Bifurcation and stability analysis in predator–prey model with a stage-structure for predator

A predator–prey system with Holling type II functional response and stage-structure for predator is presented. The stability and Hopf bifurcation of this model are studied by analyzing the associated characteristic transcendental equation. Further, an explicit formula for determining the stability and the direction of periodic solutions bifurcating from positive equilibrium is derived by the normal form theory and center manifold argument. Some numerical simulations are also given to illustrate our results.     

Biological control of a predator–prey system through provision of a super predator

In this paper, we make a systematic analysis of the dynamics of a predator–prey system with type-II functional response, in which the predator growth rate is affected by the presence of a super predator. The main aim of this research is to study the consequences of the presence of a super predator on the system dynamics. The existence and stability of the different possible equilibrium points are studied, and we conclude that the maximum consumption rate of a super predator plays a key role in determining the eventual state of the ecosystem. A detailed bifurcation analysis is carried out through numerical simulations, and we observe that theoretically it is possible to control the dynamics of the system by manipulating the consumption rate of the super predator.     

Hopf bifurcation of a predator–prey system with predator harvesting and two delays

In this paper, we consider the differential-algebraic predator–prey model with predator harvesting and two delays. By using the new normal form of differential-algebraic systems, center manifold theorem and bifurcation theory, we analyze the stability and the Hopf bifurcation of the proposed system. In addition, the new effective analytical method enriches the toolbox for the qualitative analysis of the delayed differential-algebraic systems. Finally, numerical simulations are given to show the consistency with theoretical analysis obtained here.     

Spatiotemporal dynamics of a predator–prey model

Spatial component of ecological interactions has been identified as an important factor in how ecological communities are shaped. In this paper, we consider a Holling?CTanner model with spatial diffusion. Choosing appropriate parameter values in parameter spaces, we obtain rich patterns, including spotted, black-eye, and labyrinthine patterns. The numerical results show that predator?Cprey system can exhibit complicated behavior.     

Influence of prey refuge on predator–prey dynamics

A diffusive predator–prey system with Michaelis–Menten type functional response subject to prey refuge is considered. Bifurcation analysis of Hopf and Turing are carried out in detail. In particular, Turing domain is given in the two parameters space. The obtained results show that the refuges used by prey have great influence on the pattern formation of the populations. More specifically, as prey refuge being increased, spotted pattern and coexistence of spotted and stripe-like pattern emerge. It is also proved that the pattern is not dependent on the initial conditions, which means the pattern is controlled by the intrinsic mechanism.     

Bifurcation analysis of a diffusive ratio-dependent predator–prey model

In this paper, a ratio-dependent predator–prey model with diffusion is considered. The stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied. Necessary and sufficient conditions for the stability of the positive constant equilibrium are explicitly obtained. Spatially heterogeneous steady states with different spatial patterns are determined. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. For the steady state bifurcation, the normal form shows the possibility of pitchfork bifurcation and can be used to determine the stability of spatially inhomogeneous steady states. Some numerical simulations are carried out to illustrate and expand our theoretical results, in which, both spatially homogeneous and heterogeneous periodic solutions are observed. The numerical simulations also show the coexistence of two spatially inhomogeneous steady states, confirming the theoretical prediction.     

Nonlinear dynamics of a delayed Leslie predator–prey model

The present paper is concerned with a delayed Leslie predator–prey model. The conditions of boundedness of the solutions of the system, existence, and stability of the equilibrium of the system are investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. The extensive simulations carried out show that the bifurcations arise around the positive equilibrium.     

