution to the deterministic logistic growth equation, i.e. equation (2). The figure shows that if the constant c is relatively small, the solution to (7) will stay close to the red curve. In this case the noise is mostly apparent when the population has reached the carrying capacity. If c is larger the noise will “take over” and the population will not grow exponentially to the carrying capacity. A Milstein approximation was also carried out and this (improved) approximation method didn’t yield any other result then the Euler-Maryuama method.

In Appendix 4 the stationary PDF (Definition 7) of the solution to (7) is derived, which turns

out to be a gamma function with shape parameter 2

22c

cr −=α and scale parameterr

Kc

2

2

=β .

Thus, as time progresses the expected value and variance of the solution to (7) are ( )

r

crK))t(X(E

22 2−== αβ

and ( )

2

2222

42r

crcK))t(X(V

−== αβ .

For Gause’s data, and parameter estimated from de Vries et al. (2006), table 2 lists the estimate of the expected value (denoted by E) and the standard deviation (denoted by SD) of the stationary distribution, with two different values of c.

20

Table 2. The estimated expected values and standard deviations with two different values of the noise constant c. 080.c = 8.0=c E SD E SD species 1 540.9 34.5 323.2 266.6 species 2 201.6 14.1 104.4 101.3 As can be seen in the table when the noise constant, c, is relatively small (i.e. 0.08) the estimates of the expected values are approximately the same as the estimates of the stable steady state of the deterministic equation (1), i.e. the carrying capacity. However if c is increased tenfold, then the estimated expected value is much lower and the standard deviation much higher (in particular they are almost the same). Figure 14 shows how the estimated expected values and variances change with changing c. The expected value is equal to the carrying capacity when 0=c , and when 0>c then the

expected value decreases. The variance is zero when 0=c and at it’s highest when 1≈c . In

figure 15 the stationary PDF is plotted with 080.c = and 80.c = (red and blue curves respectively). From figure 15 it is visible that when c is small the PDF for the gamma function resembles the PDF of a normal distribution.

Figure 14. The expected value and the variance as a function of the noise constant, c, for species 1 (red) and species 2 (blue).

1400120010008006004002000

0,012

0,010

0,008

0,006

0,004

0,002

0,000

X

De

nsit

y

246 2,2

1,47 220

Shape Scale

Distribution PlotGamma; Thresh=0

5004003002001000

0,030

0,025

0,020

0,015

0,010

0,005

0,000

X

De

nsit

y

205 0,98

1,06 98

Shape Scale

Distribution PlotGamma; Thresh=0

Figure 15. PDFs for the gamma distribution with two different values of the noise constant, c, 080.c = (black) and 80.c = (red). The PDFs to the left are for species 1 and those to the right are for species 2.

21

2.2 The stochastic competition model

+

−−=

+

−−=

).()(1)(

),()(1)(

222

121

2

2222

111

212

1

1111

tdWtXcdtK

Xb

K

XXrtdX

tdWtXcdtK

Xb

K

XXrtdX

y

x

(9)

Note that the first equation is an SDE driven by the Wiener process )(tWx and the second is

an SDE driven by the Wiener process )(tWy . These two Wiener processes are assumed to be

independent. Theorem 1 carries over to systems of SDE, and it can be shown that a unique solution to (9) exists. The solution was approximated with the Euler-Maryuama scheme. The constants were dealt with in the same manner as the stochastic logistic growth model. The approximate solutions, with different values of c, are plotted in figure 16.

Figure 16. Euler-Maryuama approximations to (9). The black (species 1) and red (species 2) curves are the deterministic approximate solutions, and the green (species 1) and blue (species 2) curves are the stochastic approximate solutions. The left and right pictures are drawn with different values of c; 080.c = (left) and 80.c = (right). The same behaviour as for the stochastic logistic growth model for varying c is apparent here. In figure 17 the approximate solution to (7) is plotted in the ( 21, XX )-plane. Here the same behaviour is observed as in the previous figure. If c is relatively small, the solution to (9) is close to the solution of the deterministic equation (compare with figure 11), and if c is larger, the noise “takes over”.

