Nonautonomous Predator-prey System Research Articles

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 for a hybrid non-autonomous prey–predator system with generalist predator and impulsive conditions on time scales

In this paper, we investigate the dynamical behavior for a hybrid non-autonomous predator–prey system with Holling Type II functional response, impulsive effects and generalist predator on time scales, where our proposed model commutes between a continuous-time dynamical system and discrete-time dynamical system. By using comparison theorems, we first study the permanence results of the proposed model. Also, we established the uniformly asymptotic stability for the almost periodic solution of the proposed model. Finally, in the last section, we provide some examples with numerical simulation.

Dynamics of a non-autonomous predator-prey system with Hassell-Varley-Holling Ⅱ function response and mutual interference

In this paper, we establish a non-autonomous Hassell-Varley-Holling type predator-prey system with mutual interference. We construct some sufficient conditions for the permanence, extinction and globally asymptotic stability of system by use of the comparison theorem and an appropriate Liapunov function. Then the sufficient and necessary conditions for a periodic solution of the system are obtained via coincidence degree theorem. Finally, the correctness of the previous conclusions are demonstrated by some numerical cases.

Positive Periodic Solutions of Nonautonomous Delayed Predator-Prey System with Pulse Controls

In this paper, a biological model for two predators and one prey with impulses and periodic delays is considered. By assuming that one predator consumes prey according to Holling II functional response while the other predators consume prey according to the Beddington-DeAngelis functional response, based on the coincidence degree theory, the existence of positive periodic solutions of nonautonomous predator-prey system with impulses and periodic delays is obtained under suitable conditions.

Hopf Bifurcation of a Generalized Delay-Induced Predator–Prey System with Habitat Complexity

A delay-induced nonautonomous predator–prey system with variable habitat complexity is proposed based on mathematical and ecological issues, and this system is more realistic than the published models. Firstly, the permanence of the nonautonomous predation system is studied and some sufficient conditions are obtained. Secondly, the dynamical behaviors of the corresponding autonomous predation system are investigated, including the positivity and boundedness, and local and global stabilities. Thirdly, the properties of Hopf bifurcation of the autonomous predation system without/with delay are investigated and the explicit formulas which determine the stability and the direction of periodic solutions are obtained. Finally, a numerical example is given to test our theoretical results.

Dynamics for a non-autonomous predator-prey system with generalist predator

In this paper, we study complex dynamics of a non-autonomous predator-prey system with a Holling type II functional response and predator being generalist. We first studied basic dynamics such as boundedness, positive invariance, permanence, non-persistence and globally asymptotic stability. We also provide sufficient conditions for the existence, uniqueness and globally asymptotic stability of positive periodic solutions and boundary periodic solutions of the proposed model when parameters are periodic. We give the integral conditions to prove the extinction of prey and predator and globally asymptotic stability of boundary periodic solutions, and we show that these conditions have more reasonable biological interpretation than those expressed by supremum and infimum of parameters in some literature. We also perform numerical simulations to complement our analytical results and obtain more insights. Some of our numerical simulations indicate that periodic system may promote or suppress the permanence of its autonomous version with parameters being the averages of periodic parameters.

Global stability of the boundary solution of a nonautonomous predator-prey system with Beddington-DeAngelis functional response

ABSTRACT In this paper, we consider a nonautonomous predator-prey system with Beddington-DeAngelis functional response and explore the global stability of boundary solution. Based on the dynamics of logistic equation, some new sufficient conditions on the global asymptotic stability of boundary solution are presented for general time-dependence case. Our main results indicate that (i) the long-term ineffective predation behaviour or high mortality of predator species will lead the predator species to extinction, even if the intraspecies competition of predator species is weak or no intraspecies competition; (ii) the long-term intense intraspecific competition may lead the predator species to extinction, even though the long-term accumulative predation benefit is higher than the death lose. When all parameters are periodic functions with common period, a necessary and sufficient condition on the global stability of boundary periodic solution is obtained. In addition, some numerical simulations are performed to illustrate the theoretical results.

On the applicability of the poincaré–Birkhoff twist theorem to a class of planar periodic predator-prey models

This paper studies the existence of subharmonics of arbitrary order in a generalized class of non-autonomous predator-prey systems of Volterra type with periodic coefficients. When the model is non-degenerate it is shown that the Poincare–Birkhoff twist theorem can be applied to get the existence of subharmonics of arbitrary order. However, in the degenerate models, whether or not the twist theorem can be applied to get subharmonics of a given order might depend on the particular nodal behavior of the several weight function-coefficients involved in the setting of the model. Finally, in order to analyze how the subharmonics might be lost as the model degenerates, the exact point-wise behavior of the \begin{document}$ T $\end{document} -periodic solutions of a non-degenerate model is ascertained as a perturbation parameter makes it degenerate.

Dynamical Behavior of a Delayed Predator-prey Model in Periodically Fluctuating Environments

In this paper we develop a non-autonomous predator-prey system with time delay to study the influence of water level fluctuations on the interactions between fish species living in an artificial lake. We derive persistence and extinction conditions of the species. Using coincidence degree theory, we determine conditions for which the system has at least one periodic solution. Numerical simulations are presented to illustrate theoretical results.

