Stochastic Predator-prey System Research Articles

Multiple stochastic and inverse stochastic resonances with transition phenomena in complex corporate financial systems.

This study examines the role of periodic information, the mechanism of influence, stochastic resonance, and its controllable analysis in complex corporate financial systems. A stochastic predator-prey complex corporate financial system model driven by periodic information is proposed. Additionally, we introduce signal power amplification to quantify the stochastic resonance phenomenon and develop a method for analyzing stochastic resonance in financial predator-prey dynamics within complex corporate financial systems. We optimize a simplified integral calculation method to enhance the proposed model's performance, which demonstrates superiority over benchmark models based on empirical evidence. Based on stochastic simulations and numerical calculations, we can observe multiple stochastic and multiple inverse stochastic resonances. Furthermore, variations in initial financial information, periodic information frequency, and corporate growth capacity induced stochastic resonance and inverse stochastic resonance. These variations also led to state transitions between the two resonance behaviors, indicating transition phenomena. These findings suggest the potential for regulating and controlling stochastic and inverse stochastic resonance in complex corporate finance, enabling controllable stochastic resonance behaviors.

A Levy process approach coupled to the stochastic Leslie–Gower model

This paper focuses on a two-dimensional Leslie–Grower continuous-time stochastic predator–prey system with Lévy jumps. Firstly, we prove that there exists a unique positive solution of the system with a positive initial value. Then, we establish sufficient conditions for the mean stability and extinction of the considered system. Numerical algorithms of higher order are elaborated. The obtained results show that Lévy jumps significantly change the properties of population systems.

Threshold dynamics and probability density functions of a stochastic predator–prey model with general distributed delay

Predator–prey interactions are among the most complicated interactions between biological species, in which there may be both delay digestion and stochastic fluctuation. In the literature, due to the computational complexity and the lack of available methods, few papers have directly considered the predator–prey model for the combination of general infinite time delays with stochasticity. In this sense, this work is devoted to studying the long-term qualitative behavior of a stochastic predator–prey system with general infinite distributed delay. First, we establish sufficient and necessary conditions for exponential extinction and persistence of species. Notably, not only do our results cover the predator–prey model without stochastic noises and time delays, but also reveal the impact of environmental noises on population dynamics. Moreover, there are some challenges to analyze the stationary distribution of the numbers of prey and predator under species persistence. Based on the recent work (Zhou et al., 2020) where the density function of a three-dimensional avian influenza model with non-degenerate diffusion is concerned, we further generalize the relevant theories to a five-dimensional population system with degenerate diffusion. Via solving the corresponding Fokker–Planck equations, several approximate expressions of the density function of the stochastic predator–prey system with special distribution delay kernels are obtained. Finally, we provide some numerical simulations to supplement the analytical results and study the impact of stochastic noises.

Dynamics of a Prey–Predator Model with Group Defense for Prey, Cooperative Hunting for Predator, and Lévy Jump

A stochastic predator–prey system with group cooperative behavior, white noise, and Lévy noise is considered. In group cooperation, we introduce the Holling IV interaction term to reflect group defense of prey, and cooperative hunting to reflect group attack of predator. Firstly, it is proved that the system has a globally unique positive solution. Secondly, we obtain the conditions of persistence and extinction of the system in the sense of time average. Under the condition that the environment does not change dramatically, the intensity of cooperative hunting and group defense needs to meet certain conditions to make both predators and preys persist. In addition, considering the system without Lévy jump, it is proved that the system has a stationary distribution. Finally, the validity of the theoretical results is verified by numerical simulation.

Study on Dynamic Behavior of a Stochastic Predator–Prey System with Beddington–DeAngelis Functional Response and Regime Switching

In this paper, by introducing environmental white noise and telegraph noise, we proposed a stochastic predator–prey model with the Beddington–DeAngelis type functional response and investigated its dynamical behavior. Persistence and extinction are two core contents of population model research, so we analyzed these two properties. The sufficient conditions of the strong persistence in the mean and extinction were established and the threshold between them was obtained. Moreover, we took stability into account and, by means of structuring a suitable Lyapunov function with regime switching, we proved that the stochastic system has a unique stationary distribution. Finally, numerical simulations were used to illustrate our theoretical results.

Dynamic behavior of stochastic predator-prey system

<abstract> <p>In this paper, a stochastic predator-prey system with mutual interference is studied, which provides guidance on creating appropriate biodegradable environments. By utilizing Mawhin's coincidence degree theorem and constructing a suitable Lyapunov function, a Volterra model with mutual interference is studied. A few sufficient conditions are obtained for existence, extinction and global asymptomatic stability of the positive solution of the model. Then we perform various numerical simulations to verify the stochastic and deterministic systems are global asymptotically stable. It is significant that such a model is firstly proposed with stochastic mutual interference.</p> </abstract>

双捕食者与双食饵的随机捕食系统动力学

双捕食者与双食饵的随机捕食系统动力学

Analysis of a stochastic predator–prey system with fear effect and Lévy noise

This paper studies a stochastic predator–prey model with Beddington–DeAngelis functional response, fear effect, and Lévy noise, where the fear is of prey induced by predator. First, we use Itô’s formula to prove the existence and uniqueness of a global positive solution and its moment boundedness. Next, sufficient conditions for the persistence and extinction of both species are given. We further investigate the stability in distribution of our system. Finally, we verify our analytical results by exhaustive numerical simulations.