Figure 17. Euler-Maryuama approximations to (9) plotted in the (X1,X2)-plane, where the red curve is the deterministic approximate solution, and the blue curve is the stochastic approximate solution. The left and right pictures are drawn with different values of c; 080.c = (left) and 80.c = (right).

22

A Milstein approximation was also carried out. This (improved) approximation method didn’t yield any other result than the Euler-Maryuama method. To elucidate what happens with the populations for a longer time period, then that of Gause’s experiment, the time variable is set to lie in the interval [0,100]. The approximate solution is sketched in figure 18. Here a mark of caution is required. The approximate solution (the curves in the figure) depend on the parameters estimated from Gause’s data and hence extrapolating outside the time interval of Gause’s experiment gives unreliable information.

Figure 18. Euler-Maryuama approximations to (9), with 080.c = and in the time interval [0.100]. To the left the black (species 1) and red (species 2) curves are the deterministic approximate solutions, and the green (species 1) and blue (species 2) curves are the stochastic approximate solutions. To the right the red curve is the deterministic approximate solution, and the blue curve is the stochastic approximate solution. To obtain the stationary PDF (definition 7) a ghastly PDE has to be solved, which I will not attempt to do.

.*2

*2

*

**)(0

2

2

2

222

2

212

2

22

2

21

1

121

1

21

11

121

2

2

2

212

1

121

y

py

x

px

x

pc

K

xybr

K

yryr

x

pc

K

xybr

K

xrxrpy

K

br

K

rx

K

br

K

rrr

∂∂+

∂∂+

∂∂

−−−−

∂∂

−−−−

++

+++−=

2.3 Alternative models In the stochastic analysis so far I have only considered the linear diffusion coefficient

( ) )(),( tcXttX =σ . Other diffusion coefficients are plausible. For example, the r-constant in the logistic growth model (1) can be perturbed with white noise (see Appendix 5). The result is:

)t(dWK

)t(X)t(Xrdt

K

)t(X)t(Xr)t(dX

−+

−= 11 10 . (10)

This procedure can also be applied to the competition model (3), resulting in

23

−−+

−−=

−−+

−−=

).(11)(

),(11)(

1

212

1

1121

2

121

2

22202

1

212

1

1111

1

212

1

11101

tdWK

Xb

K

XXrdt

K

Xb

K

XXrtdX

tdWK

Xb

K

XXrdt

K

Xb

K

XXrtdX

y

x

(11)

Figure 19 and 20 show Euler-Maryuama approximation of the solution to (10) and (11) respectively. The same behaviour for changing values of the noise constant as we have witnessed earlier is also apparent here. However, in these two new models the noise is mostly apparent in the exponential growth phase.

Figure 19. Two different simulations of the Euler-Maryuama approximation to (10) for species 1. The red curve is the deterministic solution, the blue and black curves are the approximations, with 401 .r = (blue) and 2.11 =r

(black).

Figure 20. Euler-Maryuama approximations to (11). The black (species 1) and red (species 2) curves are the deterministic approximate solutions, and the green (species 1) and blue (species 2) curves are the stochastic approximate solutions. The left and right pictures are drawn with different values of r11 and r21; 402111 .rr ==

(left) and 2.12111 == rr (right).

To illustrate this difference in the behaviour of the noise, the approximate solutions to (7) and (10) are plotted in figure 21.

24

Figure 21. The Euler-Maryuama approximation to equation (7) (red) and to equation (10) (blue), with 08.0=c and 400 .r = .

3. Comparison of deterministic and stochastic analysis of Gause’s data The criteria used here for comparison is how well the models predict the data. In figure 22 the solution of the deterministic competition model along with the stochastic analogue ( 080.c = ) are sketched together with the data of Gause’s experiment. The red and black curves correspond to the deterministic model and the blue and green curves correspond to the stochastic model. The crosses are the data points.

Figure 22. Gause’s data plotted together with the Euler and Euler-Maryuama approximations to the deterministic and stochastic competition models. The black (species 1) and red (species 2) curves are the deterministic approximate solutions, and the green (species 1) and blue (species 2) curves are the stochastic approximate solutions. The blue (species 1) and green (species 2) crosses are the data. In the figure it is apparent that the data vary around the deterministic curves much in the same way as the stochastic solutions do. It is also visible that the variation is mostly apparent after the exponential growth, indicating that the model of equation (9) is preferable compared to the alternative model, equation (11) (compare figure 21). These results show that, compared to the deterministic models, the stochastic models with small noise improve the models ability to predict data.