Periodic solutions of a non-autonomous predator–prey system with migrating prey and disease infection: via Mawhin’s coincidence degree theory

In this paper, we consider a non-autonomous competitive predator–prey system with migrating prey and disease infection in both species. Based on some novel estimation techniques for the prior bound and Mawhin’s coincidence degree theorem, some sufficient conditions for the permanence and existence of periodic solutions are obtained. A suitable Lyapunov function was constructed to prove the global asymptotical stability of the positive periodic solution. Finally, an example and its numerical simulations demonstrate the validity of the results.

Periodic Solution for a Stochastic Non-autonomous Predator-Prey Model with Holling II Functional Response

A biological population may be subjected to stochastic disturbance and exhibit periodicity. In this paper, a stochastic non-autonomous predator-prey system with Holling II functional response is proposed, and the existence of a unique positive solution is derived. We give sufficient conditions for extinction and strong persistence in the mean by analyzing a corresponding one-dimensional stochastic system. Also we establish the existence of positive periodic solutions for this stochastic non-autonomous predator-prey system. Finally, we use numerical simulations to illustrate our results and we present some conclusions and future directions. The results of this paper provide methods for other stochastic population models, which we hope to analyze in the future.

Periodic Solution of a Nonautonomous Predator-Prey System of Holling Type II with Strong Allee Effect and Impulsive Perturbations

The principle of this paper is to explore the existence of periodic solution of a nonautonomous predator-prey system of Holling type II with strong Allee effect and impulsive perturbations. Sufficient and realistic conditions are obtained by using Mawhin’s continuation theorem of coincidence degree and analysis techniques. Further, some numerical simulations demonstrate our results.

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.

Stationary distribution and periodic solution for stochastic predator-prey systems with nonlinear predator harvesting

In this paper, we investigate the dynamics of the stochastic autonomous and non-autonomous predator-prey systems with nonlinear predator harvesting respectively. For the autonomous system, we first give the existence of the global positive solution. Then, in the case of persistence, we prove that there exists a unique stationary distribution and it has ergodicity by constructing a suitable Lyapunov function. The result shows that, the relatively weaker white noise will strengthen the stability of the system, but the stronger white noise will result in the extinction of one or two species. Particularly, for the non-autonomous periodic system, we show that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii. Finally, numerical simulations illustrate our theoretical results.

Asymptotic behavior of a stochastic non-autonomous predator-prey system with jumps

In this paper, a two-species stochastic non-autonomous food chain predator-prey system with jumps is proposed and studied. The asymptotic properties of the system are examined. Sufficient conditions for extinction, non-persistence in the mean, weak persistence in the mean and weak persistence of the system are established. Results show that Lévy jumps are disadvantageous for the persistence of the populations. Furthermore, we also show that the solution is stochastically ultimate bounded under certain conditions. Some examples together with numerical simulations are introduced to illustrate our main results.

Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response

We investigate the dynamics of a non-autonomous and density dependent predator-prey system with Beddington-DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered. First, we derive a sufficient condition of permanence by comparison theorem, at the same time we propose a weaker condition ensuring some positive bounded set to be positive invariant. Next, we obtain two existence conditions for positive periodic solution by Brouwer fixed-point theorem and by continuation theorem, where the second condition is weaker than the first and gives the existence range of periodic solution. Further we show the global attractivity of the bounded positive solution by constructing Lyapunov function. Similarly, we have sufficient condition of global attractivity of boundary periodic solution.

The dynamical behaviors of a nonautonomous Holling III predator-prey system with impulses

In this paper, we study a Holling III nonautonomous predator-prey system with impulses. Firstly, the dynamics of single-species nonautonomous Logistic system with impulses are discussed. Further, based on these results and by impulsive differential comparison theorem, the extinction and permanence of Holling III nonautonomous predator-prey system with impulses are obtained.

Optimal Application Timing of Pest Control Tactics in Nonautonomous Pest Growth Model

Considering the effects of the living environment on growth of populations, it is unrealistic to assume that the growth rates of predator and prey are all constants in the models with integrated pest management (IPM) strategies. Therefore, a nonautonomous predator-prey system with impulsive effect is developed and investigated in the present work. In order to determine the optimal application timing of IPM tactics, the threshold value which guarantees the stability of pest-free periodic solution has been obtained firstly. The analytical formula of optimal application timings within a given period for different cases has been obtained such that the threshold value is the smallest, which is the most effective in successful pest control. Moreover, extensively numerical investigations have also been confirmed our main results and the biological implications have been discussed in more detail. The main results can guide the farmer to design the optimal pest control strategies.

Dynamics of a stochastic non-autonomous predator-prey system with Beddington-DeAngelis functional response

A stochastic non-autonomous predator-prey system with Beddington-DeAngelis functional response is proposed, the existence of a global positive solution and stochastically ultimate boundedness are derived. Sufficient conditions for extinction, non-persistence in the mean, weak persistence in the mean and strong persistence in the mean are established. The global attractiveness of the solution is also considered. Finally, numerical simulations are carried out to support our findings. MSC:92B05, 34F05, 60H10, 93E03.

Global Analysis of a Nonautonomous Predator-Prey Model with Double Delays and Stage Structure

A nonautonomous predator-prey system with stage-structure and double time delays is investigated in the present paper. Based on some comparison arguments we discuss the permanence of the species. By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for the system is established. Further, by constructing a V functional and using the result of the existence of positive periodic solution, we obtain sufficient conditions for the uniqueness and the global stability of positive periodic solution.