The Stochastic Asymptotic Stability Analysis in Two Species Lotka-Volterra Model

The asymptotic stability of two species stochastic Lotka-Volterra model is explored in this paper. Firstly, the Lotka-Volterra model with random parameter is built and reduced into the equivalent deterministic system by orthogonal polynomial approximation. Then, the linear stability theory and Routh-Hurwitz criterion for nonlinear deterministic systems are applied to the equivalent one. At last, at the aid of Lyapunov second method, we obtain that as the random intensity or statistical parameter of random variable is changed, the stability about stochastic Lotka-Volterra model is different from the deterministic system.

On the Dynamics of a Stochastic Ratio-Dependent Predator-Prey System with Infection for the Prey

In this paper, we investigate the dynamics of a stochastic predator-prey model with ratio-dependent functional response and disease in the prey. Firstly, we prove the existence and uniqueness of the positive solution for the stochastic model by using conventional methods. Then we obtain the threshold <img alt="" src="Edit_0a62b9be-7934-457b-aca3-af3420f5b5ee.png" /> for the infected prey population, that is, the disease will tend to extinction if <img alt="" src="Edit_e6cd63f6-de07-42be-a22a-8750d6c8aac9.png" />< 1, and it will exist in the long time if <img alt="" src="Edit_5964fdd8-a9fe-4dc2-b897-f4206f046f65.png" />> 1. Finally, the sufficient condition on the existence of a unique ergodic stationary distribution is obtained, which indicates that all the populations are permanent in the time mean sense. Numerical simulations are conducted to verify our analysis results.

Global Stability of A Stochastic Predator-prey Model with Stage-structure

In this paper, a stochastic predator-prey model with stage structure for predatorand ratio-dependent functional response is concerned. Sufficient conditions for the globalasymptotic stability of positive equilibrium are established. Some numerical simulations arecarried out to illustrate 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.

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.

EXISTENCE AND UNIQUENESS OF THE POSITIVE STEADY STATE SOLUTION FOR A LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH A CROWDING TERM

This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation.We obtain a critical value λ1^D(Ω0),and demonstrate that the existence of the predator inΩ0 only depends on the relationship of the growth rateμof the predator and λ1^D(Ω0),not on the prey.Furthermore,whenμ<λ1^D(Ω0),we obtain the existence and uniqueness of its positive steady state solution,while whenμ≥λ1^D(Ω0),the predator and the prey cannot coexist inΩ0.Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding regionΩ0,which is different from that of the classical Lotka-Volterra predator-prey model.

Persistence and periodic measure of a stochastic predator-prey model with Beddington-DeAngelis functional response

In this paper,we study a stochastic predator-prey model with Beddington-DeAngelis functional response and time-periodic coefficients.By analyzing the stability of the solution on the boundary and some stochastic estimates,the threshold conditions for the time-average persistence in probability and extinction of each population are established.Furthermore,the existence of a unique periodic measure of the model is also presented under the condition of the time-average persistence in probability of the model.Several numerical simulations are given to verify the effectiveness of the theoretical results and to illustrate the effects of the white noises on the persistence and periodic measure of the model.

Predicting solutions of the Lotka-Volterra equation using hybrid deep network

Prediction of Lotka-Volterra equations has always been a complex problem due to their dynamic properties.In this paper,we present an algorithm for predicting the Lotka-Volterra equation and investigate the prediction for both the original system and the system driven by noise.This demonstrates that deep learning can be applied in dynamics of population.This is the first study that uses deep learning algorithms to predict Lotka-Volterra equations.Several numerical examples are presented to illustrate the performances of the proposed algorithm,including Predator nonlinear breeding and prey competition systems,one prey and two predator competition systems,and their respective systems.All the results suggest that the proposed algorithm is feasible and effective for predicting Lotka-Volterra equations.Furthermore,the influence of the optimizer on the algorithm is discussed in detail.These results indicate that the performance of the machine learning technique can be improved by constructing the neural networks appropriately.

Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay

A delayed Lotka-Volterra two-species predator-prey system of population allelopathy with discrete delay is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs). Finally, some numerical simulations are carried out for illustrating the theoretical results.

Dynamics of stochastic predator-prey models with distributed delay and stage structure for prey

In this paper,we analyze two stochastic predator-prey models with distributed delay and stage structure for prey.For the nonautonomous periodic case of the model,by using Khasminskii’s theory of periodic solution,we show that the system has at least one positive T-periodic solution.For the model which is disturbed by both white and telegraph noises,we obtain sufficient criteria for positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.The positive recurrence implies that both prey and predator populations will be persistent in the long term.

QUALITATIVE ANALYSIS OF A STOCHASTIC PREDATOR-PREY SYSTEM WITH DISEASE IN THE PREDATOR

A stochastic predator prey system with disease in the predator population is proposed, the existence of global positive solution is derived. When the white noise is small, there is a stationary distribution. In addition, conditions of global stability for the determin- istic system are also established from the above result. By Lyapunov function, the long time behavior of solution around the disease-free equilibrium of deterministic system is derived. These results mean that stochastic system has the similar property with the corresponding deterministic system. When the white noise is small, however, large envi- ronmental noise makes the result different. Finally, numerical simulations are carried out to support our findings.

Stability and bifurcation for a stochastic differential algebraic Holling-Ⅱ predator-prey model with nonlinear harvesting and delay

In this paper,a stochastic delayed differential algebraic predator-prey model with Michaelis-Menten-type prey harvesting is proposed.Due to the influence of gestation delay and stochastic fluctuations,the proposed model displays a complex dynamics.Criteria on the local stability of the interior equilibrium are established,and the effect of gestation delay on the model dynamics is discussed.Taking the gestation delay and economic profit as bifurcation parameters,Hopf bifurcation and singularity induced bifurcation can occur as they cross through some critical values,respectively.Moreover,the solution of the model will blow up in a limited time when delay τ>τ0.Then,we calculate the fluctuation intensity of the stochastic fluctuations by Fourier transform method,which is the key to illustrate the effect of stochastic fluctuations.Finally,we demonstrate our theoretical results by numerical simulations.

Dynamics of a stochastic Holling II predator-prey model with Levy jumps and habitat complexity

This paper investigates a stochastic Holling II predator-prey model with Levy jumps and habit complexity.It is first proved that the established model admits a unique global positive solution by employing the Lyapunov technique,and the stochastic ultimate boundedness of this positive solution is also obtained.Sufficient conditions are established for the extinction and persistence of this solution.Moreover,some numerical simulations are carried out to support the obtained results.

Dynamics of a stochastic predator-prey model with mutual interference

In this paper, a stochastic predator-prey （PP） model with mutual interference is considered. Some sufficient conditions for the existence of globally positive solution, non- persistence in the mean, weak persistence in the mean, strong persistence in the mean and almost surely extinction of the the model are established. Moreover, the thresh- old between weak persistence in the mean and almost surely extinction of the prey is obtained. Some examples are given to show the feasibility of the results by numeri- cal simulation. It is significant that such a model is firstly proposed with stochastic perturbation.

PERMANENCE OF ASYMPTOTICALLY PERIODIC MULTISPECIES LOTKA-VOLTERRA COMPETITION PREDATOR-PREY SYSTEM

In this paper, we consider the permanence of asymptotically periodic mul-tispecies Lotka-Volterra competition predator-prey system. By means of the standard comparison theorem, we improve or extend the corresponding results given by Peng and Chen [1], Teng and Li [2], Zhao and Chen [3]. Also, we obtain the conditions which ensure the permanence and global attractivity of asymptotically periodic multispecies competition predator-prey system.

Permanence in Nonautonomous Predator-prey Lotka-Volterra Systems

In this paper some easily verifiable sufficient conditions on the permanence of solutions for general nonautonomous two-species predator-prey model are established. These new criteria improve and extend the results given by Ma, Wang, Teng and Teng, Yu.