Traveling wave solutions for a predator–prey system with two predators and one prey

We study a predator–prey model with two alien predators and one aborigine prey in which the net growth rates of both predators are negative. We characterize the invading speed of these two predators by the minimal wave speed of traveling wave solutions connecting the predator-free state to the co-existence state. The proof of the existence of traveling waves is based on a standard method by constructing (generalized) upper-lower-solutions with the help of Schauder’s fixed point theorem. However, in this three species model, we are able to construct some suitable pairs of upper-lower-solutions not only for the super-critical speeds but also for the critical speed. Moreover, a new form of shrinking rectangles is introduced to derive the right-hand tail limit of wave profile.     

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.     

A predator–prey model with disease in the prey and two impulses for integrated pest management

In this paper, a predator–prey model with disease in the prey is constructed and investigated for the purpose of integrated pest management. In the first part of the main results, the sufficient condition for the global stability of the susceptible pest-eradication periodic solution is obtained, which means if the release amount of infective prey and predator satisfy the condition, then the pest will be controlled. The sufficient condition for the permanence of the system is also obtained subsequently, which means if the release amount of infective prey and predator satisfy the condition, then the prey and the predator will coexist. At last, we interpret our mathematical results.     

Global dynamics of a predator–prey model with defense mechanism for prey

Recently, Venturino and Petrovskii proposed a general predator–prey model with group defense for prey species (Venturino and Petrovskii, 2013). The local dynamics had been studied and showed that the model might undergo Hopf bifurcation, and have an extinction domain. In this paper, we dedicate ourselves to the investigation of the global dynamics of the model by establishing the conditions of the nonexistence of periodic orbits, and the existence and uniqueness of limit cycles.     

The effect of predator competition on positive solutions for a predator–prey model with diffusion

A diffusive predator–prey model with predator competition is considered under Dirichlet boundary conditions. Some existence and non-existence results are firstly obtained. Then by investigating the bifurcation of positive solutions, the multiplicity of positive solutions is established for suitably large m$m$. Furthermore, by meticulously analyzing the asymptotic behaviors of positive solutions when k$k$ goes to ∞$\infty$, we find that there is at most a positive solution for any c∈R$c\in R$ when k$k$ is sufficiently large. At last, some numerical simulations are presented to supplement the analytic results in one dimension.     

Dynamics of a stage-structured predator–prey model with prey impulsively diffusing between two patches

In this work, we propose a stage-structured predator–prey model, with prey impulsively diffusing between two patches. Using the discrete dynamical system determined by the stroboscopic map, we obtain a predator-extinction periodic solution. Further, the predator-extinction periodic solution is globally attractive. By the theory on the delay and impulsive differential equation, we prove that the investigated system is permanent. Our results indicate that the discrete time delay has influence to the dynamical behaviors of the investigated system.     

Spatial propagation of a lattice predation–competition system with one predator and two preys in shifting habitats

This paper deals with a discrete diffusive predator–prey system involving two competing preys and one predator in a shifting habitat induced by the climate change. By applying Schauder's fixed-point theorem on various invariant cones via constructing several pairs of generalized super- and subsolutions, we establish four different types of supercritical and critical forced extinction waves, which describe the conversion from the state of a saturated aboriginal prey with a pair of invading alien predator–prey, two competing aboriginal coexistent preys with an invading alien predator, a pair of aboriginal coexistent predator–prey and an invading alien prey, and the coexistence of three species to the extinction state, respectively. Meanwhile, the nonexistence of some subcritical forced waves is showed by contradiction. Furthermore, some numerical simulations are given to present and promote the theoretical results.     

Stability and Hopf bifurcation for a delayed predator–prey model with disease in the prey

This paper is concerned with a mathematical model dealing with a predator–prey system with disease in the prey. Mathematical analysis of the model regarding stability has been performed. The effect of delay on the above system is studied. By regarding the time delay as the bifurcation parameter, the stability of the positive equilibrium and Hopf bifurcations are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, some numerical simulations are also included.     

Hopf bifurcation and center stability for a predator–prey biological economic model with prey harvesting

In this paper, we investigate Hopf bifurcation and center stability of a predator–prey biological economic model. By employing the local parameterization method, Hopf bifurcation theory and the formal series method, we obtain some testable results on these issues. The economic profit is chosen as a positive bifurcation parameter here. It shows that a phenomenon of Hopf bifurcation occurs as the economic profit increases beyond a certain threshold. Besides, we also find that the center of the biological economic model is always unstable. Finally, some numerical simulations are given to illustrate the effectiveness of our results.     

Chaos and Hopf bifurcation analysis for a two species predator–prey system with prey refuge and diffusion

In this paper, a delayed predator–prey model with Holling type II functional response incorporating a constant prey refuge and diffusion is considered. By analyzing the characteristic equation of linearized system corresponding to the model, we study the local asymptotic stability of the positive equilibrium of the system. By choosing the time delay due to gestation as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Further, an example is presented to illustrate our main results. Finally, recurring to the numerical method, the influences of impulsive perturbations on the dynamics of the system are also investigated.     