25

III. Summary and Concluding remarks 1. Summary The logistic growth model: The population in the deterministic model grows exponentially to the stable steady state that is the carrying capacity (figure 7). In the stochastic model the growth of the population depends on the value of the noise parameter, c. If the value of c is small, the population will grow exponentially, and upon reaching the carrying capacity the population size will fluctuate around it. However, if the value of c is large the population size will fluctuate wildly and not grow exponentially to the carrying capacity (figure 13). The stationary PDF is in this case a gamma distribution. The expected value and the standard deviation depend on c. If the value of c is small, the expected value is close to the carrying capacity and the standard deviation is relatively small. However, if the value of c is large the expected value is smaller than the carrying capacity, and the standard deviation is much higher (figure 14). The competition model: The deterministic analysis shows that the two populations will coexist (figure 11) – however if the parameter values are slightly changed, species 2 will go extinct. In the stochastic analysis the evolution of the species depend on the values of the noise parameters, c1 and c2. If their values are small, the growths of the populations will almost mimic the “deterministic growth”, with some fluctuations mostly apparent when the population sizes have stabilized. However, if the values of the c:s are large the population sizes will fluctuate wildly (figure 16, 17). The stationary joint PDF is not obtained, hence at this stage it is not apparent what the joint steady state PDF is. 2. Concluding remarks Many classical population models suffer from being too simple and lacking in certain biological features. Hence improvements are needed. Here I have taken two classical deterministic models and converted them into stochastic analogues, thereby allowing the models to reflect the random fluctuations which occur in biological populations. There exist many different ways of ”conversion”. In the main part of my analysis I have looked at SDEs with the linear multiplicative noise, )t(cX=σ , but this is by no means the only way to include noise. I also briefly discussed an alternative model of incorporating noise; see subsection 2.3 in the Results section. Many other possibilities of ”incorporation” exist. When comparing the alternative models with the models with linear multiplicative noise, it is apparent that the main difference is in which growth phase the noise is most prominent. For the alternative models this occurs in the exponential growth phase and, in contrast, for the other models the noise is most prominent when the growth has stabilised. An important question for the analysis of the stochastic models is: How big should the noise be? Here there are no simple answers and the assistance of experts in the field of population biology is greatly needed.

26

In my analysis I have shown that

If the noise is relatively small, the model resembles the deterministic model, with some random fluctuations around the growth curve; thus giving a more realistic picture of the growth of the populations then that of the deterministic model.

If the noise is too large, the model does not give an accurate representation of the growth of the populations.

However, where the line is to be drawn between small and large noise is an open question. I have looked at two different values for the noise constant, 080.c = and 80.c = , and deemed the former to be relatively small and the latter to be too large. In this paper I have looked at the classical experiment performed by Gause. This experiment was performed in the laboratory with controlled conditions. It would be very interesting to apply the methods I have used to data taken from nature, since the random fluctuations experienced by populations are greater in nature than in a laboratory environment. IV. Acknowledgements I would like to thank my supervisor Anders Muszta. Thanks for everything! (I really enjoyed our discussions).