Dynamics in two nonsmooth predator–prey models with threshold harvesting

Considering a good pest control program should reduce the pest to levels acceptable to the public, we investigate the threshold harvesting policy on pests in two predator–prey models. Both models are nonsmooth and the aim of this paper is to provide how threshold harvesting affects the dynamics of the two systems. When the harvesting threshold is larger than some positive level, the harvesting does not affect the ecosystem; when the harvesting threshold is less than the level, the model has complex dynamics with multiple coexistence equilibria, limit cycle, bistability, homoclinic orbit, saddle-node bifurcation, transcritical bifurcation, subcritical and supercritical Hopf bifurcation, Bogdanov–Takens bifurcation, and discontinuous Hopf bifurcation. Firstly, we provide the complete stability analysis and bifurcation analysis for the two models. Furthermore, some numerical simulations are given to illustrate our results. Finally, it is found that harvesting lowers the level of both species for natural enemy–pest system while raises the densities of both species for the pest–crop system. It is seen that the threshold harvesting policy of the enemy system is more effective than the crop system.     

A review of predator–prey systems with dormancy of predators

The predator–prey system has received much attention in the field of ecology and evolution. The interaction and competition between populations in nature can be described by the predator–prey system. Under large-amplitude fluctuations caused by harsh environmental conditions, the dormant progeny has been found as an effective strategy to prevent extinction. In this review paper, recent developments of dormancy in predator–prey systems are reviewed. The significant impacts of dormancy on the competition and evolution in predator–prey systems are then discussed through different models. The connections between dormancy in predator–prey systems and the game-theoretic Parrondo’s paradox are also discussed: the dormitive predator with inferior traits can outcompete the perennially active predator by switching between two losing strategies. Future outlook about the dormancy research in predator–prey systems is also discussed.

     

Ratio-dependent predator–prey model of interacting population with delay effect

A system of delay differential equation is proposed to account the effect of delay in the predator?Cprey model of interacting population. In this article, the modified ratio-dependent Bazykin model with delay in predator equation has been considered. The essential mathematical features of the proposed model are analyzed with the help of equilibria, local and global stability analysis, and bifurcation theory. The parametric space under which the system enters into a Hopf-bifurcation has been investigated. Global stability results are obtained by constructing suitable Lyapunov functions. We derive the explicit formulae for determining the stability, direction, and other properties of bifurcating periodic solutions by using normal form and central manifold theory. Using the global Hopf-bifurcation result of Wu (Trans. Am. Math. Soc., 350:4799?C4838, 1998) for functional differential equations, the global existence of periodic solutions has been established. Our analytical findings are supported by numerical experiments. Biological implication of the analytical findings are discussed in the conclusion section.     

Effort dynamics of a delay-induced prey–predator system with reserve

This paper describes a prey?Cpredator fishery system with prey dispersal in a two-patch environment, one of which is a free fishing zone and the other a protected zone. The proposed system reflects the dynamic interaction between the net economic revenue and the fishing effort used to harvest the population in presence of a suitable tax. Local as well as global stability of the system is analyzed. The optimal taxation policy is formulated and solved with the help of Pontryagin??s maximal principle. The objective of the paper is to achieve the sustainability of the fishery, keeping the ecological balance, and maximize the monetary social benefit. The dynamical behavior of the delay system is further analyzed through incorporating discrete type gestational delay of predators, and the existence of Hopf bifurcation phenomenon is checked at the interior equilibrium point. Moreover, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. Theoretical results are verified with the help of numerical examples and graphical illustrations.     

Stability and bifurcation in a generalized delay prey–predator model

The present paper considers a generalized prey–predator model with time delay. It studies the stability of the nontrivial positive equilibrium and the existence of Hopf bifurcation for this system by choosing delay as a bifurcation parameter and analyzes the associated characteristic equation. The researcher investigates the direction of this bifurcation by using an explicit algorithm. Eventually, some numerical simulations are carried out to support the analytical results.     

Dynamics of a delayed diffusive predator–prey model with hyperbolic mortality

This paper is devoted to consider a time-delayed diffusive prey–predator model with hyperbolic mortality. We focus on the impact of time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively, and we investigate the similarities and differences between them. Our conclusions show that when time delay continues to increase and crosses through some critical values, a family of homogenous and inhomogeneous periodic solutions emerge. Particularly, we find the minimum value of time delay, which is often hard to be found. We also consider the nonexistence and existence of steady state solutions to the reaction–diffusion model without time delay.     

