Dynamics of a Stochastic Predator–Prey Model with Stage Structure for Predator and Holling Type II Functional Response

In this paper, we develop and study a stochastic predator–prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results.     

Stationary distribution and extinction for a food chain chemostat model with random perturbation

In this paper, we study the dynamical behavior of a stochastic food chain chemostat model, in which the white noise is proportional to the variables. Firstly, we prove the existence and uniqueness of the global positive solution. Then by constructing suitable Lyapunov functions, we show the system has a unique ergodic stationary distribution. Furthermore, the extinction of microorganisms is discussed in two cases. In one case, both the prey and the predator species are extinct, and in the other case, the prey species is surviving and the predator species is extinct. Finally, numerical experiments are performed for supporting the theoretical results.     

Stationary distribution and extinction of a stochastic one-prey two-predator model with Holling type II functional response

In this article, we investigate a stochastic one-prey two-predator model with Holling type II functional response. We first establish sufficient conditions for persistence and extinction of prey and predator populations, then by constructing a suitable stochastic Lyapunov function, we establish sharp sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model. The results show that the smaller white noise can ensure the persistence of prey and predator populations while the larger white noise can lead to the extinction of prey and predator populations.     

Stationary distribution and extinction of a stochastic predator–prey model with distributed delay

In this paper, we focus on a stochastic predator–prey model with distributed delay. We first obtain the existence of a stationary distribution to the positive solutions by stochastic Lyapunov function method. Then we establish sufficient conditions for extinction of the predator population, that is, the prey population is survival and the predator population is extinct.     

Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects

In this paper, we propose a stochastic non-autonomous Lotka–Volterra predator–prey model with impulsive effects and investigate its stochastic dynamics. We first prove that the subsystem of the system has a unique periodic solution which is globally attractive. Furthermore, we obtain the threshold value in the mean which governs the stochastic persistence and the extinction of the prey–predator system. Our results show that the stochastic noises and impulsive perturbations have crucial effects on the persistence and extinction of each species. Finally, we use the different stochastic noises and impulsive effects parameters to provide a series of numerical simulations to illustrate the analytical results.     

Dynamical behavior of a stochastic food chain chemostat model with Monod response functions

This paper studies a food chain chemostat model with Monod response functions, which is perturbed by white noise. Firstly, we prove the existence and uniqueness of the global positive solution. Then sufficient conditions for the existence of a unique ergodic stationary distribution are established by constructing suitable Lyapunov functions. Moreover, we consider the extinction of microbes in two cases. In the first case, both the predator and prey species are extinct. In the second case, only the predator species is extinct, and the prey species survives. Finally, numerical simulations are carried out to illustrate the theoretical results.     

Dynamics of a stochastic three species prey-predator model with intraguild predation

Intraguild predation is ubiquitous in many ecological communities. This paper is concerned with a stochastic three species prey-predator model with intraguild predation. The model involves a prey, an intermediate predator which preys on only prey and an omnivorous top predator which preys on both prey and intermediate predator. First, we show the existence of a unique positive global solution of the model. Then we mainly establish the sufficient conditions for the extinction and persistence in the mean of each population. Moreover, we show that the model is stable in distribution. Finally, some numerical simulations are given to illustrate the main results.     

一种具有脉冲扰动的恒化器模型的动力学性质

在这篇文章中,我们提出并分析了一个具有捕食者,食饵和既有周期脉冲输入又有周期脉冲输出营养液的恒化器模型.我们得到了一种微生物和营养液共存的周期解,同时,也得到两种微生物都绝灭的周期解,而且建立了周期解稳定的充分条件.最后,我们给出了一个简单的讨论.     

Role of environmental disturbance in an eco‐epidemiological model with disease from external source

An eco‐epidemiological model with random environmental disturbance is proposed and analyzed. We assume that the susceptible prey population can acquire infection both from external sources and from internal transmission of the disease. It is also assumed that there is no recovery of the disease, and the consumption of diseased prey has a deleterious effect on the predator population. The conditions for the extinction of the predator and the prey populations are worked out. The most important observation of the present investigation is that oscillatory behavior of the populations observed in deterministic framework undergoes stable coexistence in the stochastic framework. Copyright © 2012 John Wiley & Sons, Ltd.     

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

Abstract

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.     

Persistence of two prey–one predator system with ratio‐dependent predator influence

In this study, we consider a mathematical model of two competing prey and one predator system where the prey species follow Lotka–Volterra‐type dynamics and the predator uptake functions are ratio dependent. We have derived the conditions for existence of different boundary equilibria and discussed their global behaviour. The sufficient condition for permanent co‐existence of all the species is derived. Finally, we have discussed the possibility of extinction of the species from the system. Copyright © 2000 John Wiley & Sons, Ltd.     

A remark on a stochastic predator–prey system with time delays

An autonomous stochastic predator–prey model with time delays is investigated. Almost sufficient and necessary conditions for stability in the mean and extinction of each population are established. Numerical simulations are introduced to support the results.     

