Predator invasion in predator–prey model with prey-taxis in spatially heterogeneous environment

We consider a predator–prey model with prey-taxis and Holling-type II functional responses in a spatially heterogeneous environment to analyze the effects of prey-taxis and the heterogeneity of an environment on predator invasion. To achieve our goal, we investigate the stability of semi-trivial solution in which the predator is absent. It is known that both the predator diffusion and the death rate contribute to the predator invasion in a heterogeneous habitat when there is no prey-taxis. In this paper, we show that predator invasion is affected by the prey-taxis and diffusions of the prey-taxis model for a certain range of predator death rates in a heterogeneous environment. Furthermore, in cases where predator invasion by predator diffusion does not occur in a particular death rate range of the predator, predator invasion can occur by prey-taxis in a spatially heterogeneous habitat. In addition, we compare this phenomenon to the corresponding predator–prey model with ratio-dependent functional responses. It is observed that none of the predator’s diffusion and prey-taxis affect the predator’s invasion, and that only the predator’s death rate contributes to predator invasion for the model with ratio-dependent functional responses.     

A Regularity Condition of 3d Axisymmetric Navier-Stokes Equations

This paper is concerned with a Variable-territory model with limited self-limitation of predator. By the bifurcation theorem, regular perturbation theorem and numerical simulation, the conditions of existence, stability and convergence of positive solutions are established. This work shows that prey and predator can be controlled by parameters in Variable-territory model, such as intrinsic growth rate, death rate, handling time and the self-limitation of predator.     

Dynamics in a ratio-dependent predator-prey model with predator harvesting

The objective of this paper is to study systematically the dynamical properties of a ratio-dependent predator-prey model with nonzero constant rate predator harvesting. It is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation), the subcritical and supercritical Hopf bifurcations. These results reveal far richer dynamics compared to the model with no harvesting and different dynamics compared to the model with nonzero constant rate prey harvesting in [D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM Appl. Math. 65 (2005) 737-753]. Biologically, it is shown that nonzero constant rate predator harvesting can prevent mutual extinction as a possible outcome of the predator prey interaction, and remove the singularity of the origin, which was regarded as “pathological behavior” for a ratio-dependent predator prey model in [P. Yodzis, Predator-prey theory and management of multispecies fisheries, Ecological Applications 4 (2004) 51-58].     

Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure

We present a predator-prey model of Beddington-DeAngelis type functional response with stage structure on prey. The constant time delay is the time taken from birth to maturity about the prey. By the uniform persistence theories and monotone dynamic theories, sharp threshold conditions which are both necessary and sufficient for the permanence and extinction of the model as well as the sufficient conditions for the global stability of the coexistence equilibria are obtained. Biologically, it is proved that the variation of prey stage structure can affect the permanence of the system and drive the predator into extinction by changing the prey carrying capacity: Our results suggest that the predator coexists with prey permanently if and only if predator's recruitment rate at the peak of prey abundance is larger than its death rate; and that the predator goes extinct if and only if predator's possible highest recruitment rate is less than or equal to its death rate; furthermore, our results also show that a sufficiently large mutual interference by predators can stabilize the system.     

A stage-structured food chain model with stage dependent predation: Existence of codimension one and codimension two bifurcations

Stage-structured predator–prey models exhibit rich and interesting dynamics compared to homogeneous population models. The objective of this paper is to study the bifurcation behavior of stage-structured prey–predator models that admit stage-restricted predation. It is shown that the model with juvenile-only predation exhibits Hopf bifurcation with the growth rate of the adult prey as the bifurcation parameter; also, depending on parameter values, a stable limit cycle will emerge, that is, the bifurcation will be of supercritical nature. On the other hand, the analysis of the model with adult-stage predation shows that the system admits a fold-Hopf bifurcation with the adult growth rate and the predator mortality rate as the two bifurcation parameters. We also demonstrate the existence of a unique limit cycle arising from this codimension-2 bifurcation. These results reveal far richer dynamics compared to models without stage-structure. Numerical simulations are done to support analytical results.     