Remarks on cannibalism and pattern formation in spatially extended prey–predator systems

The relationships between cannibalism and pattern formation in spatially extended prey–predator systems are studied with a model that degenerates, in the absence of cannibalism, into the most standard prey–predator model, known as Rosenzweig–MacArthur model. The analysis is based on the theory developed long ago by Turing in his famous paper on morphogenesis, but in a special form, which allows one to decouple the role of demographic parameters from that of diffusive dispersal. The proofs are given in terms of prey and predator nullclines because ecologists are mainly familiar with this technique. The final result of the analysis is that spatial pattern can exist only in systems with highly cannibalistic and highly dispersing predator provided the attractor of the system in the absence of cannibalism is a limit cycle. This result is more simple and more complete than that published in this journal a few years ago by Sun and coauthors.     

Bifurcation analysis of a diffusive predator–prey system with nonmonotonic functional response

Nonlinear Dynamics - In this paper, a diffusive predator–prey model with nonmonotonic functional response is investigated. The stability of the positive spatially homogeneous steady states...     

A delayed predator–prey model with strong Allee effect in prey population growth

In this paper, we consider a delayed predator-prey system with intraspecific competition among predator and a strong Allee effect in prey population growth. Using the delay as bifurcation parameter, we investigate the stability of coexisting equilibrium point and show that Hopf-bifurcation can occur when the discrete delay crosses some critical magnitude. The direction of the Hopf-bifurcating periodic solution and its stability are determined by applying the normal form method and the centre manifold theory. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using the global Hopf-bifurcation result of Wu ({Trans. Am. Math. Soc.} 350:4799?C4838, 1998) for functional differential equations, we establish the global existence of periodic solutions. Numerical simulations are carried out to validate the analytical findings.     

A mathematical study of a prey–predator model in relevance to pest control

In this paper, we propose and analyze an ecological system consisting of pest and its natural enemy as predator. Here we also consider the role of infection to the pest population and the presence of some alternative source of food to the predator population. We analyze the dynamics of this system in a systemic manner, study the dependence of the dynamics on some vital parameters and discuss the global behavior and controllability of the proposed system. The investigation also includes the use of pesticide control to the system and finally we use Pontryagin’s maximum principle to derive the optimal pest control strategy. We also illustrate some of the key findings using numerical simulations.     

The genic mutation on dynamics of a predator–prey system with impulsive effect

In this work, we consider a genic mutational predator?Cprey system with birth pulse and impulsive cutting on prey population at different moments. All the solutions of the investigated system are proved to be uniformly ultimately bounded. The conditions of the globally asymptotically stable predator-extinction boundary periodic solution of the investigated system are obtained. The permanent conditions of the investigated system are also obtained. Finally, numerical simulations are inserted to illustrate the results. Our results present that the genic mutational rate plays an important role on the permanence of the investigated system. Our results also provide reliable tactic basis for the practical biological economics management.     

Bifurcation of a predator–prey model with disease in the prey

In this paper, a predator–prey model with disease in the prey is considered. Assume that the predator eats only the infected prey, and the incidence rate is nonlinear. We study the dynamics of the model in terms of local analysis of equilibria and bifurcation analysis of a boundary equilibrium and a positive equilibrium. We discuss the Bogdanov–Takens bifurcation near the boundary equilibrium and the Hopf bifurcation near the positive equilibrium; numerical simulation results are given to support the theoretical predictions.     

Spatio-temporal dynamics of a reaction-diffusion system for a predator–prey model with hyperbolic mortality

We investigate the effects of diffusion on the spatial dynamics of a predator–prey model with hyperbolic mortality in predator population. More precisely, we aim to study the formation of some elementary two-dimensional patterns such as hexagonal spots and stripe patterns. Based on the linear stability analysis, we first identify the region of parameters in which Turing instability occurs. When control parameter is in the Turing space, we analyse the existence of stable patterns for the excited model by the amplitude equations. Then, for control parameter away from the Turing space, we numerically investigate the initial value-controlled patterns. Our results will enrich the pattern dynamics in predator–prey models and provide a deep insight into the dynamics of predator–prey interactions.