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.     

Multiple periodic solutions of a delayed stage-structured predator–prey model with non-monotone functional responses

A delayed stage-structured predator–prey model with non-monotone functional responses is proposed. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age, and that immature predators do not have the ability to attack prey. Some new and interesting sufficient conditions are obtained for the global existence of multiple positive periodic solutions of the stage-structured predator–prey model. Our method is based on Mawhin’s coincidence degree and novel estimation techniques for the a priori bounds of unknown solutions to Lx = λNx. An example is given to illustrate the feasibility of our main result.     

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.     

Periodic solutions of a nonautonomous predator–prey system with stage structure and time delays

A nonautonomous Lotka–Volterra type predator–prey model with stage structure and time delays is investigated. It is assumed in the model that the individuals in each species may belong to one of two classes: the immatures and the matures, the age to maturity is presented by a time delay, and that the immature predators do not feed on prey and do not have the ability to reproduce. By some comparison arguments we first discuss the permanence of the model. By using the continuation theorem of coincidence degree theory, sufficient conditions are derived for the existence of positive periodic solutions to the model. By means of a suitable Lyapunov functional, sufficient conditions are obtained for the uniqueness and global stability of the positive periodic solutions to the model.     

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．     

Stability and bifurcation of a stage-structured predator–prey model with both discrete and distributed delays

This paper concerns with a new delayed predator–prey model with stage structure on prey, in which the immature prey and the mature prey are preyed by predator and the delay is the length of the immature stage. Mathematical analysis of the model equations is given with regard to invariance of non-negativity, boundedness of solutions, permanence and global stability and nature of equilibria. Our work shows that the stage structure on the prey is one of the important factors that affect the extinction of the predator, and the predation on immature prey is a cause of periodic oscillation of population and can make the behaviors of the system more complex. The predation on the immature and mature prey brings both positive and negative effects on the permanence of the predator, if ignore the predation on immature prey in the system, the stage-structure on prey brings only negative effect on the permanence of the predator.     

The dynamics of an age structured predator–prey model with disturbing pulse and time delays

Many of the existing predator–prey models on stage structured populations are some ordinary differential equations (ODE) or models without a disturbing effect of human behavior. In reality, death of the juvenile during its immature stage and catching or poisoning for the prey or predator occur continuously. From this basic standpoint, we formulate a general and robust prey-dependent consumption predator–prey model with periodic harvesting (catching or poisoning) for the prey and stage structure for the predator with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and ecological study. We show that the conditions for global attractivity of the ‘predator-extinction’ (‘predator-eradication’) periodic solution and permanence of the population of the model depend on time delay, so, we call it “profitless”. We also show that constant maturation time delay and impulsive catching or poisoning for the prey can bring great effects on the dynamics of system by numerical analysis. In this paper, the main feature is that we introduce time delay and pulse into the predator–prey (natural enemy–pest) model with age structure, exhibit a new modeling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.     

Eco-epidemiology model with age structure and prey-dependent consumption for pest management

In this paper, a prey-dependent consumption predator–prey (natural enemy-pest) model with age structure for the predators and infectious disease in the prey, is considered. Infectious pests, immature natural enemies and mature natural enemies are released impulsively. By using Floquet’s theorem, small-amplitude perturbation skills and comparison theorem, we obtain both the sufficient conditions for the global asymptotical stability of the susceptible pest-eradication periodic solution and the permanence of the system. The results provide a reliable theoretical tactics for pest management.     

一类具有时滞、脉冲效应和阶段结构的捕食者-食饵模型（英文）

A delayed predator-prey model concerning impulsive spraying pesticides and releasing natural enemies is proposed and investigated,in which both the prey and the predator have a history that takes them through two stages:immature and mature.The global attractiveness of the pest-eradication periodic solution is discussed,and sufficient condition is obtained for the permanence of the system.Further,numerical simulations show that there is a characteristic sequence of bifurcations leading to a chaotic dynamics,which implies that the system with constant periodic impulsive perturbations admits rich and complex dynamics.     

A Lotka-Volterra type food chain model with stage structure and time delays

A three-species Lotka-Volterra type food chain model with stage structure and time delays is investigated. It is assumed in the model that the individuals in each species may belong to one of two classes: the immatures and the matures, the age to maturity is presented by a time delay, and that the immature predators (immature top predators) do not have the ability to feed on prey (predator). By using some comparison arguments, we first discuss the permanence of the model. By means of an iterative technique, a set of easily verifiable sufficient conditions are established for the global attractivity of the nonnegative equilibria of the model.     