27

V. References Arató, M. (2003). A Famous Nonlinear Stochastic Equation (Lotka-Volterra Model with Diffusion). Mathematical and Computer Modelling, Vol. 38, 709-726 Britton, Nicholas F. (2005). Essential Mathematical Biology. London: Springer. Edelstein-Keshet, Leah (1988). Mathematical Models in Biology. New York: Random House, Inc. Grimmett, Geoffrey and Stirzaker, David (2006). Probability and Random Processes. 3rd edition. Oxford: Oxford University Press. Khasminskii, R. Z. and Klebaner, F. C. (2001). Long Term Behavior of Solutions of the Lotka-Volterra System under Small Random Perturbations. The Annals of Applied Probability, Vol. 11, No. 3, 952-963 Klebaner, Firma C. (2007). Introduction to Stochastic Calculus with Applications. 2nd edition. London: Imperial College Press. Kloeden, Peter E. and Platen, Eckhard (1999). Numerical Solution of Stochastic Differential Equations. Berlin: Springer. Leslie, P.H. (1957). An Analysis of the Data for Some Experiments Carried out by Gause with Populations of the Protozoa, Paramecium Aurelia and Paramecium Caudatum. Biometrika, Vol. 44, No. 3/4, 314-327 Nagle, R. Kent, Saff, Edward B. and Snider, Arthur David (2004). Fundamentals of Differential Equations and Boundary Value Problems. 4th edition. Boston: Pearson Education, Inc. Pascual, Miguel A. and Karieva, Peter (1996). Predicting the Outcome of Competition Using Experimental Data: Maximum Likelihood and Bayesian Approaches. Ecology, Vol. 77. No. 2, 337-349. Roberts, A.J. (2009). Elementary Calculus of Financial Mathematics. Philadelphia: SIAM. de Vries, Gerda, Hillen, Thomas, Lewis, Mark, Müller, Johannes and Schönfisch, Birgitt (2006). A course in Mathematical Biology. Philadelphia: SIAM.

28

Appendix 1 The data from Gause’s experiment

Isolation Competition

Day P.aurelia P.caudatum P.aurelia P.caudatum

0 2 2 2 2

2 14 10 10 10

3 34 10 21 11

4 56 11 58 29

5 44 21 92 50

6 189 56 202 88

7 266 104 163 102

8 330 137 221 124

9 416 165 293 93

10 507 194 236 80

11 580 217 303 66

12 610 194 302 83

13 513 201 340 55

14 593 182 387 67

15 557 192 335 52

16 560 179 363 55

17 522 190 323 40

18 565 206 358 48

19 517 209 308 47

20 500 196 350 50

21 585 195 330 40

22 500 234 350 20

23 495 210 350 20

24 525 210 330 35

25 510 180 350 20

29

Appendix 2

Theorem 1: [Existence and uniqueness] If the following conditions are satisfied, then there exists a unique strong solution X(t) to the SDE (4).

(i) [Local Lipschitz condition] N,TL,N,T ∃∀∀ [ ]T,t,Ny,x 0∈∀≤∀∀ ,

yxL NTtytxtytx −≤−+− ,,,,, σσµµ .

(ii) [Linear growth condition] NTLNT ,,, ∃∀∀ [ ]T,t,Ny,x 0∈∀≤∀∀ ,

( )xL NTtxtx +≤+ 1,,, σµ .

(iii) X0 independent of ( )Tt,Wt ≤≤0 and ( ) ∞<20XE .

In order to check if a unique strong solution to the SDE (7) exists, I show that the conditions stated in Theorem 1 are satisfied. Here I will make use of the Mean-Value Theorem (from calculus), which states that

)yx()z('f)y(f)x(f −⋅=− , where z is a number somewhere between x and y. The first condition is proven separately for µ and for σ: According to the Mean-Value Theorem:

yx)z(')y()x( −⋅µ=µ−µ

yx)z(')y()x( −⋅σ=σ−σ

Hence if )z('µ and )z('σ are bounded by N,TL then yxL)y()x( N,T −≤µ−µ

and .)()( , yxLyx NT −≤−σσ

.222

)(' NK

rrz

K

rrz

K

rrz ⋅+<⋅+≤⋅−=µ

The last inequality comes from the fact that .Nxz ≤≤

Hence )z('µ is bounded by NK

rrL N,T ⋅+= 2

and hence yxL)y()x( N,T −≤µ−µ

c)z(' =σ hence if c is bounded by N,TL then yxL)y()x( N,T −≤σ−σ

30

Since both yxL)y()x( N,T −≤µ−µ and yxL)y()x( N,T −≤σ−σ then

yxL)y()x()y()x( N,T −≤σ−σ+µ−µ and condition (i) is thus satisfied.

