BIFURCATION AND COMPLEXITY IN A RATIO-DEPENDENT PREDATOR-PREY CHEMOSTAT WITH PULSED INPUT

In this paper, a three dimensional ratio-dependent chemostat model with periodically pulsed input is considered. By using the discrete dynamical system determined by the stroboscopic map and Floquet theorem, an exact periodic solution with positive concentrations of substrate and predator in the absence of prey is obtained. When β is less than some critical value the boundary periodic solution （xs（t）, O, zs（t）） is locally stable, and when β is larger than the critical value there are periodic oscillations in substrate, prey and predator. Increasing the impulsive period T the system undergoes a series of period-doubling bifurcation leading to chaos, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex.     

Existence of Positive Periodic Solution of Periodic Time-Dependent Predator-Prey System with Impulsive Effects

The general system of differential equations describing predator-prey dynamics with impulsive effects is modified by the assumption that the coefficients are periodic functions of time. By use of standard techniques of bifurcation theory, it is known that this system has a positive periodic solution provided the time average of the predator‘s net uninhibited death rate is in a suitable range.The bifurcation is from the periodic solution of the time-dependent logistic equation for the prey (which results in the absence of predator).     

PERMANENCE AND GLOBAL STABILITY OF A DELAYED RATIO-DEPENDENT PREDATOR-PREY MODEL WITH STAGE STRUCTURE

A ratio-dependent predator-prey system with stage structure and time delays for both prey and predator is considered in this paper. Both the predator and prey have two stages,immature stage and mature stage,and the growth of them is of Lotka-Volterra nature. It is assumed that immature individuals and mature individuals of each species are divided by a fixed age,and that mature predators attack immature prey only. The global stability of three nonnegative equilibria and permanence are presented.     

Analysis of a Prey-predator Model with Disease in Prey

In this paper, a system of reaction-diffusion equations arising in ecoepidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. Local stability of the constant positive solution is considered. A number of existence and non-existence results about the nonconstant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the prey with disease is treated as a bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steadystate solution under some conditions.     

The Dynamics of a Predator-prey Model with Ivlev＇s Functional Response Concerning Integrated Pest Management

A mathematical model of a predator-prey model with Ivlev‘s functional response concerning integrated pest management (IPM) is proposed and analyzed. We show that there exists a stable pest-eradication periodic solution when the impulsive period is less than some critical values, Further more, the conditions for the permanence of the system are giverl. By using bifurcation theory, we show the existence and stability of a positive periodic solution. These results are quite different from those of the corresponding system without impulses. Numerical simulation shows that the system we consider has more complex dynamical behaviors.Finally, it is proved that IPM stragey is more effective than the classical one.     

PERMANENCE OF A NONLINEAR PERIODIC PREDATOR-PREY SYSTEM WITH PREY DISPERSAL AND PREDATOR DENSITY-INDEPENDENCE

In this paper,a set of suffcient conditions which ensure the permanence of a nonlinear periodic predator-prey system with prey dispersal and predator density-independence are obtained,where the prey species can disperse among n patches,while the density-independent predator is confined to one of the patches and cannot disperse. Our results generalize some known results.     

The Dynamics of a Predator-prey Model with Ivlev''''s Functional Response Concerning Integrated Pest Management

A mathematical model of a predator-prey model with Ivlev's functional response concerning inte-grated pest management(IPM)is proposed and analyzed.We show that there exists a stable pest-eradicationperiodic solution when the impulsive period is less than some critical values.Further more,the conditions forthe permanence of the system are given.By using bifurcation theory,we show the existence and stability ofa positive periodic solution.These results are quite different from those of the corresponding system withoutimpulses.Numerical simulation shows that the system we consider has more complex dynamical behaviors.Finally,it is proved that IPM stragey is more effective than the classical one.     

DYNAMICS OF A NONLINEAR NON-AUTONOMOUS n-PATCHES PREDATOR-PREY DISPERSION-DELAY MODEL

In this paper,a nonlinear nonautonomous predator-prey dispersion model with continuous distributed delay is studied,where all parameters are time-dependent.In this system consisting of n-patches the prey species can disperse among n-patches,but the predator species is confined to one patch and cannot disperse.It is proved that the system is uniformly persistent under any dispersion rate effect.Furthermore,some sufficient conditions are established for the existence of a unique almost periodic solution of the system.The example shows that the criteria in the paper are new,general and easily verifiable.     

