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# Numerical solution of a stochastic population growth model in a closed system

- Morteza Khodabin
^{1}Email author, - Khosrow Maleknejad
^{1}and - Mahnaz Asgari
^{1}

**2013**:130

https://doi.org/10.1186/1687-1847-2013-130

© Khodabin et al.; licensee Springer 2013

**Received:**18 May 2012**Accepted:**17 April 2013**Published:**6 May 2013

## Abstract

In this paper, we introduce a stochastic population model in a closed system. This model is a nonlinear stochastic integro-differential equation. At first, we solve this problem *via* the stochastic *θ*-method. Then we solve it by using the Bernstein polynomials and collocation method. This method reduces integro-differential equation to a system of nonlinear algebraic equations. The results demonstrate applicability and accuracy of this method.

## Keywords

- Brownian Motion
- Collocation Method
- Bernstein Polynomial
- Nonlinear Algebraic Equation
- Random Excitation

## 1 Introduction

Several phenomena in life and sciences, especially in mechanics, engineering and, since recently, in finance, have been found to depend on random excitations. It therefore seems natural that a current trend in describing and studying these phenomena is focused on the use of stochastic mathematical models rather than deterministic ones.

Having in mind that in many cases random excitations are of the Gaussian white noise type, which is mathematically described as a formal derivative of the Brownian motion, all such phenomena are mathematically modeled and essentially represented by complex stochastic differential or integro-differential equations of the Itô type. In mathematical literature, many population models have been considered, from deterministic and stochastic population models, where the population size is represented by a discrete random variable, to very complex continuous stochastic models [1–3].

This article deals with a mathematical model of the accumulated effect of toxins on a population living in a closed system [4]. We obtain a stochastic model of it. Then we apply numerical methods to solve the problem. At first, we convert it to a stochastic differential equation and solve with the stochastic *θ*-method. Then we convert the problem to a stochastic integral equation (SIE) and introduce Bernstein polynomials for solving the SIE.

Bernstein polynomials are differentiable and integrable piecewise polynomials. In recent years, these polynomials have been used for solving differential and integral equations [5–8]. We use them to solve a nonlinear stochastic integro-differential equation (SIDE) that arises in a population growth model in a closed system. By using Bernstein polynomials and their derivatives along with the collocation method, SIDE is converted to nonlinear algebraic equations.

In Section 2, we review deterministic population growth in a closed system. Section 3 introduces the stochastic population growth in a closed system. Section 4 solves the problem by using the stochastic *θ*-method. In Section 5, we introduce Bernstein polynomials and convert SIDE to a nonlinear algebraic system. Finally, the conclusion is given in Section 6.

## 2 Preliminaries

**Definition 2.1** (Brownian motion process)

- (i)
the process has independent increments for $0\le {t}_{0}\le {t}_{1}\le \cdots \le {t}_{n}\le T$,

- (ii)
for all $t\ge 0$, $h>0$, $B(t+h)-B(t)$ is normally distributed with mean 0 and variance

*h*, - (iii)
function $t\u27f6B(t)$ is continuous a.s.

**Definition 2.2**Suppose $0\le s\le T$, let $D=D(s,T)$ be the class of functions

- (i)
The function $(t,\omega )\to f(t,\omega )$ is $\beta \times \u03dc$ measurable, where

*β*is the Borel algebra on $[0,\mathrm{\infty})$ and*Ϝ*is the*σ*-algebra on Ω. - (ii)
*f*is adapted to ${\u03dc}_{t}$, where ${\u03dc}_{t}$ is the*σ*-algebra generated by the random variables $B(s)$; $s\le t$ and adapted means that*f*is determined by $B(s)$; $s\le t$. - (iii)
$E[{\int}_{s}^{T}f{(t,\omega )}^{2}\phantom{\rule{0.2em}{0ex}}dt]<\mathrm{\infty}$.

**Definition 2.3** (The Itô integral)

*f*is defined by

See [9].