The second condition is also proven separately for µ and for σ:

+⋅⋅<−⋅⋅=

−=µ xK

NrKxK

xrKxK

rKx)x(111

The inequality comes from the condition Nx ≤

When 1<K then ( )xL)x( N,T +≤µ 1 where NrKL N,T ⋅=

( )xccx)x( +⋅<=σ 1 thus ( )xL)x( N,T +≤σ 1 where cL N,T =

Since both ( )xL)x( N,T +≤µ 1 and ( )xL)x( N,T +≤σ 1 then ( )xL)x()x( N,T +≤σ+µ 1 and

condition (ii) is thus satisfied. The third condition can be formulated as: the random starting value, X0, can’t have anything to do with the Wiener process and it can’t assume extreme values. These conditions are met here and hence condition (iii) is satisfied. Thus, since all three conditions of theorem 1 are satisfied, there exists a unique strong solution to the SDE (7).

31

Appendix 3 Here the expected values of (a) the integral in the denominator of equation (8) and (b) the whole equation (8) are calculated.

(a) Let .)(0

)(2

2

∫+

−

=t scWs

cr

dsetI

Then by Fubini’s theorem( ) ( )∫

−

=t

)s(cWs

cr

dseEe)t(IE0

2

2

, but

( ))s(cWeE is the moment generating function for W(s)~N(0, s ), and thus is ( ) 2

2sc)s(cW eeE = .

Hence ( )r

edsedsee)t(IE

rttrs

t scsc

r 1

00

22

22

−=== ∫∫

−

.

(b) I use the Delta method of approximating the expected value of a quotient of two dependent random variables, i.e.

( )( ).ZE

YE

Z

YE ≈

Hence

( ) ( )( ) .

)()(

0

)(20

0

)(2

0

)(2

02

2

2

tIrEXK

eEKeX

dserXKE

KeXE

tXEtcW

tc

r

t scWsc

r

tcWtc

r

+=

+

≈

−

+

−

+

−

∫

The expected value( ))t(IE is calculated in (a) and the expected value ( ))t(cWeE is the

moment generating function for W(s)~N(0, t ), see (a), thus

( ) ( ) ( ).11)(

0

0

0

220

22

−+=

−+≈

−

rt

rt

rt

tctc

r

eXK

KeX

eXK

eKeXtXE

32

Appendix 4 Here the stationary PDF of the solution to the SDE (7) is derived from

( )

∂∂+−

∂∂=

∂∂+−

∂∂=

220

22

2

2 p

xp

x

p

xp

x

σµσµ

Integrating yields

σ∂∂+µ−=

2

2 p

xpttancons

The constant in the left hand side is zero since in order for the integral of p to be one the derivative of the density has to vanish for large enough x. Thus

∂∂=

2

2 p

xp

σµ (A4.1)

By the product rule the right hand side is

x

p*

x*

pp

x ∂∂+

∂∂=

∂∂

222

222 σσσ.

Since 222 xc=σ the right-hand side becomesx

p*

xcxc*p

∂∂+

2

222 .

Putting this and

−=K

xrx 1µ into equation (A4.1) yields

.21)(2

2

*2

1*2

2*1

22

2

22

2

222

222

p

dpdx

Kc

r

xc

cr

p

dpdx

xc

cK

rxrx

x

pxcxc

K

xrxp

x

pxcxcpp

K

xrx

=

−⋅−⇔=

−−

⇔∂∂=

−

−⇔∂∂+=

−

Integrating yields plnAxKc

rxln

c

)cr( =+−−22

2 22, where A is a constant.

Solving for p yields the density of a gamma distribution

pBexx

Kc

r

c

)cr(

=⋅⋅−−

22

2 22

(A4.2)

where AeB = .

33

To see this, let the random variable Y~Gamma(α,β), where α is the shape parameter and β the scale parameter. Then the PDF, p, for Y is

)(

exp

y

αΓβ⋅= α

β−

−α 1

Hence for (A4.2) the shape parameter is 22

2 21

2c

r

c

)cr( =+−=α and the scale parameter is

2

2Kc

r=β . The constant B is )(

BαΓβ

= α1

.

34

Appendix 5 To perturb the r-constant in the logistic growth model, let )t(rrr ξ10 += where )t(ξ is the

white noise process. Then (1) becomes

( )

).(1111

111

1010

1010

tdWK

NNrdt

K

NNrdt

K

NNrdt

K

NNrdN

K

NNr

K

NNr

K

NNrr

dt

dN

−+

−=

−+

−=

⇔

−⋅+

−⋅=

−⋅+=

ξ

ξξ

The last equality follows from the definition of the white noise process )t(ξ .