The dynamical behavior of a predator–prey system with Gompertz growth function and impulsive dispersal of prey between two patches

In this paper, we investigate a predator–prey model with Gompertz growth function and impulsive dispersal of prey between two patches. Using the dynamical properties of single‐species model with impulsive dispersal in two patches and comparison principle of impulsive differential equations, necessary and sufficient criteria on global attractivity of predator‐extinction periodic solution and permanence are established. Finally, a numerical example is given to illustrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment

This paper intends to develop a new method to obtain the threshold of an impulsive stochastic chemostat model with saturated growth rate in a polluted environment. By using the theory of impulsive differential equations and stochastic differential equations, we obtain conditions for the extinction and the permanence of the microorganisms of the deterministic chemostat model and the stochastic chemostat model. We develop a new numerical computation method for impulsive stochastic differential system to simulate and illustrate our theoretical conclusions. The biological results show that a small stochastic disturbance can cause the microorganism to die out, that is, a permanent deterministic system can go to extinction under the white noise stochastic disturbance. The theoretical method can also be used to explore the threshold of some impulsive stochastic differential equations.     

STOCHASTIC OPTIMIZATION FOR MULTISPECIES FISHERIES IN THE BARENTS SEA

We present a multispecies stochastic model that suggests optimal fishing policy for two species in a three‐species predator–prey ecosystem in the Barents Sea. We employ stochastic dynamic programming to solve a three‐dimensional model, in which the catch is optimized by using a multispecies feedback strategy. Applying the model to the cod, capelin, and herring ecosystem in the Barents Sea shows that the optimal catch for the stochastic interaction model is more conservative than that implied by the deterministic model. We also find that stochasticity has a stronger effect on the optimal exploitation policy for prey (capelin) than for predator (cod).     

Migratory effect of middle predator in a tri‐trophic food chain model

A tri‐trophic food chain model in a two‐patch environment is considered. Although tri‐trophic food chain model is well studied, the study considering migration of middle predator is lacking. To the best of our knowledge, the present investigation is the first study in this direction. Both prey and predator density‐dependent migrations are considered to observe the effects on stability and persistence of the system. We observe that migration of middle predator has the ability to control chaos in tri‐trophic food chain model. Our results indicate that the chance of predator extinction enhances for prey density‐dependent middle predator migration. Copyright © 2010 John Wiley & Sons, Ltd.     

Extinction and permanence in a stochastic non-autonomous population system

A stochastic non-autonomous predator–prey system with Holling II functional response is investigated. Sufficient criteria for extinction and uniform weak persistence in the mean for each species are established. The acute persistence–extinction thresholds for each species are obtained in many cases.     

ACCOUNTING FOR TEMPERATURE IN PREDATOR FUNCTIONAL RESPONSES

ABSTRACT. A rational mechanism that integrates temperature‐mediated activity cycles into standard predator functional responses is presented. Daily temperature variations strongly influence times that predators can search for prey, and they affect the activity periods of prey, thereby modifying their detection by predators. Thus, key parameters in the functional response, the search time and the detection, become temperature‐dependent. These temperature mediated responses are included in discrete‐time population growth models, and it is shown how environmental temperature variations, such as those that may occur under global climate change, can affect population levels. As an illustration, a logistic growth model with a stochastic, temperature‐dependent predation term is examined, and the response to both average temperature levels and temperature variability is quantified. We infer, through simulations, that predation and prey abundance are strongly affected by mean temperature, temperature amplitudes, and increasing uncertainty in predicting temperature levels and variation, thus confirming many qualitative conclusions in the ecological literature. In particular, we show that increased temperature variability increases oscillations in the system and leads to increased probability of extinction of the prey.     

Stochastic persistence and stability analysis of a modified Holling–Tanner model

The article aims to study the basic dynamical features of a modified Holling–Tanner prey–predator model with ratio‐dependent functional response. We have proved the global existence of the solution for the deterministic model. The parametric restriction for persistence of both species is also obtained along with the proof of local asymptotic stability of the interior equilibrium point(s). Conditions for local bifurcations of interior equilibrium points are provided. The global dynamic behavior is examined thoroughly with supportive numerical simulation results. Next, we have formulated the stochastic model by perturbing the intrinsic growth rates of prey and predator populations with white noise terms. The existence uniqueness of solutions for stochastic model is established. Further, we have derived the parametric restrictions required for the persistence of the stochastic model. Finally, we have discussed the stochastic stability results in terms of the first and second order moments. Numerical simulation results are provided to support the analytical findings. Copyright © 2012 John Wiley & Sons, Ltd.     

A ratio‐dependent eco‐epidemiological model of the Salton Sea

Ratio‐dependent models set up a challenging issue for their rich dynamics incomparison to prey‐dependent models. Little attention has been paid so far to describe the importance of transmissible disease in ecological situation by considering ratio‐dependent models. In this paper, by assuming the predator response function as ratio‐dependent, we consider a model of a system of three non‐linear differential equations describing the time evolution of susceptible and infected Tilapia fish population and their predator, the Pelican. Existence and stability analysis of different equilibria of the system lead to different realistic thresholds in terms of system parameters. The condition for extinction of the species is also worked out. Our analytical and numerical studies may be helpful to chalk out suitable control strategies for minimizing the extinction of the Pelicans. We also suggest that supply of alternative food source for predator population may be used as a possible solution to save the predator from their extinction. Copyright © 2005 John Wiley & Sons, Ltd.