Stability and Hopf bifurcation of a diffusive predator‐prey model with hyperbolic mortality

The dynamics of a reaction‐diffusion predator‐prey model with hyperbolic mortality and Holling type II response effect is considered. The stability of the positive equilibrium and the existence of Hopf bifurcation are investigated by analyzing the distribution of eigenvalues without diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system which are spatially homogeneous. To verify our theoretical results, some numerical simulations are also presented. © 2015 Wiley Periodicals, Inc. Complexity 21: 34–43, 2016     

Zero‐Hopf bifurcation in the Volterra‐Gause system of predator‐prey type

We prove that the Volterra‐Gause system of predator‐prey type exhibits 2 kinds of zero‐Hopf bifurcations for convenient values of their parameters. In the first, 1 periodic solution bifurcates from a zero‐Hopf equilibrium, and in the second, 4 periodic solutions bifurcate from another zero‐Hopf equilibrium. This study is done using the averaging theory of second order.     

Turing instability and Hopf bifurcation in a diffusive Leslie–Gower predator–prey model

In this paper, a reaction‐diffusion predator–prey system that incorporates the Holling‐type II and a modified Leslie‐Gower functional responses is considered. For ODE, the local stability of the positive equilibrium is investigated and the specific conditions are obtained. For partial differential equation, we consider the dissipation and persistence of solutions, the Turing instability of the equilibrium solutions, and the Hopf bifurcation. By calculating the normal form, we derive the formulae, which can determine the direction and the stability of Hopf bifurcation according to the original parameters of the system. We also use some numerical simulations to illustrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamics of a ratio‐dependent stage‐structured predator‐prey model with delay

In this paper, we investigate the dynamics of a time‐delay ratio‐dependent predator‐prey model with stage structure for the predator. This predator‐prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. 26 We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.     

Bifurcations in a diffusive predator–prey model with predator saturation and competition response

A diffusive predator–prey model with predator saturation and competition response subject to homogeneous Neumann boundary conditions is considered in this paper. We find that the spatially homogeneous and non‐homogeneous periodic solutions through all parameters of the system are spatially homogeneous. To verify our theoretical results, some numerical simulations are also carried out. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

A simple Gause‐type predator–prey model considering social predation

The consumer–resource relationships are among the most fundamental of all ecological relationships and have been the focus of ecology since its beginnings. Usually are described by nonlinear differential equation systems, putting the emphasis in the effect of antipredator behavior (APB) by the prey; nevertheless, a minor quantity of articles has considered the social behavior of predators. In this work, two predator–prey models derived from the Volterra model are analyzed, in which the equation of predators is modified considering cooperation or collaboration among predators. It is well known that competition among predators produces a stabilizing effect on system describing the model, since there exists a wide set in the parameter space where the system has a unique equilibrium point in the phase plane, which is globally asymptotically stable. Meanwhile, the cooperation can originate more complex and unusual dynamics. As we will show, it is possible to prove that for certain subset of parameter values the predator population sizes tend to infinite when the prey population goes to extinct. This apparently contradicts the idea of a realistic model, when it is implicitly assumed that the predators are specialist, ie, the prey is its unique source of food. However, this could be a desirable effect when the prey constitutes a plague. To reinforce the analytical result, numerical simulations are presented.     

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter.     

Hopf bifurcation in a diffusive predator-prey model with herd behavior and prey harvesting

In this paper, the dynamics of a diffusive delayed predator-prey model with herd behavior and prey harvesting subject to the homogeneous Neumann boundary condition is considered. Firstly, choosing the harvesting term as a bifurcation parameter, then we obtain the existence and the stability of the equilibrium by analyzing the distribution of the roots of associated characteristic equation. Secondly, time delay is regarding as a bifurcation parameter, and the use of the normal form theory and center manifold theorem, the existence, stability and direction of bifurcating periodic solutions are all demonstrated detailly. Finally, summarizing some numerical simulations to illustrate the theoretical analysis.     

The effect of prey refuge and time delay on a diffusive predator‐prey system with hyperbolic mortality

In this article, we investigate the effect of prey refuge and time delay on a diffusive predator‐prey system with Holling II functional response and hyperbolic mortality subject to Neumann boundary condition. More precisely, we study Turing instability of positive equilibrium by using refuge as parameter, instability and Hopf bifurcation induced by time delay. In addition, by the theory of normal form and center manifold, we derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution. © 2016 Wiley Periodicals, Inc. Complexity 21: 446–459, 2016     

Stability and Hopf bifurcation in a predator–prey model with stage structure for the predator

A predator–prey system with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. Further, the existence of a Hopf bifurcation at the positive equilibrium is also studied. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. As a result, the threshold is obtained for the permanence and extinction of the system. Numerical simulations are carried out to illustrate the main results.     

