Dynamic complexities of a food chain model with impulsive perturbations and Beddington–DeAngelis functional response

In this paper, the dynamic complexities of a three trophic level food chain system with impulsive perturbations and Beddington–DeAngelis functional responses is studied. Conditions for extinction of predator are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics in the positive octant.     

Analysis of a predator–prey model with Holling II functional response concerning impulsive control strategy

According to biological and chemical control strategy for pest control, we investigate the dynamic behavior of a Holling II functional response predator–prey system concerning impulsive control strategy-periodic releasing natural enemies and spraying pesticide at different fixed times. By using Floquet theorem and small amplitude perturbation method, we prove that there exists a stable pest-eradication periodic solution when the impulsive period is less than some critical value. Further, the condition for the permanence of the system is also given. Numerical results show that the system we consider can take on various kinds of periodic fluctuations and several types of attractor coexistence and is dominated by periodic, quasiperiodic and chaotic solutions, which implies that the presence of pulses makes the dynamic behavior more complex. Finally, we conclude that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently.     

具有HollingⅡ型功能反应和Allee效应的捕食系统模型

利用计算机模拟方法研究一类离散种群相互作用模型的动态复杂性.通过理论推导建立食饵具有Allee效应和HollingⅡ型功能反应的自治捕食系统模型,用Matlab软件模拟离散种群的生长状态,探索研究参数的变化对种群大小的影响,阐释Allee效应及HollingⅡ型功能反应在种群间相互作用模型中的重要性.研究结果表明:1)当处理时间处于有效区间内时,处理时间越大种群的稳定共存参数域越大;2)Allee效应的引入使种群的动态行为更为复杂,从而增加了捕食者种群的灭绝风险;3)系统受强Allee效应的影响,种群会出现提前分叉现象,如果继续增加Allee效应就会导致种群灭绝;4)强Allee效应更容易使种群趋向灭绝.所得结论在丰富生态学理论的同时,提出了保护生态学的重要依据.     

The dynamical behaviors of a food chain model with impulsive effect and Ivlev functional response

In this paper, a food chain model with Ivlev functional response and impulsive effect of top predator is investigated. Conditions for extinction of mid-level predator are given. By using the Floquet theory of linear τ-period impulsive differential equation and small amplitude perturbation skills, we show that the lowest-level prey and the mid-level predator extinction periodic solution is unstable, while the mid-level predator eradication periodic solution is stable, and meanwhile, we prove that the system is permanent if the impulsive period is larger than some critical value. Furthermore, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which displays complicated behavior including a sequence of direct and inverse cascade of period doubling, period halfing as well as chaos.     

Dynamics of multi-species competition-predator system with impulsive perturbations and Holling type III functional responses

We investigate the dynamics of a class of multi-species predator-prey interaction models with Holling type III functional responses based on systems of nonautonomous differential equations with impulsive perturbations. Sufficient conditions for existence of a positive periodic solution are investigated by using a continuation theorem in coincidence degree theory, which have been extensively applied in studying existence problems in differential equations and difference equations. In addition, sufficient criteria are established for the global stability and the globally exponential stability of the system by using the comparison principle and the Lyapunov method.     

Existence of limit cycles in a tritrophic food chain model with Holling functional responses of type II and III

We are interested in the coexistence of three species forming a tritrophic food chain model. Considering a linear grow for the lowest trophic species, Holling III and Holling II functional response for the predator and the top‐predator, respectively. We prove that this model has stable periodic orbits for adequate values of its parameters. Copyright © 2016 John Wiley & Sons, Ltd.     

On the explosive instability in a three‐species food chain model with modified Holling type IV functional response

In earlier literature, a version of a classical three‐species food chain model, with modified Holling type IV functional response, is proposed. Results on the global boundedness of solutions to the model system under certain parametric restrictions are derived, and chaotic dynamics is shown. We prove that in fact the model possesses explosive instability, and solutions can explode/blow up in finite time, for certain initial conditions, even under the parametric restrictions of the literature. Furthermore, we derive the Hopf bifurcation criterion, route to chaos, and Turing bifurcation in case of the spatially explicit model. Lastly, we propose, analyze, and simulate a version of the model, incorporating gestation effect, via an appropriate time delay. The delayed model is shown to possess globally bounded solutions, for any initial condition. Copyright © 2017 John Wiley & Sons, Ltd.     

Hopf bifurcation analysis in a multiple delayed innovation diffusion model with Holling II functional response

A nonlinear mathematical model with Holling II functional response describing the dynamics of nonadopter and adopters population in a stage structured innovation diffusion model, which incorporates the evaluation stage (multiple delays), is proposed. Firstly, we study the stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays at the positive equilibrium by analyzing the distribution of the roots of the corresponding exponential characteristic equation obtained through the variational matrix. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined with the help of normal form theory and center manifold theorem. Meanwhile, various cases are discussed to examine the effect of different delays on the stability of delayed innovation diffusion system and are also established numerically. It is also observed that the cumulative density of external influences has a significant role in developing maturity stage (adoption stage) in the system. Finally, numerical simulations are carried out to support and supplement the analytical findings.     