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.     

POSITIVE STEADY STATES OF A DIFFUSIVE PREDATOR-PREY SYSTEM WITH PREDATOR CANNIBALISM

The purpose of this paper is to investigate positive steady states of a diffusive predator-prey system with predator cannibalism under homogeneous Neumann boundary conditions. With the help of implicit function theorem and energy integral method, nonexistence of non-constant positive steady states of the system is obtained, whereas coexistence of non-constant positive steady states is derived from topological degree theory. The results indicate that if dispersal rate of the predator or prey is sufficiently large, there is no nonconstant positive steady states. However, under some appropriate hypotheses, if the dispersal rate of the predator is larger than some positive constant, for certain ranges of dispersal rates of the prey, there exists at least one non-constant positive steady state.     

Challenges of living in the harsh environments: A mathematical modeling study

We propose and analyze a mathematical model, which mimics community dynamics of plants and animals in harsh environments. The mathematical model exploits type IV functional responses whose idiosyncrasies have been recognized only in recent years. The interaction of the middle predator with the top predator is cast into Leslie-Gower scheme. Linear and non-linear stability analyses are performed to get an idea of the stability behavior of the model food chain. It turns out that carrying capacity of the prey and the immunity parameter of the middle predator are two crucial parameters governing the model. Availability of alternative food options to the generalist predator also plays a key role in deciding the model dynamics.Simulation runs performed on this model provide insight into population dynamics of monkeys of macaque family found in northern Japan. These monkeys are social animals which reproduce sexually. The characteristic feature of the model dynamics is that the generalist predator (macaque monkeys) is able to avoid impending extinction frequently and recovers at a rate which falsify threats from exogenous external forces; extreme weather conditions, etc.     

Classical predator–prey system with infection of prey population—a mathematical model

The present paper deals with the problem of a classical predator–prey system with infection of prey population. A classical predator–prey system is split into three groups, namely susceptible prey, infected prey and predator. The relative removal rate of the susceptible prey due to infection is worked out. We observe the dynamical behaviour of this system around each of the equilibria and point out the exchange of stability. It is shown that local asymptotic stability of the system around the positive interior equilibrium ensures its global asymptotic stability. We prove that there is always a Hopf bifurcation for increasing transmission rate. To substantiate the analytical findings, numerical experiments have been carried out for hypothetical set of parameter values. Our analysis shows that there is a threshold level of infection below which all the three species will persist and above which the disease will be epidemic. Copyright © 2003 John Wiley & Sons, Ltd.     

Extinction and permanence of the predator–prey system with stocking of prey and harvesting of predator impulsively

In this paper, a predator–prey system with stocking of prey and harvesting of predator impulsively is studied. Here, the prey population is stocked with a constant quantity and the predator population is harvested at a rate proportional to the species itself at fixed moments. Under some conditions, the existence and global asymptotic stability of the boundary periodic solution are proved, which implies that the system will be extinct; and given some different restrictions, ultimate positive upper and lower bounds of all solutions are obtained, showing the system being permanent. At last, two examples are given to illustrate our results. Copyright © 2005 John Wiley & Sons, Ltd.     

Global dynamics of a predator-prey model

This paper deals with the dynamics of a predator-prey model with Hassell-Varley-Holling functional response. First, we show that the predator coexists with prey if and only if predator's growth ability is greater than its death rate. Second, using a blow-up technique, we prove that the origin equilibrium point is repelling and extinction of both predator and prey populations is impossible. Third, the local and global stability of the positive steady state coincide when the predator interference is large. Finally, for a typical biological case, we show instability of the positive equilibrium implies global stability of the limit cycle. Numerical simulations are carried out for a hypothetical set of parameter values to substantiate our analytical findings.     

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.     