Global stability of a periodic Holling‐Tanner predator‐prey model

This paper is concerned with the global dynamics of a Holling‐Tanner predator‐prey model with periodic coefficients. We establish sufficient conditions for the existence of a positive solution and its global asymptotic stability. The stability conditions are first given in average form and afterward as pointwise estimates. In the autonomous case, the previous criteria lead to a known result.     

A fractional‐order model for chronic lymphocytic leukemia and immune system interactions

In this study, a mathematical, fractional‐order model was developed for B cell chronic lymphocytic leukemia, with immune system, and then analyzed. Interactions between B leukemia cells, natural killer cells, cytotoxic T cells, and T‐helper cells are considered to be incorporated into a system consisting of four fractional differential equations. For estimation of the parameters, clinical data of six patients were used. By numerical solution of the system, the interactions between the leukemia cell population and the immune system cell populations for values of α ∈ (0,1) at different times were explained. By determining points of equilibrium and stability of the system were met. Bifurcation analysis showed that use of the fractional‐order model, figure out unpredictable behaviors of the system such as saddle‐node, bistability and hysteresis phenomenon occurred in the system by changing the values of some of the parameters, it was predictable.     

A predator–prey model with disease in the prey species only

A predator–prey model with transmissible disease in the prey species is proposed and analysed. The essential mathematical features are analysed with the help of equilibrium, local and global stability analyses and bifurcation theory. We find four possible equilibria. One is where the populations are extinct. Another is where the disease and predator populations are extinct and we find conditions for global stability of this. A third is where both types of prey exist but no predators. The fourth has all three types of individuals present and we find conditions for limit cycles to arise by Hopf bifurcation. Experimental data simulation and brief discussion conclude the paper. Copyright © 2006 John Wiley & Sons, Ltd.     

Global bifurcation of solutions for a predator–prey model with prey‐taxis

We study pattern formations in a predator–prey model with prey‐taxis. It is proved that a branch of nonconstant solutions can bifurcate from the positive equilibrium only when the chemotactic is repulsive. Furthermore, we find the stable bifurcating solutions near the bifurcation point under suitable conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamics and pattern formations in diffusive predator‐prey models with two prey‐taxis

A reaction‐diffusion two‐predator‐one‐prey system with prey‐taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing preys. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any small prey‐taxis sensitivity coefficient are investigated by the semigroup theory. The spatial pattern formation induced by the prey‐taxis is characterized by the Turing type linear instability of homogeneous state; it is shown that prey‐taxis can both compress and prompt the spatial patterns produced through diffusion‐induced instability in two‐predator‐one‐prey systems.     

An ecoepidemiological predator‐prey model with standard disease incidence

This investigation accounts for epidemics spreading among interacting populations. The infective disease spreads among the prey, of which only susceptibles reproduce, while infected prey do not grow, recover, reproduce nor compete for resources. The model is general enough to describe a large number of ecosystems, on land, in the air or in the water. The main results concern the boundedness of the trajectories, the analysis of local and global stability, system's persistency and a threshold property below which the infection disappears. A sufficiently strong disease in the prey may avoid predators extinction and its presence can destabilize an otherwise stable predator‐prey configuration. The occurrence of transcritical, saddle‐node and Hopf‐bifurcations is also shown. Copyright © 2008 John Wiley & Sons, Ltd.     

Limit cycles in a Gause‐type predator–prey model with sigmoid functional response and weak Allee effect on prey

The goal of this work is to examine the global behavior of a Gause‐type predator–prey model in which two aspects have been taken into account: (i) the functional response is Holling type III; and (ii) the prey growth is affected by a weak Allee effect. Here, it is proved that the origin of the system is a saddle point and the existence of two limit cycles surround a stable positive equilibrium point: the innermost unstable and the outermost stable, just like with the strong Allee effect. Then, for determined parameter constraints, the trajectories can have different ω ? limit sets. The coexistence of a stable limit cycle and a stable positive equilibrium point is an important fact for ecologists to be aware of the kind of bistability shown here. So, these models are undoubtedly rather sensitive to disturbances and require careful management in applied contexts of conservation and fisheries. Copyright © 2012 John Wiley & Sons, Ltd.     

Dynamics and bifurcations of a modified Leslie‐Gower–type model considering a Beddington‐DeAngelis functional response

In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie‐Gower model, considering a Beddington‐DeAngelis functional response. It generates a complex dynamics of the predator‐prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and its local behavior, and the existence of a trapping set is proved. We describe different types of bifurcations (such as Hopf, Bogdanov‐Takens, and homoclinic bifurcation), and the existence of limit cycles is shown. Analytic proofs are provided for all results. Ecological implications and a set of numerical simulations supporting the mathematical results are also presented.     

Bogdanov‐Takens bifurcations of codimensions 2 and 3 in a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting

The Bogdanov‐Takens bifurcations of a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339‐366,” Gupta et al proved that the Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov‐Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.