DYNAMIC BEHAVIORS OF A NONAUTONOMOUS RATIO-DEPENDENT LESLIE SYSTEM INCORPORATING A PREY REFUGE

A nonautonomous ratio-dependent Leslie system incorporating a prey refuge is studied in this paper. By applying the comparison theorem of diferential equations and constructing a suitable Lyapunov function, a set of sufcient conditions which guarantee the persistent property and global attractivity of the system is obtained. Also, by applying the comparison theorem of diferential equations and Fluctuation Lemma, a set of sufcient conditions which ensure the extinction of the prey species and the global attractivity of predator species is obtained. This result shows that for the Lotka-Volterra type predator-prey system, when the value of prey refuge increases, predator species will be driven to extinction due to the lack of food. Our study shows that the alternative food resource predator species is always permanent, which means that prey refuge has no infuence on the permanence of predator species. However, refuge plays an important role in the persistent property of the prey species: large enough prey refuge could keep the persistent property of the prey species, while small enough refuge could lead to the extinction of prey species. Numerical simulations show the feasibility of the main results.     

一类具有脉冲效应和HollingII型功能性反应的阶段结构的捕食系统

A stage-structured predator-prey system with impulsive effect and Holling type-II functional response is investigated. By the Floquet theory and small amplitude perturbation skills, it is proved that there exists a global stable pest-eradication periodic solution when the impulsive period is less than some critical values. Farther, the conditions for the permanence of system are established. Numerical simulations are carried out to illustrate the impulsive effect on the dynamics of the system.     

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.     

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.     

一种具有脉冲扰动的恒化器模型的动力学性质

在这篇文章中,我们提出并分析了一个具有捕食者,食饵和既有周期脉冲输入又有周期脉冲输出营养液的恒化器模型.我们得到了一种微生物和营养液共存的周期解,同时,也得到两种微生物都绝灭的周期解,而且建立了周期解稳定的充分条件.最后,我们给出了一个简单的讨论.     

Dynamics of an impulsive control system which prey species share a common predator

In an ecosystem multiple prey species often share a common predator and the interactions between the preys are neutral. In view of these facts and based on a multiple species prey–predator system with Holling IV and II functional responses, an impulsive differential equation to model the process of periodically releasing natural enemies and spraying pesticides at different fixed times for pest control is proposed and investigated. It is proved that there exists a locally asymptotically stable pest-eradication periodic solution under the assumption that the impulsive period is less than some critical value (or the release amount of the predator is greater than another critical value). Permanence conditions are established when the impulsive period is greater than another critical value (or the release amount of the predator is less than some critical value). Numerical results show that the system we consider has more complex dynamics including period solution, quasi-periodic oscillation, chaos, intermittency and crises.     

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.     

Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain

An attempt has been made to understand the role of top predator interference and gestation delay on the dynamics of a three species food chain model involving intermediate and top predator populations. Interaction between the prey and an intermediate predator follows the Volterra scheme (with Holling type IV functional response), while that between the top predator and its prey depends on Beddington–DeAngelis type functional response. Stability switches and Hopf-bifurcation occurs when the delay crosses some critical value. Model system exhibits irregular behavior when the interference is high or gestation period is larger than its critical value. Furthermore, the direction of Hopf-bifurcation and the stability of the bifurcating periodic solutions are determined using the center manifold theorem and normal form theory. Computer simulations have been carried out to illustrate the analytical findings. Different diagnostic tests, like, initial sensitivity, Lyapunov exponent, recurrence plot tests ensure the complex dynamical behavior of the model system. Finally, we observed the subcritical Hopf-bifurcation phenomena in the designed model system and the bifurcating periodic solution is unstable for the considered set of parameter values.     

Qualitative behaviour of a ratio-dependent predator–prey system

This paper deals with the qualitative properties of an autonomous system of differential equations, modeling ratio-dependent predator–prey interactions.This model differs from traditional ratio dependent models essentially in the predator mortality term, the death rate of the predator is not constant but instead increases when there is overcrowding.We incorporate delay(s) into the system. The most important observation is that as the delay(s) is (are) increased the originally asymptotic stable interior equilibrium loses its stability. Furthermore at a certain critical value a Hopf bifurcation takes place: small amplitude periodic solutions arise.     

Analysis of a Monod–Haldene type food chain chemostat with periodically varying substrate

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with periodically varying 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. Furthermore, we numerically simulate a model with sinusoidal input, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

比率型-捕食者-两竞争食饵模型的动力学行为

本文研究比率型非自治的捕食者 -食饵模型 .该系统是两个具有竞争关系的食饵种群被一个捕食种群捕食 .我们研究其动力学行为 ,包括持久性 ,全局渐近稳定性 ,周期解 ,概周期解的存在唯一性     

Bifurcation and Turing pattern formation in a diffusive ratio-dependent predator–prey model with predator harvesting

The ratio-dependent predator–prey model exhibits rich dynamics due to the singularity of the origin. Harvesting in a ratio-dependent predator–prey model is relatively an important research project from both ecological and mathematical points of view. In this paper, we study the temporal, spatial and spatiotemporal dynamics of a ratio-dependent predator–prey diffusive model where the predator population harvest at catch-per-unit-effort hypothesis. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction–diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibit Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the existence and non-existence of positive non-constant steady-state solutions are established. Moreover, numerical simulations are performed to visualize the complex dynamic behavior.