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.     

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.     

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.     

Dynamical behavior of a delayed diffusive predator–prey model with competition and type III functional response

In this paper, a delayed diffusive predator–prey model with competition and type III functional response is investigated. By using inequality analytical technique, some sufficient conditions which ensure the permanence of the model have been derived. By Lyapunov functional method, a series of sufficient conditions which assure the global asymptotic stability of the system are established. The paper ends with some numerical simulations that illustrate our analytical predictions.     

Modeling and analysis of a prey–predator system with disease in the prey

A three dimensional ecoepidemiological model consisting of susceptible prey, infected prey and predator is proposed and analysed in the present work. The parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected. Mathematically we analyze the dynamics of the system such as, boundedness of the solutions, existence of non-negative equilibria, local and global stability of interior equilibrium point. Next we choose delay as a bifurcation parameter to examine the existence of the Hopf bifurcation of the system around its interior equilibrium. Moreover we use the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Some numerical simulations are carried out to support the analytical results.     

Chaotic dynamics of a discrete prey–predator model with Holling type II

A discrete-time prey–predator model with Holling type II is investigated. For this model, the existence and stability of three fixed points are analyzed. The bifurcation diagrams, phase portraits and Lyapunov exponents are obtained for different parameters of the model. The fractal dimension of a strange attractor of the model was also calculated. Numerical simulations show that the discrete model exhibits rich dynamics compared with the continuous model, which means that the present model is a chaotic, and complex one.     

Impulsive perturbations in a predator–prey model with dormancy of predators

In this paper, a predator–prey model consisting of active and dormant states of predators with impulsive control strategy is established. Using Floquet theories, the small amplitude perturbation technique and the piecewise Lyapunov function method, the conditions of local and global asymptotical orbital stability of the prey-eradication periodic solution are obtained. The boundness and permanence of the impulsive system are proved by the comparison principle. Through numerical simulations, the effects of the impulsive perturbation on the inherent oscillation are investigated, which implies that the impulsive perturbation can lead to period-doubling bifurcation, chaos, and period-halving bifurcation. Moreover, the effects of the impulsive perturbation and hatching rate on the chaos of the system are comparatively studied by numerical simulation. These obtained results can be useful for ecosystem management and for explaining complex phenomena of ecosystems.     

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.     

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.     

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 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 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.     

Dynamics of a predator–prey system with inducible defense and disease in the prey

Establishing and researching a population dynamical model based on the differential equation is of great significance. In this paper, a predator–prey system with inducible defense and disease in the prey is built from biological evolution and Eco-epidemiology. The effect of disease on population stability in the predator–prey system with inducible defense is studied. Firstly, we verify the positivity and uniform boundedness of the solutions of the system. Then the existence and stability of the equilibria are studied. There are no more than nine equilibrium points in the system. We use a sophisticated parameter transformation to study the properties of the coexistence equilibrium points of the system. A sufficient condition is established for the existence of Hopf bifurcation. Numerical simulations are performed to make analytical studies more complete.     

Dynamics of impulsive reaction–diffusion predator–prey system with Holling type III functional response

An impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator–prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given at the end.     

Effects of prey refuge on a ratio-dependent predator–prey model with stage-structure of prey population

In this paper, a stage-structured predator–prey model is proposed and analyzed to study how the type of refuges used by prey population influences the dynamic behavior of the model. Two types of refuges: those that protect a fixed number of prey and those that protect a constant proportion of prey are considered. Mathematical analyses with regard to positivity, boundedness, equilibria and their stabilities, and bifurcation are carried out. Persistence condition which brings out the useful relationship between prey refuge parameter and maturation time delay is established. Comparing the conclusions obtained from analyzing properties of two types of refuges using by prey, we observe that value of maturation time at which the prey population and hence predator population go extinct is greater in case of refuges which protect a constant proportion of prey.     

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.     

Bifurcations in a predator–prey model in patchy environment with diffusion

In this paper we formulate a predator–prey system in two patches in which the per capita migration rate of each species is influenced only by its own density, i.e. there is no response to the density of the other one. Numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation, i.e. the stable constant steady state loses its stability and spatially non-constant stationary solutions, a pattern emerge.