Stochastic Predator-prey Model Research Articles

Large deviations for stochastic predator–prey model with Lévy noise

This paper discusses the large deviations for stochastic predator–prey model driven by multiplicative Lévy noise. Using Galerkin approximation, we initially prove the existence and uniqueness of solution. Due to the equivalence between Laplace principle and large deviation principle under a Polish space, the method of weak convergence has been followed in order to establish our results for this coupled system of equations.

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.

Dynamical bifurcation of a stochastic Holling-II predator–prey model with infinite distributed delays

This paper serves as a continuation and expansion of the previous achievement by B. Han and D. Jiang in their article (Han and Jiang, 2024). Our initial step involves transforming the more complex stochastic Holling-II model featuring a stronger kernel into a simplified yet degenerate stochastic system composed of five interconnected equations. For the deterministic component of this model, we delve into an extensive examination of the local asymptotic stability of its positive equilibrium state. Subsequently, for the stochastic model, we derive critical conditions that determine the thresholds for exponential extinction and persistence of both predator and prey populations. Importantly, our findings not only encompass scenarios where there are no stochastic disturbances but also shed light on how environmental noise impacts the population dynamics within the predator–prey system. Through the exploration of the homologous Fokker–Planck equations, we present approximate representations characterizing the probability density function of the stochastic predator–prey model. To substantiate these theoretical advancements, several illustrative examples are provided, offering numerical elucidations of our proposed results and emphasizing the profound effects of stochastic noises and some important parameters on the behaviors of the stochastic model.

Dynamical Analysis of Stochastic Predator-prey Model with Scavenger

In this paper, we studied the dynamic properties of predator-prey and scavenger three species system by using ergodic invariant measures. Pengyu Ma. find the five points of dynamical bifurcation of the stochastic model, which happened between extinction and survival of each species. Environmental noise was added and proved by the fact that driving force produced by environmental noise influence the system and it was find that system may extinct or partially extinct. Here, we have analysed the stochastic bifurcation phenomena of the prey-predator with scavenger system from the nature of dynamic bifurcation. The phase plots and time diagram plotted for the different values of parameters. We have verified all the results by numerical simulations.

Continuous dependence of stationary distributions on parameters for stochastic predator–prey models

AbstractThis research studies the robustness of permanence and the continuous dependence of the stationary distribution on the parameters for a stochastic predator–prey model with Beddington–DeAngelis functional response. We show that if the model is extinct (resp. permanent) for a parameter, it is still extinct (resp. permanent) in a neighbourhood of this parameter. In the case of extinction, the Lyapunov exponent of predator quantity is negative and the prey quantity converges almost to the saturated situation, where the predator is absent at an exponential rate. Under the condition of permanence, the unique stationary distribution converges weakly to the degenerate measure concentrated at the unique limit cycle or at the globally asymptotic equilibrium when the diffusion term tends to 0.

Dynamics caused by the mean-reverting Ornstein–Uhlenbeck process in a stochastic predator–prey model with stage structure

There are many species that go through different life stages in nature, they have different ecological characteristics in different stages, besides, stochastic variation exists in the natural environment. Based on this biological phenomenon, we develop and investigate dynamics in a predator–prey system with stage structure and the mean-reverting Ornstein–Uhlenbeck processes. We prove that this system satisfies the existence and uniqueness of globally positive equilibrium. The sufficient conditions for the existence of a stationary Markov process in stochastic system are attained. We deal with explicit expression and the existence of density function of the system which make a great contribution to biological intrinsic essence for the system. In addition, the sufficient criteria for extinction of the predator populations is derived. Finally, numerical simulations are furnished to illustrate the analysis results.

