The Logistic Growth SDE

Motivation  In population biology the logistic growth model is one of the simplest models of population dynamics.  To begin studying Stochastic Differential Equations (SDE) we begin by studying the effects of adding a stochastic term to well known deterministic models.

Terminology  Stochastic Process: Let I denote an arbitrary nonempty index set and let {Ω,U, P} denote a probability space. A family of R n – valued random variables is a stochastic process.  Markov Property: with probability 1.

Terminology  Diffusion Process: A Markov process with continuous sample paths such that its probability density function satisfies for any a(x,t) is the infinitesimal mean and is called the drift vector and B(x,t) is the infinitesimal variance and is called the diffusion matrix.

Terminology  Wiener process: a stochastic process where W(t) depends continuously on t, and the following hold:

Ito’s Integral Stochastic dynamics yields differential equations of the form where is Gaussian white noise. The goal is to transform (1) into an integral equation and solve for X(t).

Ito’s Integral The second integral in (2) is undefined. It can be shown that the Wiener Process is the derivative of the white noise term. Using (3) in (2)

Ito’s Integral  The first integral in (4) is the deterministic term and is a regular integral.  The second integral in (4) is the stochastic term and must be defined Take We want to define the integral:

Ito’s Integral To examine the behavior of (6) we start by assuming it to be Riemann-Stieltjes integral and integrating. This yields The partial sums are defined as they converge with finer partitions and arbitrary choice of the intermediate points.

Ito’s Integral The approximation sums converge in mean square. They can be written as Convergence depends on the choice of the intermediate point. Choose.

Ito’s Integral Equation (6) then becomes By convention an Ito SDE is written as and satisfies the integral equation

Ito’s Formula

Example: Exponential Growth Consider the SDE, exponential growth with environmental variation, where c and r are positive constants. Let Applying Ito’s formula and integrating from 0 to t, and solving for X(t) yields:

Example: Logistic Growth Consider the SDE logistic growth with environmental variation, where c, K and r are positive constants. Let Applying Ito’s formula and integrating from 0 to t, and solving for X(t) yields:

Example: Bimodal Equations Consider the SDE logistic growth with environmental variation, where c, K and r are positive constants. Let. Then applying Ito’s formula and integrating from 0 to t, and solving for X(t) yields:

Graphs: The Basic Equations

Graphs:

Considerations  Effects of the coefficient of the stochastic term.  How to determine the correct coefficients for a specific problem  Expected Gaussian versus graphed Levy distribution

References  An Introduction to Stochastic Process with Applications to Biology  Linda J.S. Allen  Stochastic Differential Equations  Ludwig Arnold  Introduction to Stochastic Differential Equations  Thomas Gard