Dynamics of a delayed predator-prey model with Allee effect and Holling type II functional response

In this paper, a delayed with Holling type II functional response (Beddington-DeAngelis) and Allee effect predator-prey model is considered. The growth of the prey is affected by the parameter $M$, which defines the Allee effect. In addition, the delay $\tau$ also influences the logistic growth of the prey, which can be interpreted as the maturity time or the gestation period. In the study of the characteristic equation, we observe that the delay $\tau$ also depends on the parameter $M$, which affects the dynamics in the prey population. Considering the delay as a bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. On the other hand, we find that the system can also suffer a Hopf bifurcation in the positive equilibrium when the delay passes through a sequence of critical values. In particular, we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions, an explicit algorithm is provided applying the normal form theory and center manifold reduction for the functional differential equations. Finally, numerical simulations that support the theoretical analysis are included.     

A prey-predator model with Holling II functional response and the carrying capacity of predator depending on its prey

One prey-predator model is formulated and the global behavior of its solution is analyzed. In this model, the carrying capacity of predator depends on the amount of its prey, and the Holling II functional response is involved. This model may have four classes of positive equilibriums and limit cycle. The positive equilibriums may be stable, or a saddle-node, or a saddle, or a degenerate singular point. In alpine meadow ecosystem, the dynamics of vegetation and plateau pika can be described by this model. Through simulating with virtual parameters, the cause of alpine meadow degradation and effective recovery strategy is investigated. Increasing grazing rate or decreasing plateau pika mortality may cause alpine meadow degradation. Correspondingly, reducing grazing rate and increasing plateau pika mortality may recover the degraded alpine meadow effectively.     

Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge

We study a predator-prey model with Holling type II functional response incorporating a prey refuge under homogeneous Neumann boundary condition. We show the existence and non-existence of non-constant positive steady-state solutions depending on the constant m∈(0,1], which provides a condition for protecting (1−m)u of prey u from predation. Moreover, we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions.     

Dynamics of a simple food chain model with a ratio-dependent functional response

This paper considers a diffusive ratio-dependent simple food chain model. The sufficient conditions for the existence and non-existence of coexistence states are provided. In addition, this paper investigates the uniqueness of coexistence states and examines the global attractor of the time-dependent system. Moreover, in view of extinction results, the domino effect and biological control are discussed.     

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.     

具有脉冲扰动和非单调功能反应的三种群捕食系统的分析

研究一类具有脉冲效应和非单调功能反应的两个捕食者一个食饵害虫控制系统．通过脉冲微分方程的Floquet理论和小幅扰动方法,证明了当脉冲周期小于某个临界值时,系统存在一个渐近稳定的害虫根除周期解,否则系统是持续生存的．最后,通过数值实例,给出了一简单讨论．     

Chaos in periodically forced Holling type IV predator–prey system with impulsive perturbations

The effect of periodic forcing and impulsive perturbations on predator–prey model with Holling type IV functional response is investigated. The periodic forcing is affected by assuming a periodic variation in the intrinsic growth rate of the prey. The impulsive perturbations are affected by introducing periodic constant impulsive immigration of predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can easily give rise to complex dynamics, including (1) quasi-periodic oscillating, (2) period doubling cascade, (3) chaos, (4) period halfing cascade.     

Dynamics of impulsive reaction–diffusion predator–prey system with Holling type III functional response

An impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator–prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given at the end.     

Dynamics of n+1-dimensional impulsive reaction-diffusion predator-prey system with Holling type III functional response

An n+1-dimensional impulsive reaction-diffusion periodic predator-prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions on the ultimate boundedness and permanence of the predator-prey system are established based on the comparison theory of differential equation and upper and lower solution method. By constructing appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given in the end of the paper.     

Bifurcation behaviours in a delayed three-species food-chain model with Holling type-II functional response

This article is concerned with the local stability of a positive equilibrium and the Hopf bifurcation of a delayed three-species food-chain system with the Holling type-II functional response. Some new sufficient conditions ensuring the local stability of a positive equilibrium and the existence of Hopf bifurcation for the system are established. Some explicit formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions using the normal form theory and the centre manifold theory. Numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are included.     

Global stability and Hopf bifurcation of a delayed eco‐epidemiological model with Holling type II functional response

In this paper, a delayed eco‐epidemiological model with Holling type II functional response is investigated. By analyzing corresponding characteristic equations, the local stability of each of the feasible equilibria and the existence of Hopf bifurcations at the disease‐free equilibrium, the susceptible predator‐free equilibrium and the endemic‐coexistence equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are derived for the global stability of the endemic‐coexistence equilibrium, the disease‐free equilibrium, the susceptible predator‐free equilibrium and the predator‐extinction equilibrium of the system, respectively. Numerical simulations are carried out to illustrate the theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.