Dynamical behavior for a class of predator–prey system with general functional response and discontinuous harvesting policy

In this paper, we introduce a class of predator–prey system with general functional response, whose harvesting policy is modeled by a discontinuous function. Based on the differential inclusions theory, topological degree theory in set‐valued analysis and generalized Lyapunov approach, we analyze the existence, uniqueness and global asymptotic stability of positive periodic solution. In particular, a series of useful criteria on existence, uniqueness and global asymptotic stability of the positive equilibrium point are established for the autonomous system corresponding to the non‐autonomous biological and mathematical model with a discontinuous right‐hand side. Moreover, some new sufficient conditions are provided to guarantee the global convergence in measure of harvesting solution and convergence in finite time of any positive solution for the autonomous discontinuous biological system. The obtained results, which improve and generalize previous works on dynamical behavior in the literature, are of interest for understanding and designing biological system with not only continuous or even Lipschitz continuous but also discontinuous harvesting function. Finally, we give three examples with numerical simulations to show the applicability and effectiveness of our main results. Copyright © 2015 John Wiley & Sons, Ltd.     

具有收获和阶段结构的时滞脉冲的捕食-食饵模型

研究了食饵分布在不同斑块,捕食者具有阶段结构和收获的时滞脉冲的捕食-食饵模型.利用离散动力系统的频闪映射,得到了捕食者灭绝周期解的存在性和它的精确表达式.使用比较原理,得到了捕食者灭绝周期解全局渐近稳定的充分条件和系统的持久性.最后,用Matlab软件进行数值仿真验证了获得的结果.     

Multiplicity of positive almost periodic solutions in a delayed Hassell–Varley‐type predator–prey model with harvesting on prey

In this paper, we consider a delayed Hassell–Varley‐type predator–prey model with harvesting on prey. By means of Mawhin's continuation theorem of coincidence degree theory, some new sufficient conditions are obtained for the existence of at least two positive almost periodic solutions for the aforementioned model. To the best of the author's knowledge, so far, the result of this paper is completely new. An example is employed to illustrate the result of this paper. Copyright © 2013 John Wiley & Sons, Ltd.     

On almost periodicity of delayed predator–prey model with mutual interference and discontinuous harvesting policies

The objective of this paper is to investigate the almost periodic dynamics for a class of delayed predator–prey model with mutual interference and Beddington–DeAngelis type functional response, in which the harvesting policies are modeled by discontinuous functions. Based on the theory of functional differential inclusions theory and set‐valued analysis, the solution in sense of Filippov of system with the discontinuous harvesting policies is given, and the local and global existence of positive the solution in sense of Filippov of the system is studied. By employing generalized differential inequalities, some useful Lemmas are obtained. After that, sufficient conditions which guarantee the permanence of the system are obtained in view of the constructed Lemmas. By constructing some suitable generalized Lyapunov functional, a series of useful criteria on existence, uniqueness, and global attractivity of the almost positive periodic solution to the system are derived in view of functional differential inclusions theory and nonsmooth analysis theory. Some suitable examples together with their numeric simulations are given to substantiate the theoretical results and to illustrate various dynamical behaviors of the system. Copyright © 2016 John Wiley & Sons, Ltd.     

Global stability of a periodic Holling‐Tanner predator‐prey model

This paper is concerned with the global dynamics of a Holling‐Tanner predator‐prey model with periodic coefficients. We establish sufficient conditions for the existence of a positive solution and its global asymptotic stability. The stability conditions are first given in average form and afterward as pointwise estimates. In the autonomous case, the previous criteria lead to a known result.     

A delayed prey–predator model with Crowley–Martin‐type functional response including prey refuge

In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin‐type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

Dynamics of a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects

This paper is concerned with a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects. By using the fixed point theory based on monotone operator, some simple conditions are obtained for the existence of at least one positive periodic solution of the model. The existence result of this paper implies that the functional response on prey does not influence the existence of positive periodic solution of the model, which completes some results given in recent years. Further, by applying the comparison theorem in impulsive differential equations and constructing a suitable Lyapunov functional, the permanence and global attractivity of the model are also investigated. The main results in this paper extend, complement, and improve the previously known result. And some examples and numerical simulations are given to illustrate the feasibility and effectiveness of the main results. Copyright © 2015 John Wiley & Sons, Ltd.     

