1 An Introduction to Stochastic Population Models

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1 1 An Introduction to Stochastic Population Models References [1] J. H. Matis and T. R. Kiffe. Stochastic Population Models, a Compartmental Perective. Springer-Verlag, New York, [2] E. Renshaw. Modelling Biological Populations in Space and Time. Cambridge University Press, Cambridge, Some Ecological Examples 1. Spread of muskrats in the Netherlands 2. Invasion by Africanized honey bee 3. Infestations of honey bees by varroa mites Chapter 1: Introduction to Stochastic Population Models Copyright c 2003 by Thomas E. Wehrly Slide 1

2 1.2 A Quick Contrast Between Deterministic and Stochastic Models We consider the linear birth-death model where each individual gives birth at rate a 1 and dies at rate a 2. We let X(t) be the number of individuals in the population at t Deterministic Model The differential equation for the deterministic model is dx(t) dt = (a 1 a 2 )X(t), with solution X(t) = X(0)e (a 1 a t )t. The deterministic model results in either exponential growth or exponential decay of the population. Chapter 1: Introduction to Stochastic Population Models Copyright c 2003 by Thomas E. Wehrly Slide 2

3 1.2.2 Stochastic Model For the stochastic model, we make assumptions concerning events in a small interval of (t, t + t) of length t. We suppose that each individual gives birth with probability a 1 t and dies with probability a 2 t. This leads to the assumptions: P (X(t + t) = x + 1 X(t) = x) = a 1 x t + o( t) P (X(t + t) = x 1 X(t) = x) = a 2 x t + o( t) P (X(t + t) = x X(t) = x) = 1 (a 1 + a 2 )x t + o( t) We let p x (t) = P [X(t) = x]. The above assumptions imply that p x (t + t) = p x (t)[1 (a 1 + a 2 )x t] + p x 1 (t)(x 1)a 1 t + p x+1 (t)(x + 1)a 2 t since X(t) = x can be reached from X(t) = x 1, x, x + 1 in a small interval. Letting t 0, we obtain the system of differential equations, called the Kolmogorov forward equations, for p x (t): ṗ x (t) = a 1 (x 1)p x 1 (t) [(a 1 +a 2 )x]p x (t)+a 2 (x+1)p x+1 (t) for x > 0 and ṗ 0 (t) = a 2 p 1 (t) Chapter 1: Introduction to Stochastic Population Models Copyright c 2003 by Thomas E. Wehrly Slide 3

4 The solution to these equations can be solved using standard differential equation techniques. We now focus on the stochastic aects. Since the individuals behave independently, we can view the population as comprising X 0 separate populations, each of size 1. When X 0 = 1, the population size X(t) follows a geometric distribution with pmf p 0 (t) = α(t) p x (t) = [1 α(t)][1 β(t)][β(t)] x 1, x = 1, 2,... where α(t) = a 2(e (a 1 a 2 )t 1) a 1 e (a 1 a 2 )t a 2 β(t) = a 1(e (a 1 a 2 )t 1) a 1 e (a 1 a 2 )t a 2 Standard results for the geometric distribution yield the mean and variance functions for the population size: E[X(t)] = X 0 e (a 1 a 2 )t V [X(t)] = X 0 [ a 1 +a 2 a 1 a 2 ] e (a 1 a 2 )t (e (a 1 a 2 )t 1) The mean function agrees with the deterministic model, however the variance function depends on the magnitudes of the birth and death rates as well as on their difference. Chapter 1: Introduction to Stochastic Population Models Copyright c 2003 by Thomas E. Wehrly Slide 4

5 For the deterministic model, there is exponential growth if the birth rate exceeds the death rate. However, for the stochastic model, there is a probability of extinction even in this case: p 0 (t) = α(t) X 0 We now look at the probability of ultimate extinction: If a 1 < a 2, p 0 ( ) = 1 If a 1 > a 2, p 0 ( ) = (a 2 /a 1 ) X 0. If a 1 = a 2, p 0 (t) = [a 2 t/(1 + a 2 t)] X 0 1 as t. It is easy to simulate the stochastic model. By examining sample paths, one can see how single realizations of a process can give misleading results. One can show that the between events is exponentially distributed with parameter R(X) = (a 1 + a 2 )X(t). A birth occurs at this with probability a 1 /(a 1 + b 1 ). Otherwise, a death occurs. Chapter 1: Introduction to Stochastic Population Models Copyright c 2003 by Thomas E. Wehrly Slide 5

6 Simulation with a 1 = 5, a 2 = 1, X 0 = Chapter 1: Introduction to Stochastic Population Models Copyright c 2003 by Thomas E. Wehrly Slide 6

7 Simulation with a 1 = 1, a 2 = 5, X 0 = Chapter 1: Introduction to Stochastic Population Models Copyright c 2003 by Thomas E. Wehrly Slide 7

8 Simulation with a 1 = 5, a 2 = 5, X 0 = Chapter 1: Introduction to Stochastic Population Models Copyright c 2003 by Thomas E. Wehrly Slide 8

9 Simulation with a 1 = 5, a 2 = 4, X 0 = Chapter 1: Introduction to Stochastic Population Models Copyright c 2003 by Thomas E. Wehrly Slide 9