Optimal harvesting for a stochastic competition system with stage structure and distributed delay

A stochastic competition system with harvesting and distributed delay is investigated, which is described by stochastic differential equations with distributed delay. The existence and uniqueness of a global positive solution are proved via Lyapunov functions, and an ergodic method is used to obtain that the system is asymptotically stable in distribution. By using the comparison theorem of stochastic differential equations and limit superior theory, sufficient conditions for persistence in mean and extinction of the stochastic competition system are established. We thereby obtain the optimal harvest strategy and maximum net economic revenue by the optimal harvesting theory of differential equations

Analysis of a Holling-type IV stochastic prey-predator system with anti-predatory behavior and Lévy noise

In this paper, we investigate a stochastic prey-predator model with Holling-type IV functional responses, anti-predatory behavior (referring to prey resistance to predator), gestation time delay of prey and Lévy noise. We investigate the existence and uniqueness of global positive solutions through Itô's formulation and Lyapunov's method. We also provide sufficient conditions for the persistence and extinction of prey-predator populations. Additionally, we examine the stability of the system distribution and validate our analytical findings through detailed numerical simulations. Our paper concludes with the implications of our results

Analysis of a stochastic delay competition system driven by Lévy noise under regime switching

This paper is concerned with a stochastic delay competition system driven by Lévy noise under regime switching. Both the existence and uniqueness of the global positive solution are examined. By comparison theorem, sufficient conditions for extinction and non-persistence in the mean are obtained. Some discussions are made to demonstrate that the different environment factors have significant impacts on extinction. Furthermore, we show that the global positive solution is stochastically ultimate boundedness under some conditions, and an important asymptotic property of system is given. In the end, numerical simulations are carried out to illustrate our main results

Nonlinear Stochastic Systems And Controls: Lotka-Volterra Type Models, Permanence And Extinction, Optimal Harvesting Strategies, And Numerical Methods For Systems Under Partial Observations

This dissertation focuses on a class of stochastic models formulated using stochastic differential equations with regime switching represented by a continuous-time Markov chain, which also known as hybrid switching diffusion processes. Our motivations for studying such processes in this dissertation stem from emerging and existing applications in biological systems, ecosystems, financial engineering, modeling, analysis, and control and optimization of stochastic systems under the influence of random environments, with complete observations or partial observations. The first part is concerned with Lotka-Volterra models with white noise and regime switching represented by a continuous-time Markov chain. Different from the existing literature, the Markov chain is hidden and canonly be observed in a Gaussian white noise in our work. We use a Wonham filter to estimate the Markov chain from the observable evolution of the given process, and convert the original system to a completely observable one. We then establish the regularity, positivity, stochastic boundedness, and sample path continuity of the solution. Moreover, stochastic permanence and extinction using feedback controls are investigated. The second part develops optimal harvest strategies for Lotka-Volterra systems so as to establish economically, ecologically, and environmentally reasonable strategies for populations subject to the risk of extinction. The underlying systems are controlled regime-switching diffusions that belong to the class of singular control problems. We construct upper bounds for the value functions, prove the finiteness of the harvesting value, and derive properties of the value functions. Then we construct explicit chattering harvesting strategies and the corresponding lower bounds for the value functions by using the idea of harvesting only one species at a time. We further show that this is a reasonable candidate for the best lower bound that one can expect. In the last part, we study optimal harvesting problems for a general systems in the case that the Markov chain is hidden and can only be observed in a Gaussian white noise. The Wonham filter is employed to convert the original problem to a completely observable one. Then we treat the resulting optimal control problem. Because the problem is virtually impossible to solve in closed form, our main effort is devoted to developing numerical approximation algorithms. To approximate the value function and optimal strategies, Markov chain approximation methods are used to construct a discrete-time controlled Markov chain. Convergence of the algorithm is proved by weak convergence method and suitable scaling