Stochastic Predator-prey Model Research Articles

Threshold behavior in a stochastic predator–prey model with general functional response

In this work we study a stochastic predator–prey model with general functional response and time periodic coefficients. The threshold conditions for the persistence and extinction of each population are established. An example and its numerical simulations are given to verify the effectiveness of the theoretical results.

Analysis for a Delayed Three-Species Predator-Prey Model with Feedback Controls and Prey Diffusion

In this paper, we study a class of 3-species multidelay Lotka–Volterra ratio-dependent predator-prey model with feedback controls and prey diffusion. By using the theory of delay differential inequalities and developing some new analysis methods as well as constructing a suitable Lyapunov function, some sufficient conditions are obtained to guarantee the permanence of the system and the global attractivity of the positive solution for the predator-prey system. Furthermore, the corresponding periodic system is discussed, and some conditions are established about the existence, uniqueness, and stability of the positive periodic solution for the periodic system by using the fixed-point theory and theoretical analysis. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria. Finally, the corresponding stochastic time-delay predator-prey model with multiplicative noise sources is solved numerically, and some interesting new dynamics behavior is obtained.

Dynamical behavior of a stochastic predator-prey model with stage structure for prey

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings.

Survival analysis of a stochastic predator–prey model with prey refuge and fear effect

In this paper, we propose and investigate a stochastic Holling type-II predator–prey model with prey refuge and fear effect. We first prove the existence and uniqueness of the global positive solution. Then we perform the survival analysis of the model, including the existence of a unique ergodic stationary distribution and the extinction of the model. Numerical simulations are carried out to validate our analytical results. Our findings indicate that the white noise is adverse to the growth of predator and prey populations, and the increase of fear effect will lead to the decrease of predator density, but with no obvious effect on prey density.

DYNAMICAL PROPERTIES OF A STOCHASTIC PREDATOR-PREY MODEL WITH FUNCTIONAL RESPONSE

A stochastic prey-predator model with functional response is investigated in this paper. A complete threshold analysis of coexistence and extinction is obtained. Moreover, we point out that the stochastic predator-prey model undergoes a stochastic Hopf bifurcation from the viewpoint of numerical simulations. Some numerical simulations are carried out to support our results.

Periodic solution and stationary distribution for stochastic predator-prey model with modified Leslie-Gower and Holling type II schemes

In this paper, a stochastic predator-prey system with modified Leslie-Gower and Holling type II schemes is studied. For the autonomous case, we prove that the system has a stationary distribution under some parametric restrictions. We also obtain conditions for the non-persistence of the system, and the results are illustrated by computer simulations. For the non-autonomous system with continuous periodic coefficients, sufficient conditions which guarantee the existence of periodic solution of the system are established.

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 Levy noise and impulsive toxicant input. We study the stability in distribution of solutions by inequality techniques and ergodic method. By comparison method and Ito’s formula, we obtain the sufficient conditions for the survival of each species. Some numerical simulations are introduced to show the theoretical results.

Erratum to the paper “Extinction and ergodic stationary distribution of a Markovian-switching prey-predator model with additional food for predator”

In this work we have studied a stochastic predator-prey model where the prey grows logistically in the absence of predator. All parameters but carrying capacity have been perturbed with telephone noise. The prey’s growth rate and the predator’s death rate have also been perturbed with white noises. Both of these noises have been proved extremely useful to model rapidly fluctuating phenomena Dimentberg (1988). The conditions under which extinction of predator and prey populations occur have been established. We also give sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution of the positive solution, red which in stochastic predator-prey systems means that the predator and prey populations can be persistent, that is to say, the predator and prey populations can be sustain a quantity that is neither too much nor too little. In our analysis, it is found that the environmental noise plays an important role in extinction as well as coexistence of prey and predator populations. It is shown in numerical simulation that larger white noise intensity will lead to the extinction of the population, while telephone noise may delay or reduce the risk of species extinction.

Extinction and ergodic stationary distribution of a Markovian-switching prey-predator model with additional food for predator

In this work we have studied a stochastic predator-prey model where the prey grows logistically in the absence of predator. All parameters but carrying capacity have been perturbed with telephone noise. The prey’s growth rate and the predator’s death rate have also been perturbed with white noises. Both of these noises have been proved extremely useful to model rapidly fluctuating phenomena Dimentberg (1988). The conditions under which extinction of predator and prey populations occur have been established. We also give sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution of the positive solution, red which in stochastic predator-prey systems means that the predator and prey populations can be persistent, that is to say, the predator and prey populations can be sustain a quantity that is neither too much nor too little. In our analysis, it is found that the environmental noise plays an important role in extinction as well as coexistence of prey and predator populations. It is shown in numerical simulation that larger white noise intensity will lead to the extinction of the population, while telephone noise may delay or reduce the risk of species extinction.