Stability Analysis of a Diffusive Predator-prey Model with Hunting Cooperation

In this paper, we are concerned with the dynamics of a diffusive predator-prey model that incorporates the functional response concerning hunting cooperation. First, we investigate the stability of the semi-trivial steady state. Then, we investigate the influence of the diffusive rates on the stability of the positive constant steady state. It is shown that there exists diffusion-driven Turing instability when the diffusive rate of the predator is smaller than the critical value, which is dependent on the diffusive rate of the prey, and the semi-trivial steady state and the positive constant steady state are both locally asymptotically stable when the diffusive rate of the predator is larger than the critical value. Finally, the nonexistence of nonconstant steady states is discussed.     

A two-patch prey-predator model with food-gathering activity

In insect ecosystem, the dynamics of prey and predator is regulated by complex interactions between them. Insect pests are spatially aggregated in patches forming a spatial pattern in the environment. An efficient predator dynamically changes its strategies and time for its random search movements to concentrate on higher resource patches based on the benefit of assessment. This food-gathering activity of both prey and predator plays a major role in stabilizing the system by influencing the per unit food consumption. Extending Holling time-budget argument by migration, here we formulate a two patch prey-predator model and show that how several foraging parameters such as handling time, dispersal rate can have important consequences in stability of prey-predator system. Specifically, the ratio between timings that a predator remains mobile in searching and handling their food, is the most important one and simulation on this suggests that the stabilizing effect continues to operate when the dispersal process is modeled more realistically. Thus we conclude that the migration submodel is an important constituent of a spatial predator-prey system. These results are shown to have important implications for possible biological control.     

Nonexistence of Nonconstant Positive Steady States of a Diffusive Predator-prey Model with Fear Effect

In this paper, we investigate a diffusive predator-prey model with fear effect. It is shown that, for the linear predator functional response case,the positive constant steady state is globally asymptotically stable if it ex-ists. On the other hand, for the Holling type II predator functional response case, it is proved that there exist no nonconstant positive steady states for large conversion rate. Our results limit the parameters range where complex spatiotemporal pattern formation can occur.     

A Lyapunov functional for a stage-structured predator–prey model with nonlinear predation rate

We consider the dynamics of a general stage-structured predator–prey model which generalizes several known predator–prey, SEIR, and virus dynamics models, assuming that the intrinsic growth rate of the prey, the predation rate, and the removal functions are given in an unspecified form. Using the Lyapunov method, we derive sufficient conditions for the local stability of the equilibria together with estimations of their respective domains of attraction, while observing that in several particular but important situations these conditions yield global stability results. The biological significance of these conditions is discussed and the existence of the positive steady state is also investigated.     

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.     

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.     

Predator–prey harvesting model with fatal disease in prey

A theoretical eco‐epidemiological model of a prey–predator interaction system with disease in prey species is studied. Predator consumes both susceptible and infected prey population, but predator also feeds preferentially on many numerous species, which are over represented in the predator's diet. Equilibrium points of the system are determined, and the dynamic behaviour of the system is investigated around equilibrium points. Death rate of predator species is considered as a bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighbourhood of the coexisting equilibria. Numerical simulations are carried out to support the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Positive Solutions for a Lotka–Volterra Prey–Predator Model with Cross-Diffusion of Fractional Type

In this paper we consider a Lotka–Volterra prey–predator model with cross-diffusion of fractional type. The main purpose is to discuss the existence and nonexistence of positive steady state solutions of such a model. Here a positive solution corresponds to a coexistence state of the model. Firstly we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system. Secondly we derive some necessary conditions to ensure the existence of positive solutions, which demonstrate that if the intrinsic growth rate of the prey is too small or the death rate (or the birth rate) of the predator is too large, the model does not possess positive solutions. Thirdly we study the sufficient conditions to ensure the existence of positive solutions by using degree theory. Finally we characterize the stable/unstable regions of semi-trivial solutions and coexistence regions in parameter plane.