Dynamics of a delayed predator–prey model with predator migration

Based on the availability of prey and a simple predator–prey model, we propose a delayed predator–prey model with predator migration to describe biological control. We first study the existence and stability of equilibria. It turns out that backward bifurcation occurs with the migration rate as bifurcation parameter. The stability of the trivial equilibrium and the boundary equilibrium is delay-independent. However, the stability of the positive equilibrium may be delay-dependent. Moreover, delay can switch the stability of the positive equilibrium. When the positive equilibrium loses stability, Hopf bifurcation can occur. The direction and stability of Hopf bifurcation is derived by applying the center manifold method and the normal form theory. The main theoretical results are illustrated with numerical simulations.     

Dynamics for a diffusive prey–predator model with different free boundaries

To understand the spreading and interaction of prey and predator, in this paper we study the dynamics of the diffusive Lotka–Volterra type prey–predator model with different free boundaries. These two free boundaries, which may intersect each other as time evolves, are used to describe the spreading of prey and predator. We investigate the existence and uniqueness, regularity and uniform estimates, and long time behaviors of global solution. Some sufficient conditions for spreading and vanishing are established. When spreading occurs, we provide the more accurate limits of $(u,v)$ as $t\to \infty$, and give some estimates of asymptotic spreading speeds of $u,v$ and asymptotic speeds of $g,h$. Some realistic and significant spreading phenomena are found.     

Global dynamics of a predator–prey model with time delay and stage structure for the prey

A stage-structured predator–prey system with Holling type-II functional response and time delay due to the gestation of predator is investigated. By analyzing the characteristic equations, the local stability of each of feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when both the predator-extinction equilibrium and the coexistence equilibrium are not feasible, and that the predator-extinction equilibrium is globally asymptotically stable if the coexistence equilibrium does not exist, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

Complex patterns in a predator–prey model with self and cross-diffusion

In this paper, we present a theoretical analysis of processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with self as well as cross-diffusion in a Beddington–DeAngelis-type predator–prey model. The instability of the uniform equilibrium of the model is discussed, and the sufficient conditions for the instability with zero-flux boundary conditions are obtained. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self as well as cross-diffusion in the model, and find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to stripes-spots, but also to hot/cold spots, stripes and wave pattern replication. This may enrich the pattern formation in cross-diffusive predator–prey model.     

Permanence of periodic predator–prey system with two predators and stage structure for prey

A stage-structured three-species predator–prey system with Beddington–DeAngelis and Holling IV functional response is proposed and analyzed. Based on the comparison theorem, some sufficient and necessary conditions are derived for permanence of the system. Finally, two examples are presented to illustrate the application of our main results.     

Stability and Hopf bifurcation of a diffusive predator–prey model with predator saturation and competition

This article discusses a predator–prey system with predator saturation and competition functional response. The local stability, existence of a Hopf bifurcation at the coexistence equilibrium and stability of bifurcating periodic solutions are obtained in the absence of diffusion. Further, we discuss the diffusion-driven instability, Hopf bifurcation for corresponding diffusion system with zero flux boundary condition and Turing instability region regarding the parameters are established. Finally, numerical simulations supporting the theoretical analysis are also included.     

Optimal harvesting policy for a stochastic predator–prey model

Optimal harvesting of a stochastic predator–prey model is considered in this paper. Sufficient and necessary criteria for the existence of optimal harvesting strategy are obtained. At the same time, the optimal harvest effort and the maximum of sustainable yield are given.     

Stability and bifurcation analysis for a fractional prey–predator scavenger model

In this study, we consider a fractional prey–predator scavenger model as well as harvesting by a predator and scavenger. We prove the positivity and boundedness of the solutions in this system. The model undergoes a Hopf bifurcation around one of the existing equilibria where the conditions are met for the occurrence of a Hopf bifurcation. The results show that chaos disappears in this biological model. We conclude that the fractional system is more stable compared with the classical case and the stability domain can be extended under fractional order. In addition, a suitable amount of prey harvesting and a fractional order derivative can control the chaotic dynamics and stabilize them. We also present an extended numerical simulation to validate the results.     

Stability of a predator–prey model with refuge effect

We consider a predator–prey model, where some prey are completely free from predation within a temporal or spacial refuge. The most common type of spacial refuge, that we investigate here, takes the form where a constant proportion of the prey population is protected. The model is a modification of the classical Nicholson–Bailey host-parasitoid model. In this paper, we study the effect of the presence of refuge on the stability and bifurcation of the system. Moreover, we provide a detailed analysis of the Neimark–Sacker bifurcation of the model.     

A Leslie–Gower predator–prey model with disease in prey incorporating a prey refuge

In this paper, a predator–prey Leslie–Gower model with disease in prey has been developed. The total population has been divided into three classes, namely susceptible prey, infected prey and predator population. We have also incorporated an infected prey refuge in the model. We have studied the positivity and boundedness of the solutions of the system and analyzed the existence of various equilibrium points and stability of the system at those equilibrium points. We have also discussed the influence of the infected prey refuge on each population density. It is observed that a Hopf bifurcation may occur about the interior equilibrium taking refuge parameter as bifurcation parameter. Our analytical findings are illustrated through computer simulation using MATLAB, which show the reliability of our model from the eco-epidemiological point of view.