Persistence in nonautonomous predator–prey systems with infinite delays

This paper studies the general nonautonomous predator–prey Lotka–Volterra systems with infinite delays. The sufficient and necessary conditions of integrable form on the permanence and persistence of species are established. A very interesting and important property of two-species predator–prey systems is discovered, that is, the permanence of species and the existence of a persistent solution are each other equivalent. Particularly, for the periodic system with delays, applying these results, the sufficient and necessary conditions on the permanence and the existence of positive periodic solutions are obtained. Some well-known results on the nondelayed periodic predator–prey Lotka–Volterra systems are strongly improved and extended to the delayed case.     

A nonautonomous predator–prey system with stage structure and double time delays

In the present paper we study a nonautonomous predator–prey model with stage structure and double time delays due to maturation time for both prey and predator. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the immature prey. Based on some comparison arguments we discuss the permanence of the species. By virtue of the continuation theorem of coincidence degree theory, we prove the existence of positive periodic solution. By means of constructing an appropriate Lyapunov functional, we obtain sufficient conditions for the uniqueness and the global stability of positive periodic solution. Two examples are given to illustrate the feasibility of our main results.     

Dynamics behaviors of a delayed stage-structured predator-prey model with impulsive effect

In this paper, we analyzed a delayed stage-structured predator-prey model with impulsive stocking on prey and continuous harvesting on predator. Sufficient conditions for the global attractivity of ‘predator-eradication’ periodic solution and permanence of the system are obtained. The results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system.     

Dynamical study of fractional order differential equations of predator‐pest models

To explore the impact of pest‐control strategy through a fractional derivative, we consider three predator‐prey systems by simple modification of Rosenzweig‐MacArthur model. First, we consider fractional‐order Rosenzweig‐MacArthur model. Allee threshold phenomena into pest population is considered for the second case. Finally, we consider additional food to the predator and harvesting in prey population. The main objective of the present investigation is to observe which model is most suitable for the pest control. To achieve this goal, we perform the local stability analysis of the equilibrium points and observe the basic dynamical properties of all the systems. We observe fractional‐order system has the ability to stabilize Rosenzweig‐MacArthur model with low pest density from oscillatory state. In the numerical simulations, we focus on the bistable regions of the second and third model, and we also observe the effect of the fractional order α throughout the stability region of the system. For the third model, we observe a saddle‐node bifurcation due to the additional food and Allee effect to the pest densities. Also, we numerically plot two parameter bifurcation diagram with respect to the harvesting parameter and fractional order of the system. We finally conclude that fractional‐order Rosenzweig‐MacArthur model and the modified Rosenzweig‐MacArthur model with additional food for the predator and harvested pest population are more suitable models for the pest management.     

Analysis of a nonautonomous eco‐epidemic diffusive model with disease in the prey

This paper deals with the behavior of positive solutions to a nonautonomous reaction‐diffusion system with homogeneous Neumann boundary conditions, which describes a two‐species predator‐prey system in which there is an infectious disease in prey. The sufficient condition on the permanence of the prey and the predator is established by combining the comparison principle with the results related to the corresponding ODE system. Some sufficient conditions for the spreading and vanishing of the disease are obtained. The global attractivity is also discussed by constructing a Lyapunov functional. Our results show that the disease is spreading if the transmission rate is suitably large, while if the transmission rate is small, the disease must be vanishing.     