A stochastic predator–prey system with modified LG-Holling type II functional response

In this paper, a stochastic two-predator one-prey system with modified Leslie–Gower and Holling-type II functional response is proposed, which is randomly disturbed by the well-known mean-reverting Ornstein–Uhlenbeck process. By Itoˆ’s integral formula, stochastic comparison theorem, the strong law of large number theorem for martingales, and modeling and analysis methods in stochastic differential equations, the existence and uniqueness of the global positive solution for the system are discussed. Then, the additional conditions for the persistence in the mean and extinction of the system are obtained, respectively. Besides, the effects of the speed of reversion and the intensity of volatility in the Ornstein–Uhlenbeck process on the dynamics of the system are investigated. Furthermore, the ergodic stationary distribution of the system under a low-level intensity of stochastic noise is also derived, which indicates that x, y1 and y2 will be persist and fluctuate around the positive values. Finally, a series of numerical examples are provided to verify the correctness of the theoretical analysis.

Periodic measure of a stochastic non-autonomous predator–prey system with impulsive effects

In this paper, we study a stochastic predator–prey system with impulsive effects and time periodic coefficients. By the properties of periodic Markov processes, Krylov–Bogolyubov’s method and Doob’s Theorem, the existence and uniqueness of periodic measure of the system is presented under the condition of the stochastic persistence of the system. Numerical simulations are given to verify the effectiveness of the theoretical results and to show the effects of the stochastic perturbation and impulsive perturbation on the persistence and periodic measure of the system.

Dynamics and optimal harvesting of a stochastic predator-prey system with regime switching, S-type distributed time delays and Lévy jumps

<abstract><p>This work is concerned with a stochastic predator-prey system with S-type distributed time delays, regime switching and Lévy jumps. By use of the stochastic differential comparison theory and some inequality techniques, we study the extinction and persistence in the mean for each species, asymptotic stability in distribution and the optimal harvesting effort of the model. Then we present some simulation examples to illustrate the theoretical results and explore the effects of regime switching, distributed time delays and Lévy jumps on the dynamical behaviors, respectively.</p></abstract>

Dynamics of an Impulsive Stochastic Predator–Prey System with the Beddington–DeAngelis Functional Response

Taking impulsive effects into account, an impulsive stochastic predator–prey system with the Beddington–DeAngelis functional response is proposed in this paper. First, the impulsive system is transformed into an equivalent system without pulses. Then, by constructing suitable functionals and applying the extreme-value theory of quadratic functions, sufficient conditions on the existence of periodic Markovian processes are provided. The uniform continuity and global attractivity of solutions are also investigated. Additionally, we investigate the extinction and permanence in the mean of all species with the help of comparison methods and inequality techniques. Sufficient conditions on the existence and ergodicity of the stationary distribution of solutions for the autonomous and non-impulsive case are given. Finally, numerical simulations are performed to illustrate the main results.

Dynamics of stochastic predator-prey systems with continuous time delay

The present paper deals with the problem of a stochastic predator-prey model with continuous time delay. In the case that the integral kernel is the function ae−at, the persistence in the mean and extinction of this solution are derived by making use of Lyapunov analysis methods. Numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.

Long-Time Behaviours of a Stochastic Predator-Prey System with Holling-Type II Functional Response and Regime Switching

Considering the impacts of white noise, Holling-type II functional response, and regime switching, we formulate a stochastic predator-prey model in this paper. By constructing some suitable functionals, we establish the sufficient criteria of the stationary distribution and stochastic permanence. By numerical simulations, we illustrate the results and analyze the influence of regime switching on the dynamics.

Analysis of a stochastic predator-prey system with mixed functional responses and Lévy jumps

In this paper, we consider a stochastic two predator-one prey system consisting of prey, intermediate predator and top predator with Levy jumps. Here we consider Ratio-dependent function response between intermediate predator and top predator and other function responses are assumed to be linear. Firstly, we prove that the existence and boundedness of pth moment of the positive solution. Then under some assumptions, we establish sufficient criteria for the extinction of the system. The results reveal an important property that the Levy jumps are unfavorable for the existence of species. Furthermore, we establish sufficient condition for the asymptotically stable in distribution under certain conditions. Finally, some numerical simulations are introduced to demonstrate the theoretical results.

Threshold behavior of a stochastic predator–prey system with prey refuge and fear effect

In this paper, we investigate the threshold behavior of a stochastic predator–prey system with prey refuge, fear effect and non-constant mortality rate. First, we obtain thresholds which can determine the extinction and persistence in the mean of prey and predator. Then we establish the threshold for the existence of a unique ergodic stationary distribution via constructing suitable Lyapunov functions. We can draw conclusions: (1) the survival rate of prey can be improved by increasing the strength of refuge, decreasing the cost of fear or reducing the intensity of white noise; (2) the survival rate of predator can be improved by weakening the strength of refuge, increasing the cost of fear or reducing the intensity of white noise.

Influence of the fear factor on the dynamics of a stochastic predator–prey model

In this paper, we construct and analyze a stochastic predator–prey system where apart from direct predation the prey population is affected by the fear induced from predators. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological interpretation, the existence of a stationary distribution indicates that both the prey and predator species will be persistent and coexistent in the long term.

Coexistence and extinction of a periodic stochastic predator\u2013prey model with general functional response

This note deals with a stochastic predator–prey system with periodic coefficients and general functional response, and provides the threshold between coexistence and extinction. The result refines and evolves some prior investigations.

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.

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.