**Lemma 2.1**

*Let*$f(t)$

*be a regular adapted process such that with probability one*${\int}_{0}^{T}{f}^{2}(t)\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty}$.

*Then the Itô integral*${\int}_{0}^{T}f(t)\phantom{\rule{0.2em}{0ex}}dB(t)$

*is defined and can be approximated by*

*where*, $\{{t}_{i}\}$ *is a partition of* $[0,T]$ *with* ${\delta}_{n}=max({t}_{i+1}-{t}_{i})\to 0$ *as* $n\to \mathrm{\infty}$.

*Proof* See [10]. □

## 3 Deterministic population growth in a closed system

where $a>0$ is the birth rate coefficient, $b>0$ is the crowding coefficient and the last term contains the integral indicating the ‘total metabolism’ or a total amount of toxins produced since time zero. Since the system is closed, the presence of the toxins term always causes the population level to fall to zero in the long run. Several analytical and numerical methods have been proposed to solve the classical population growth model [12–14].

This model is a first-order integro-differential equation. In [11], the author considered two cases $k=\frac{ab}{c}$ small and $k=\frac{ab}{c}$ large. He showed that for the case $k\gg 1$, where population is weakly sensitive to toxins, a rapid rise occurs along the logistic curve that will reach a peak and then is followed by a slow exponential decay. And for small k, where populations is strongly sensitive to toxins, the solutions are proportional to ${sech}^{2}(t)$.

## 4 Stochastic population growth in a closed system

*k*is not completely definite and depends on some random environment effects. We may replace this coefficient by an average value plus a random function term

*α*is a nonrandom coefficient that shows the intensity of noise at time

*t*. So, the stochastic form of (1) is given by

with $x(0)={x}_{0}$, and $B=\{B(t),t\ge 0\}$ is a standard Brownian motion defined on a probability space $\{\mathrm{\Omega},\u03dc,{\{{\u03dc}_{t}\}}_{t\ge 0},P\}$ with a filtration ${\{{\u03dc}_{t}\}}_{t\ge 0}$ that is right continuous.

## 5 Numerical solution of SIDE

*θ*-method, we get

By substituting (4), (5) and (7) into (6), the model converts to quadratic for ${x}_{n+1}$ and can be solved by the quadratic equation.

## 6 Bernstein polynomials and function approximation

*n*th degree over the interval $[a,b]$ is defined by

*n*and satisfy the following properties

- (i)
${\beta}_{i,n}(t)=0$ if $i<0$ or $i>n$.

- (ii)
${\sum}_{0}^{n}{\beta}_{i,n}(t)=1$.

- (iii)
${\beta}_{i,n}(a)={\beta}_{i,n}(b)=0$, $1\le i\le n-1$.

**Theorem 6.1** *For all function* *f* *in* $C[0,1]$, *the sequence* $\{{\beta}_{n}(f);n=1,2,\dots \}$ *converges uniformly to* *f*.

*Proof* See [15]. □

One of the many remarkable properties of the Bernestein approximation is that the derivatives of ${\beta}_{n}(f)$ of any order converge to the corresponding derivatives of *f* [16].

## 7 The numerical method based on Bernstein polynomials

*C*and $\mathrm{\Phi}(t)$ are $(n+1)\times 1$ vectors given by

*C*as follows:

We use Lemma 2.1 to calculate Itô integrals. By solving the nonlinear system (17), we find the unknown coefficient. Then we get the approximate solution $y(t)$ and $x(t)$.

The figures show the results of a numerical solution generated by the stochastic *θ*-method with $\theta =0.5$ and the Bernstein approximation with $n=12$ for two cases of *r*. The results show a rapid rise along the logistic curve and then a fast exponential decay to zero for big *r*.

They also illustrate a comparison between the numerical solutions of the deterministic and the stochastic models.

## 8 Conclusion

*r*, the problem becomes very stiff and requires very small steps in the numerical methods. In such a case, it is practical to implement the Bernstein collocation method which is effective and easy to use.

## Declarations

## Authors’ Affiliations

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