Numerical solution of a stochastic population growth model in a closed system
© Khodabin et al.; licensee Springer 2013
Received: 18 May 2012
Accepted: 17 April 2013
Published: 6 May 2013
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.
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 . 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.
Definition 2.1 (Brownian motion process)
the process has independent increments for ,
for all , , is normally distributed with mean 0 and variance h,
function is continuous a.s.
The function is measurable, where β is the Borel algebra on and Ϝ is the σ-algebra on Ω.
f is adapted to , where is the σ-algebra generated by the random variables ; and adapted means that f is determined by ; .
Definition 2.3 (The Itô integral)
where, is a partition of with as .
Proof See . □
3 Deterministic population growth in a closed system
where is the birth rate coefficient, 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 , the author considered two cases small and large. He showed that for the case , 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 .
4 Stochastic population growth in a closed system
with , and is a standard Brownian motion defined on a probability space with a filtration that is right continuous.
5 Numerical solution of SIDE
By substituting (4), (5) and (7) into (6), the model converts to quadratic for and can be solved by the quadratic equation.
6 Bernstein polynomials and function approximation
if or .
Theorem 6.1 For all function f in , the sequence converges uniformly to f.
Proof See . □
One of the many remarkable properties of the Bernestein approximation is that the derivatives of of any order converge to the corresponding derivatives of f .
7 The numerical method based on Bernstein polynomials
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 and .
The figures show the results of a numerical solution generated by the stochastic θ-method with and the Bernstein approximation with 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.
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