A predator–prey system with a stage structure for the prey

This paper considers a periodic predator–prey system where the prey has a life history that takes the prey through two stages: immature and mature. We provide a sufficient and necessary condition to guarantee permanence of the system. It is shown that the system is permanent if and only if the growth of the predator by foraging the prey minus its death rate is positive on average during the period.     

Rich dynamic of a stage-structured prey–predator model with cannibalism and periodic attacking rate

The dynamic behavior of a stage-structure prey–predator model with cannibalism for prey and periodic attacking rate for predator is investigated. Firstly, the permanence, locally and globally asymptotic stability analyses of the model with constant attacking rate are explored. After that, sufficient conditions for the permanence of the corresponding nonautonomous system with periodic attacking rate are obtained. Furthermore, numerical simulations are presented to illustrate the effects of periodic attacking rate. Simulation results show that the system with periodic attacking rate shows a rich behaviors, including period-doubling and period-having bifurcations, chaos and windows of periodicity.     

Analysis of periodic solutions in an eco-epidemiological model with saturation incidence and latency delay

In the present work, a mathematical model of predator–prey ecological interaction with infected prey is investigated. A saturation incidence function is used to model the behavioral change of the susceptible individuals when their number increases or due to the crowding effect of the infected individuals [V. Capasso, G. Serio, A generalization of the Kermack–McKendrick deterministic epidemic model, Math. Biosci. 42 (1978) 41–61]. Stability criteria for the infection-free and the endemic equilibria are deduced in terms of system parameters. The basic model is then modified to incorporate a time delay, describing a latency period. Stability and bifurcation analysis of the resulting delay differential equation model is carried out and ranges of the delay inducing stability and as well as instability for the system are found. Finally, a stability analysis of the bifurcating solutions is performed and the criteria for subcritical and supercritical Hopf bifurcation derived. The existence of a delay interval that preserves the stability of periodic orbits is demonstrated. The analysis emphasizes the importance of differential predation and a latency period in controlling disease dynamics.     

Periodic solutions in non autonomous predator prey system with delays

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

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.     

Bifurcation and complexity of Monod type predator–prey system in a pulsed chemostat

In this paper, we introduce and study a model of a predator–prey system with Monod type functional response under periodic pulsed chemostat conditions, which contains with predator, prey, and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey

A nonautonomous eco-epidemic model with disease in the prey is formulated and studied. Some sufficient and necessary conditions on the permanence and extinction of the infective prey are established by introducing the new research method. Some sufficient conditions on the global attractivity of the model are presented by constructing a Lyapunov function. Finally, an example is given to show that the periodic model is global attractivity if the infective prey is permanent.     

Dynamic complexities of predator–prey system in a pulsed chemostat

In this paper, we study a food chain model with Holling III and Monod type functional response under periodic pulsed conditions, which contains with predator, prey and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the boundary periodic solution. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in prey and predator. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the system shows two kinds of bifurcations, whose are period-doubling and period-halving.     

Analysis of a stage structured predator–prey Gompertz model with disturbing pulse and delay

In this paper, we introduce a general and robust prey-dependent consumption predator–prey Gompertz model with periodic harvesting for the prey and stage structure for the predator with constant maturation time delay and perform a systematic mathematical and ecological study. Sufficient conditions which guarantee the global attractivity of predator-extinction periodic solution and permanence of the system are obtained. We also prove that constant maturation time delay and impulsive catching or poisoning for the prey can bring great effects on the dynamics of system by numerical analysis. Our results provide reliable tactic basis for the practical pest management.     

Coexistence in seasonally varying predator–prey systems with Allee effect

A general seasonally-varying predator–prey model with Allee effect in the prey growth is investigated. The analysis is performed only on the basis of some properties determining the shape of the prey growth rate and the functional responses. General conditions for coexistence are determined, both in the case of weak and strong Allee effect. Finally, a modified Leslie–Gower predator–prey model with Allee effect is investigated. Numerical results illustrate the qualitative behaviors of the system, in particular the presence of periodic orbits.