Dynamical Analysis of a Stochastic Predator-Prey System with Lévy Noise and Impulsive Toxicant Input

This paper established a modified Leslie-Gower and Holling-type IV stochastic predator-prey model with Lévy noise and impulsive toxicant input. We study the stability in distribution of solutions by inequality techniques and ergodic method. By comparison method and Itô’s formula, we obtain the sufficient conditions for the survival of each species. Some numerical simulations are introduced to show the theoretical results.

Analysis of a Stochastic Ratio-Dependent Predator-Prey System with Markovian Switching and Lévy Jumps

In this paper, the dynamics of a stochastic ratio-dependent predator-prey system with markovian switching and Lévy noise is studied. Firstly, we show the existence condition of global positive solution under the given positive initial value. Secondly, sufficient conditions for system extinction and persistence are obtained through some assumptions. Then, the sufficient conditions of stochastically persistence are obtained by combining stochastic analysis technique and M-matrix analysis. In addition, under appropriate conditions, we demonstrate the existence of a unique stationary distribution for a system without Lévy jumps. Finally, the empirical and Mlistein methods are used to verify the theoretical results through numerical simulation.

Global Dynamics of a Stochastic Two-Prey One-Predator Model with <i>S</i>-Type Distributed Delays and Lévy Noises

In this paper, a stochastic two-prey one-predator model with S-type distributed time delays and Lévy noises is considered. Using the comparison theorem and Ito’s formula, sufficient conditions of persistence in the mean and extinct for each population are established. Then, conditions of global attractivity and stability in distribution by Barbalat’s conclusion are also obtained. Furthermore, Euler numerical simulation method is given to demonstrate our conclusions.

Analysis of a Delayed Stochastic One-Predator Two-Prey Population Model in a Polluted Environment

This paper is concerned with the dynamics of a delayed stochastic one-predator two-prey population model in a polluted environment. We show that there exists a unique positive solution that is permanent in time average under certain conditions. Moreover, the global attractively of system is studied. Finally, some numerical simulations are given to illustrate the main results.

Periodic Solution for a Stochastic Predator-Prey Model with Impulses and Holling-II Functional Response

Considering the mutual interference between species, a stochastic predator-prey model with impulses and Holling-II functional response is proposed in this paper. Firstly, by constructing an equivalent system without impulses, the existence of a globally unique positive solution is proved. Secondly, in cases of the mutual coefficient m = 1 and 0 m < 1, by constructing suitable Lyapunov functional, the existence of T-periodic solution is investigated under some certain conditions. Finally, numerical simulation is introduced to verify our main results.

The Dynamics of an Impulsive Competitive System with Infinite Delay and Diffusion

In this paper, we consider an impulsive competitive system with infinite delay and diffusion. Firstly, on basis of inequality estimation techniques and comparison theorem of impulsive differential equations, we obtain some sufficient conditions for the permanence and extinction of the system. Then, we establish sufficient conditions for the globally attractive of the system by constructing appropriate Lyapunov function. Besides, under different impulsive conditions, we discuss the effect of time delay and diffusion on dynamic behavior of the competitive system.

Periodic Solution for Stochastic Predator-Prey Systems with Nonlinear Harvesting and Impulses

In this paper, astochastic predator-prey systems with nonlinear harvesting and impulsive effect are investigated. Firstly, we show the existence and uniqueness of the global positive solution of the system. Secondly, by constructing appropriate Lyapunov function and using comparison theorem with an impulsive differential equation, we study that a positive periodic solution exists. Thirdly, we prove that system is globally attractive. Finally, numerical simulations are presented to show the feasibility of the obtained results.

Dynamics of a Stochastic Delayed Predator-Prey System with Beddington-DeAngelis Functional Response

This paper is concerned with a stochastic predator-prey system with Beddington-DeAngelis functional response and time delay. Firstly, we show that this system has a unique positive solution as this is essential in any population dynamics model. Secondly, the validity of the stochastic system is guaranteed by stochastic ultimate boundedness of the analyzed solution. Finally, by constructing suitable Lyapunov functions, the asymptotic moment estimation of the solution was given. These properties of the solution can provide theoretical support for biological resource management.

Fear and delay effects on a food chain system with two kinds of different functional responses

For food chain system with three populations,direct predation is the basic interaction between species.Different species often have different predation functional responses,so a food chain system with Holling-II response for middle predator and Beddinton-DeAngelis response for top predator is proposed.Apart from direct predation,predator population can significantly impact the survival of prey population by inducing the prey's fear,but the impact often possesses a time delay.This paper is concentrated to explore how the fear and time delay affect the system stability and the species persistence.By use of Lyapunov functional method and bifurcation theory,the positiveness and boundedness of solutions,local and global behavior of species,the system stability around the equilibrium states and various kinds of bifurcation are investigated.Numerically,some simulations are carried out to validate the main findings and the critical values of the bifurcation parameters of fear and conversion rate are obtained.It is observed that fear and delay can not only stabilize,but also destabilize the system,which depends on the magnitude of the fear and delay.The system varies from unstable to stable due to the continuous increase of the prey's fear by middle predator.Small fear induced by top predator or small delay of the prey's fear can stabilize the system,while they are sufficiently large,the system stability is to be destroyed.Simultaneously,the conversion rate can also change the stability and even make the species to be extinct.Some rich dynamics like multiple stabilities and various types of bistability behaviors are also exhibited,which results in the convergence of the species from one stable equilibrium to another.