Bayesian Inference for Functional Response of Stochastic Predator-Prey Model using Non-Informative Prior

Bayesian Inference for Functional Response of Stochastic Predator-Prey Model using Non-Informative Prior

A stochastic predator–prey model with distributed delay and Ornstein–Uhlenbeck process: Characterization of stationary distribution, extinction, and probability density function

As the evolution of species relies on not only the current state but also the past information, it is more reasonable and realistic to take delay into an ecological model. This paper deals with a stochastic predator–prey model that considers the distribution delay and assume that the intrinsic growth rate and the death rate in the model are governed by Ornstein–Uhlenbeck process to simulate the random factors in the environment. Based on the existence and uniqueness of the global solution to the model and the boundedness of the order moments of the solution, several conditions are established to analyze the survival of the species. Specifically, a criteria for the existence of the stationary distribution to the stochastic system is established by constructing some suitable Lyapunov functions. And the analytical expression of the probability density function of the model around the quasi‐equilibrium is obtained. Furthermore, the extinction of species in the model is also explored. Finally, numerical simulations are carried out to illustrate the theoretical results obtained in this paper.

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.

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.

Impact of Delay on Stochastic Predator–Prey Models

Ordinary differential equations (ODE) have long been an important tool for modelling and understanding the dynamics of many real systems. However, mathematical modelling in several areas of the life sciences requires the use of time-delayed differential models (DDEs). The time delays in these models refer to the time required for the manifestation of certain hidden processes, such as the time between the onset of cell infection and the production of new viruses (incubation periods), the infection period, or the immune period. Since real biological systems are always subject to perturbations that are not fully understood or cannot be explicitly modeled, stochastic delay differential systems (SDDEs) provide a more realistic approximation to these models. In this work, we study the predator–prey system considering three time-delay models: one deterministic and two types of stochastic models. Our numerical results allow us to distinguish between different asymptotic behaviours depending on whether the system is deterministic or stochastic, and in particular, when considering stochasticity, we see that both the nature of the stochastic systems and the magnitude of the delay play a crucial role in determining the dynamics of the system.

Autonomous and non-autonomous modified Leslie-type stochastic predator-prey model with foraging arena scheme.

This paper aims to investigate how the external random noise affects the dynamics of the predator-prey model with a modified Leslie and foraging arena scheme. The autonomous and non-autonomous systems are both considered. First, some asymptotic behaviors of two species are explored including the threshold point. Then, the existence of an invariant density is deduced, based on the theory elaborated in Pike and Luglato (1987). Moreover, the famous LaSalle-type theorem is applied to investigate weak extinction, which requires weaker parametric restrictions. A numerical study is conducted to illustrate our theory.

A stochastic predator–prey model with Ornstein–Uhlenbeck process: Characterization of stationary distribution, extinction and probability density function

This paper concerns with a stochastic system modeling the population dynamical behavior of one prey and two predators. In this paper, we adopt a special method to simulate the effect of the environmental interference to the system instead of using the linear functions of white noise, i.e., the growth rate of the prey and the death rates of the two predators are all governed by Ornstein–Uhlenbeck processes, which is more practical and interesting. The dynamical behavior among the three species in the model is discussed in detail. The existence and uniqueness, moment boundedness and long time asymptotic behavior of the unique global solution are investigated. Through rigorous proof and theoretical derivation, some sharp sufficient conditions for the persistence and extinction of the three species have been established. Furthermore, we obtain that when one or two predators die out, the remaining species can be stable in time average under same conditions. Moreover, by constructing some suitable Lyapunov functions, the sufficient conditions for the existence of the stationary distribution are explicitly determined. Under the same parametric restrictions, it is worth noting that we obtain the six-dimensional probability density function of the stochastic one-prey two-predator model around the quasi-equilibrium E∗, which can reflect most statistical properties. Finally, several numerical simulations are carried out to illustrate the theoretical results.

A Note on a Strong Persistence of Stochastic Predator-Prey Model with Jumps

We study the non-autonomous stochastic predator-prey model with a modified version of Leslie-Gower term and Holling-type II functional response driven by the system of stochastic differential equations with white noise, centered and non-centered Poisson noises. The sufficient conditions of strong persistence in the mean of the solution to the considered system are obtained.