连续收获捕食者与脉冲存放食饵的阶段结构捕食-食饵模型的全局吸引和一致持久

研究了一个捕食者具连续收获与食饵具脉冲存放的阶段结构时滞捕食-食饵模型．根据生物资源管理的实际,改进了捕食者具阶段结构的捕食-食饵模型,即原来假设每个捕食者个体都具有相同的捕食食饵的能力．假设捕食者按年龄分为两个阶段,即幼体和成体,而且幼体无能力捕食食饵．得到了捕食者灭绝周期解全局吸引和系统持久的充分条件．结论说明了脉冲存放食饵对系统的持久起了重要的作用,并且为生物资源管理提供了策略基础．数值分析也进一步说明了系统的动力学性质．     

具有Holling Ⅲ功能反应和阶段结构的Gompertz生态系统的持久性

以生态学与微分方程的理论和方法为基础,建立了一类具有HollingⅢ功能反应和阶段结构的生态Gompertz模型.利用频闪映射,获得了捕食者灭绝周期解,分析了此周期解的全局吸引性.在对食饵进行脉冲收获和捕食者具有成长期时滞条件下,运用脉冲微分方程比较定理和小振幅扰动技巧,获得了系统一致持续生存的条件.     

Homoclinic bifurcation for a general state-dependent Kolmogorov type predator–prey model with harvesting

In this paper, a general Kolmogorov type predator–prey model is considered. Together with a constant-yield predator harvesting, the state dependent feedback control strategies which take into account the impulsive harvesting on predators as well as the impulsive stocking on the prey are incorporated in the process of population interactions. We firstly study the existence of an order-1 homoclinic cycle for the system. It is shown that an order-1 positive periodic solution bifurcates from the order-1 homoclinic cycle through a homoclinic bifurcation as the impulsive predator harvesting rate crosses some critical value. The uniqueness and stability of the order-1 positive periodic solution are derived by applying the geometry theory of differential equations and the method of successor function. Finally, some numerical examples are provided to illustrate the main results. These results indicate that careful management of resources and harvesting policies is required in the applied conservation and renewable resource contexts.     

Traveling wave solutions of n‐dimensional delayed reaction–diffusion systems and application to four‐dimensional predator–prey systems

This paper deals with the existence of traveling wave solutions for n‐dimensional delayed reaction–diffusion systems. By using Schauder's fixed point theorem, we establish the existence result of a traveling wave solution connecting two steady states by constructing a pair of upper–lower solutions that are easy to construct. As an application, we apply our main results to a four‐dimensional delayed predator–prey system and obtain the existence of traveling wave solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Permanence and periodicity of a delayed ratio-dependent predator–prey model with Holling type functional response and stage structure

A periodic and delayed ratio-dependent predator–prey system with Holling type III functional response and stage structure for both prey and predator is investigated. It is assumed that immature predator and mature individuals of each species are divided by a fixed age, and immature predator do not have the ability to attack prey. Sufficient conditions are derived for the permanence and existence of positive periodic solution of the model. Numerical simulations are presented to illustrate the feasibility of our main results.     

收获捕食者的时滞脉冲捕食模型

讨论了与可再生生物资源管理相关的食饵具脉冲扰动与捕食者具连续收获的阶段结构时滞捕食-食饵模型,得到了捕食者灭绝周期解的全局吸引和系统持久的充分条件.也证明了系统的所有解的一致完全有界.结论为现实的可再生生物资源管理提供了可靠的策略依据.     

Dynamics of a generalized predator‐prey model with delay and impulse via the basic reproduction number

In this paper, we study a generalized predator‐prey model with delay and impulse. The existence of the predator‐free periodic solution is investigated. We employ the approach and techniques coming from epidemiology and calculate the basic reproduction number for the predator. Using the basic reproduction number, we consider the global attraction of the predator‐free periodic solution and permanence of the model. As for application, an example is discussed. Furthermore, some numerical simulations are given to illustrate our results.     

Survival analysis for a periodic predator–prey model with prey impulsively unilateral diffusion in two patches

In this paper, we study a periodic predator–prey system with prey impulsively unilateral diffusion in two patches. Firstly, based on the results in [41], sufficient conditions on the existence, uniqueness and globally attractiveness of periodic solution for predator-free and prey-free systems are presented. Secondly, by using comparison theorem of impulsive differential equation and other analysis methods, sufficient and necessary conditions on the permanence and extinction of prey species x with predator have other food source are established. Finally, the theoretical results both for non-autonomous system and corresponding autonomous system are confirmed by numerical simulations, from which we can see some interesting phenomena happen.