Abstract
In this paper, we study some properties of the solutions of a stochastic Lotka–Volterra predator-prey model, namely, the boundedness in the mean of numerical solutions, the strong convergence for this kind of solutions, and the turnpike property of solutions of an optimal control problem in a population modelled by a Lotka–Volterra system with stochastic environmental fluctuations. Even though there are numerous results in the deterministic case, there are few results for the behavior of numerical solutions in a population dynamic with random fluctuations. First, we show, using the Euler–Maruyama scheme, that the boundedness of numerical solutions and the convergence of the scheme are preserved in the stochastic case. Second, we analyze a property of the long-term behavior of a Lotka–Volterra system with stochastic environmental fluctuations known as turnpike property. In optimal control theory, the optimal solutions dwell mostly in the neighborhood of a balanced equilibrium path, corresponding to the optimal steady-state solution. Our study shows, by means of the Stochastic Maximum Principle, that this turnpike property is preserved, when the noise in the system is small. Numerical simulations are implemented to support our results.
Highlights
IntroductionIn 1925, Vito Volterra and Alfred Lotka obtained simultaneously a mathematical model on population dynamics and competition systems [1, 2]. eir models are based on the increment, noted by Umberto D’Ancona [3], of fish population due to reduction in fishing during World War I and the subsequent growth of sharks after the war
A novelty of our analysis is the use of two controls in the populations and the extension of the work of [18] to a stochastic case, by using the Stochastic Maximum Principle. us, we have combined some techniques of the Geometric Control eory with the property of exponential stability of the numerical solution of our stochastic model
We analyze the stability of optimal-trajectory turnpike property of the solutions of the stochastic Lotka–Volterra model. e turnpike property means that the most important fact about the behavior of solutions is the optimality criterion considered and the choice of time interval or the data used are irrelevant, for times far from the endpoints of the time interval. is property is a characteristic of the turnpike theory which was introduced by Samuelson in mathematical economics and recently has been reconsidered in Control eory by the authors of [18, 22]. e turnpike property of a solution in an optimal control problem means that an optimal trajectory for most of the time could stay in a neighborhood of a balanced equilibrium path, corresponding to the optimal steady-state solution
Summary
In 1925, Vito Volterra and Alfred Lotka obtained simultaneously a mathematical model on population dynamics and competition systems [1, 2]. eir models are based on the increment, noted by Umberto D’Ancona [3], of fish population due to reduction in fishing during World War I and the subsequent growth of sharks after the war. If we denote by x1(t) and x2(t) the differentiable functions meaning the density of the population of prey and predator, respectively, the deterministic model is given by dx1(t) αx1(t) − βx1(t)x2(t)dt,. In the study and modeling of the interaction between populations, it is important to consider some stochastic factors that have impact on their growth, persistence, and extinction. E existence and uniqueness of a global positive solution of the system and the conditions for which extinction occurs for a stochastic preypredator model have been extensively studied by various methods [9,10,11], including the Stochastic Maximum Principle (see [12]). In [13], the stochastic uniform boundedness of the solution and the existence of a globally unique positive solution are obtained, for a predator and prey model that incorporates disease invasion and sudden catastrophic shocks. We will consider an equidistant discretization of the time and ΔWk ≤ l, with l constant
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