A ratio-dependent predator–prey model with diffusion

This paper deals with the existence and nonexistence of nonconstant positive steady-state solutions to a ratio-dependent predator–prey model with diffusion and with the homogeneous Neumann boundary condition. We demonstrate that there exists $a_0(b)$ satisfying $0<a_0(b)<m_1$ for $0<b<m_1$, such that if $0<b<m_1$ and $a_0(b)<a<m_1$, then the diffusion can create nonconstant positive steady-state solutions; whereas the diffusion cannot do provided $a>m_1$.     

A predator–prey system with stage-structure for predator and nonlocal delay

This paper deals with the behavior of solutions to the reaction–diffusion system under homogeneous Neumann boundary condition, which describes a prey–predator model with nonlocal delay. Sufficient conditions for the global stability of each equilibrium are derived by the Lyapunov functional and the results show that the introduction of stage-structure into predator positively affects the coexistence of prey and predator. Numerical simulations are performed to illustrate the results.     

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.     

A modified Leslie–Gower predator–prey model with prey infection

The disease effect on ecological systems is an important issue from mathematical and experimental point of view. In this paper, we formulate and analyze a predator–prey model for the susceptible population, infected population and their predator population with modified Leslie–Gower (or Holling–Tanner) functional response. Mathematical analysis of the model equations with regard to invariance of nonnegativity and boundedness of solutions, local and global stability of the biological feasible equilibria and permanence of the system are presented. When the rate of infection crosses a critical value, we determine that the strictly positive interior equilibrium undergoes Hopf bifurcation. From our numerical simulations, we observe that the predation rate also plays an important role on the dynamic behavior of our system.     

A predator–prey–disease model with immune response in infected prey

In this paper, a predator–prey–disease model with immune response in the infected prey is formulated. The basic reproduction number of the within-host model is defined and it is found that there are three equilibria: extinction equilibrium, infection-free equilibrium and infection-persistent equilibrium. The stabilities of these equilibria are completely determined by the reproduction number of the within-host model. Furthermore, we define a basic reproduction number of the between-host model and two predator invasion numbers: predator invasion number in the absence of disease and predator invasion number in the presence of disease. We have predator and infection-free equilibrium, infection-free equilibrium, predator-free equilibrium and a co-existence equilibrium. We determine the local stabilities of these equilibria with conditions on the reproduction and invasion reproduction numbers. Finally, we show that the predator-free equilibrium is globally stable.     

The spreading speed for a predator–prey model with one predator and two preys

We study a predator–prey model with two preys and one predator. Our main concern is the invasion process of the predator into the habitat of two aborigine preys. We consider the case when the two preys are weak competitors in the absence of predator. Under certain conditions, we are able to characterize the asymptotic spreading speed by the parameters of the model.     

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.     

A bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting

In this paper, we propose a bioeconomic differential algebraic predator–prey model with Holling type II functional response and nonlinear prey harvesting. As the nonlinear prey harvesting is introduced, the proposed model displays a complex dynamics in the predator–prey plane. Taking into account of the economic factor, our predator–prey system is established by bioeconomic differential algebraic equations. The effect of economic profit on the proposed model is analyzed by viewing it as a bifurcation parameter. By jointly using the normal form of differential algebraic models and the bifurcation theory, the stability and bifurcations (singularity induced bifurcation, Hopf bifurcation) are discussed. These results obtained here reveal richer dynamics of the bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting, and suggest a guidance for harvesting in the practical word. Finally, numerical simulations are given to demonstrate the results.     

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.     

Two-patches prey impulsive diffusion periodic predator–prey model

In this paper, we study a periodic predator–prey system with prey impulsive diffusion in two patches. On the basis of comparison theorem of impulsive differential equation and other analysis methods, sufficient and necessary conditions on the predator–prey system where predator have not other food source are established. Two examples and numerical simulations are presented to illustrate the feasibility of our results. A conclusion is given in the end.     

On the dynamics of a predator–prey model with nonconstant death rate and diffusion

The main goal of this paper is to describe the global dynamic of a predator–prey model with nonconstant death rate and diffusion. We obtain necessary and sufficient conditions under which the system is dissipative and permanent. We study the global stability of the nontrivial equilibrium, when it is unique. Finally, we show that there are no nontrivial steady state solutions for certain parameter configuration.     

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.     

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.     

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.     

Stochastic predator–prey model with Allee effect on prey

We study a predator–prey model with the Allee effect on prey and whose dynamics is described by a system of stochastic differential equations assuming that environmental randomness is represented by noise terms affecting each population. More specifically, we consider a term that expresses the variability of the growth rate of both species due to external, unpredictable events. We assume that the intensities of these perturbations are proportional to the population size of each species. With this approach, we prove that the solutions of the system have sample pathwise uniqueness and bounded moments. Moreover, using an Euler–Maruyama-type numerical method we obtain approximated solutions of the system with different intensities for the random noise and parameters of the model. In the presence of a weak Allee effect, we show that long-term survival of both populations can occur. On the other hand, when a strong Allee effect is considered, we show that the random perturbations may induce the non-trivial attracting-type invariant objects to disappear, leading to the extinction of both species. Furthermore, we also find the Maximum Likelihood estimators for the parameters involved in the model.     

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.     

Traveling waves for a reaction–diffusion–advection predator–prey model

In this paper we study a reaction–diffusion–advection predator–prey model in a river. The existence of predator-invasion traveling wave solutions and prey-spread traveling wave solutions in the upstream and downstream directions is established and the corresponding minimal wave speeds are obtained. While some crucial improvements in theoretical methods have been established, the proofs of the existence and nonexistence of such traveling waves are based on Schauder’s fixed-point theorem, LaSalle’s invariance principle and Laplace transform. Based on theoretical results, we investigate the effect of the hydrological and biological factors on minimal wave speeds and hence on the spread of the prey and the invasion of the predator in the river. The linear determinacy of the predator–prey Lotka–Volterra system is compared with nonlinear determinacy of the competitive Lotka–Volterra system to investigate the mechanics of linear and nonlinear determinacy.