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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.

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Motivation
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
Terminology

  • Stochastic Process: Let I denote an arbitrary nonempty index set and let {Ω,U, P} denote a probability space. A family of Rn – valued random variables is a stochastic process.

  • Markov Property: with probability 1.


Terminology1
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.


Terminology2
Terminology

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


Ito s integral
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 integral1
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 integral2
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 integral3
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 integral4
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 integral5
Ito’s Integral

Equation (6) then becomes

By convention an Ito SDE is written as

and satisfies the integral equation



Example exponential growth
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
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
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:






Considerations
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
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


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