Dynamical behavior of stochastic predator-prey models with distributed delay and general functional response

In this article, we study two stochastic predator-prey models with distributed delay and general functional response. For the nonautonomous periodic case of the system, by utilizing Khasminskii’s theory of periodic solution, we prove that the system admits a positive T-periodic solution. For the system which is perturbed by both white and colored noises, we establish sufficient conditions for the existence of positive recurrence of the solutions to the system. The existence of positive recurrence implies stochastic weak stability.

Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays

In this work a stochastic Holling-Ⅱ type predator-prey model with infinite delays and feedback controls is investigated. By constructing a Lyapunov function, together with stochastic analysis approach, we obtain that the stochastic controlled predator-prey model admits a unique global positive solution. We then utilize graphical method and stability theorem of stochastic differential equations to investigate the globally asymptotical stability of a unique positive equilibrium for the stochastic controlled predator-prey system. If the stochastic predator-prey system is globally stable, then we show that using suitable feedback controls can alter the position of the unique positive equilibrium and retain the stable property. If the predator-prey system is destabilized by large intensities of white noises, then by choosing the appropriate values of feedback control variables, we can make the system reach a new stable state. Some examples are presented to verify our main results.

Stationary distribution and extinction of a stochastic ratio-dependent predator–prey system with stage structure for the predator

In this paper, we deal with a stochastic predator–prey model with stage structure for predator population and ratio-dependent functional response. The proposed mathematical model consists of a system of three stochastic differential equations to stimulate the interactions between prey population, immature predator and mature predator population. We first establish sufficient conditions for the existence and uniqueness of the positive solutions by constructing an appropriate Lyapunov function. Then we extend the existence of stationary distribution under certain parametric restrictions. We also obtain the sufficient conditions for extinction of the predator populations. Finally, numerical simulations have been carried out to validate our analytical findings.

Analysis of a stochastic predator–prey model with prey subject to disease and Lévy noise

We consider a non-autonomous predator–prey model, with prey subject to the disease and Lévy noise. We show the existence of global positive solution and stochastic boundedness. Then, we examine the asymptotic properties of the solution. Finally, we offer sufficient conditions for persistence and extinction.

Dynamics of a stochastic predator–prey model with distributed delay and Markovian switching

In this paper, we study a stochastic predator–prey model with distributed delay and Markovian switching. Firstly, we obtain sufficient criteria for weak persistence of the predator population. Then in the case of weak persistence, we establish sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. Finally, we establish sufficient conditions for extinction of the predator population.

An approach for measuring corporation financial stability by Econophysics and Bayesian method

Corporation financial stability was investigated by the method of Econophysics and Bayesian approach in this paper. A stochastic predator–prey model was built to describe the corporation financial condition. The mean limiting extinction time was proposed to measure the corporation financial stability. The model parameters were estimated by using Bayesian approach with real corporation financial data. Experimental results show that: (i) an increase of system risk corresponds to a decrease of the corporation stability; (ii) the optimal initial financial condition enhances the corporation stability to its maximum extent; (iii) under the optimal initial financial condition, the optimal system parameters and system risk strength can be observed.

Analysis of a stochastic predator\u2013prey population model with Allee effect and jumps

This paper is concerned with a stochastic predator–prey model with Allee effect and Lévy noise. First, by the comparison theorem of stochastic differential equations, we prove that the model has a unique global positive solution starting from the positive initial value. Then we investigate the asymptotic pathwise behavior of the model by the generalized exponential martingale inequality and the Borel–Cantelli lemma. Next, we establish the conditions under which predator and prey populations are extinct. Furthermore, we show that the global positive solution is stochastically ultimate bounded under some conditions by using the Bernoulli equation and Chebyshev’s inequality. At last, we introduce some numerical simulations to support the main results obtained. The results in this paper generalize and improve the previous related results.

On a Multi-Delay Lotka-Volterra Predator-Prey Model with Feedback Controls and Prey Diffusion

This article is focusing on a class of multi-delay predator-prey model with feedback controls and prey diffusion. By developing some new analysis methods and using the theory of differential inequalities as well as constructing a suitable Lyapunov function, we establish a set of easily verifiable sufficient conditions which guarantee the permanence of the system and the globally attractivity of positive solution for the predator-prey system. Furthermore, some conditions for the existence, uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria are new, general, and easily verifiable. Finally, we still solve numerically the corresponding stochastic predator-prey models with multiplicative noise sources, and obtain some new interesting dynamical behaviors of the system.

Stationary distribution of a stochastic predator–prey model with distributed delay and higher order perturbations

In this paper, we study a stochastic predator–prey model with distributed delay and higher order perturbations. By the linear chain technique, we transform the stochastic system with weak kernel case into an equivalent degenerate system which contains three equations. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model.

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.

STATIONARY DISTRIBUTION AND PERSISTENCE OF A STOCHASTIC PREDATOR-PREY MODEL WITH A FUNCTIONAL RESPONSE

A stochastic predator-prey model with a functional response is investigated in this paper. The asymptotic properties of the stochastic model are considered here. Under some conditions, we show that the stochastic model is persistent in mean. Moreover, the existence of stationary distribution to the model is obtained. Simulations are also carried out to confirm our analytical results.