A Stochastic Holling-Type II Predator-Prey Model with Stage Structure and Refuge for Prey

In this paper, we study a stochastic Holling-type II predator-prey model with stage structure and refuge for prey. Firstly, the existence and uniqueness of the global positive solution of the system are proved. Secondly, the stochastically ultimate boundedness of the solution is discussed. Next, sufficient conditions for the existence and uniqueness of ergodic stationary distribution of the positive solution are established by constructing a suitable stochastic Lyapunov function. Then, sufficient conditions for the extinction of predator population in two cases and that of prey population in one case are obtained. Finally, some numerical simulations are presented to verify our results.

Dynamics of a single predator multiple prey model with stochastic perturbation and seasonal variation

Considering various factors are stochastic rather than deterministic in the evolution of populations growth, in this paper, we propose a single predator multiple prey stochastic model with seasonal variation. By using the method of solving an explicit solution, the existence of global positive solution of this model are obtained. The method is more convenient than Lyapunov analysis method for some population models. Moreover, the stochastically ultimate boundedness are considered by using the comparison theorem of stochastic differential equation. Further, some sufficient conditions for the extinction and strong persistence in the mean of populations are discussed, respectively. In addition, by constructing some suitable Lyapunov functions, we show that this model admits at least one periodic solution. Finally, numerical simulations clearly illustrate the main theoretical results and the effects of white noise and seasonal variation for the persistence and extinction of populations.

Dynamics of a stochastic periodic SIRS model with time delay

Dynamical behaviors of a stochastic periodic SIRS epidemic model with time delay are investigated. By constructing suitable Lyapunov functions and applying Itô’s formula, the existence of the global positive solution and the property of stochastically ultimate boundedness of model (1.1) are proved. Moreover, the extinction and the persistence of the disease are established. The results are verified by numerical simulations.

Extinction Analysis of Stochastic Predator–Prey System with Stage Structure and Crowley–Martin Functional Response

In this paper, we researched some dynamical behaviors of a stochastic predator–prey system, which is considered under the combination of Crowley–Martin functional response and stage structure. First, we obtained the existence and uniqueness of the global positive solution of the system. Then, we studied the stochastically ultimate boundedness of the solution. Furthermore, we established two sufficient conditions, which are separately given to ensure the stochastic extinction of the prey and predator populations. In the end, we carried out the numerical simulations to explain some cases.

Dynamics behaviors of a delayed competitive system in a random environment

In the real world, the population systems are often subject to white noises and a system with such stochastic perturbations tends to be suitably modeled by stochastic differential equations. This paper is concerned with the dynamic behaviors of a delay stochastic competitive system. We first obtain the global existence of a unique positive solution of system. Later, we show that the solution of system will be stochastically ultimate boundedness. However, large noises may make the system extinct exponentially with probability one. Also, sufficient conditions for the global attractivity of system are established. Finally, illustrated examples are given to show the effectiveness of the proposed criteria.

Effects of Toxicants on a Non-Autonomous Predator-Prey Model with Random Perturbation

A new non-autonomous predator-prey model in a polluted environment with stochastic perturbation is considered in this paper. The existence of a global positive solution and stochastically ultimate boundedness are derived. Furthermore, some sufficient and necessary criteria for extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean are obtained. At last, a series of numerical simulations to illustrate our mathematical findings are presented.

Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion

This paper studies the effect of jump-diffusion random environmental perturbations on the asymptotic behaviour and extinction of Lotka-Volterra population dynamics with delays. The contributions of this paper lie in the following: (a) to consider delay stochastic differential equation with jumps, we introduce a proper initial data space, in which the initial data may be discontinuous function with downward jumps; (b) we show that the delay stochastic differential equation with jumps associated with our model has a unique global positive solution and give sufficient conditions that ensure stochastically ultimate boundedness, moment average boundedness in time, and asymptotic polynomial growth of our model; (c) the sufficient conditions for the extinction of the system are obtained, which generalized the former results and showed that the sufficiently large random jump magnitudes and intensity (average rate of jump events arrival) may lead to extinction of the population.

Stochastically Ultimate Boundedness and Global Attraction of Positive Solution for a Stochastic Competitive System

A stochastic competitive system is investigated. We first show that the positive solution of the above system does not explode to infinity in a finite time, and the existence and uniqueness of positive solution are discussed. Later, sufficient conditions for the stochastically ultimate boundedness of positive solution are derived. Also, with the help of Lyapunov function, sufficient conditions for the global attraction of positive solution are established. Finally, numerical simulations are presented to justify our theoretical results.

Stochastic Logistic Systems with Jumps

This paper is concerned with a stochastic nonautonomous logistic model with jumps. In the model, the martingale and jump noise are taken into account. This model is new and more feasible and applicable. Sufficient criteria for the existence of global positive solutions are obtained; then asymptotic boundedness inpth moment, stochastically ultimate boundedness, and asymptotic pathwise behavior are to be considered.

Stochastic Delay Population Dynamics under Regime Switching: Global Solutions and Extinction

This paper is concerned with a delay Lotka-Volterra model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall inequality and Young’s inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Finally, an example is given to illustrate the main results.