Three-Species Predator–Prey Stochastic Delayed Model Driven by Lévy Jumps and with Cooperation among Prey Species

Our study focuses on analyzing the behavior of a stochastic predator–prey model with a time delay and logistic growth of prey, influenced by Lévy noise. Initially, we establish the existence, uniqueness, and boundedness of a positive solution that spans globally. Subsequently, we explore the conditions under which extinction occurs, and identify adequate criteria for persistence. Finally, we validate our theoretical findings through numerical simulations, which also helps illustrate the dynamics of the stochastic delayed predator–prey model based on different criteria.

A stochastic predator–prey model with two competitive preys and Ornstein–Uhlenbeck process

In this paper, a stochastic predator–prey model with two competitive preys and Ornstein–Uhlenbeck process is formulated and analysed, which is used to obtain a better understanding of the population dynamics. At first, we validate that the stochastic system has a unique global solution with any initial value. Then we analyse the stochastic dynamics of the model in detail, including pth moment boundedness, asymptotic pathwise estimation in turn. After that, we obtain sufficient conditions for the existence of a stationary distribution of the system by adopting stochastic Lyapunov function methods. In addition, under some mild conditions, we derive the specific form of covariance matrix in the probability density near the quasi-positive equilibrium of the stochastic system. Finally, numerical illustrative examples are depicted to confirm our theoretical findings.

Stationary distribution and global stability of stochastic predator-prey model with disease in prey population

In this paper, a new stochastic four-species predator-prey model with disease in the first prey is proposed and studied. First, we present the stochastic model with some biological assumptions and establish the existence of globally positive solutions. Moreover, a condition for species to be permanent and extinction is provided. The above properties can help to save the dangered population in the ecosystem. Through Lyapunov functions, we discuss the asymptotic stability of a positive equilibrium solution for our model. Furthermore, it is also shown that the system has a stationary distribution and indicating the existence of a stable biotic community. Finally, our results of the proposed model have revealed the effect of random fluctuations on the four species ecosystem when adding the alternative food sources for the predator population. To illustrate our theoretical findings, some numerical simulations are presented.

DYNAMICAL BEHAVIORS OF A STOCHASTIC PREDATOR-PREY MODEL WITH ANTI-PREDATOR BEHAVIOR

In this paper, a stochastic predator-prey model is proposed and studied, where the model has anti-predator behavior. By constructing a suitable Lyapunov function, combined with the Itô’s formula and the stochastic comparison theorem, the existence and uniqueness of the global positive solution of the system are proved. Then the stochastic boundedness of the system is established, and we discussed the asymptotic behavior of the solution which fluctuates around the equilibrium point of the deterministic model. Moreover, we provide sufficient conditions for the persistence and extinction of the predator and prey. Finally, the results obtained in this paper are verified by numerical simulation, and the anti-predator behavior and stochastic perturbation are analyzed as well.

Вимирання та виживання у залежній від щільності популяціії хижака стохастичній моделі хижак-жертва із стрибками

The non-autonomous stochastic density dependent predator-prey model with Holling-type II functional response disturbed by white noise, centered and non-centered Poisson noises is investigated. Corresponding system of stochastic differential equations has a unique, positive, global (no explosions in a finite time) solution. Sufficient conditions are obtained for extinction, non-persistence in the mean, weak and strong persistence in the mean of a predator and prey population densities in the considered stochastic predator-prey model.

Analysis of stochastic disease including predator-prey model with fear factor and Lévy jump.

In this paper, we investigate the dynamical properties of a stochastic predator-prey model with a fear effect. We also introduce infectious disease factors into prey populations and distinguish prey populations into susceptible prey and infected prey populations. Then, we discuss the effect of Lévy noise on the population considering extreme environmental situations. First of all, we prove the existence of a unique global positive solution for this system. Second, we demonstrate the conditions for the extinction of three populations. Under the conditions that infectious diseases are effectively prevented, the conditions for the existence and extinction of susceptible prey populations and predator populations are explored. Third, the stochastic ultimate boundedness of system and the ergodic stationary distribution without Lévy noise are also demonstrated. Finally, we use numerical simulations to verify the conclusions obtained and summarize the work of the paper.