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.

     

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.     

Effect of delay on a predator–prey model with parasitic infection

In the paper an eco-epidemic system with delay and parasitic infection in the prey is investigated. The conditions for asymptotic stability of steady states are derived and the length of the delay preserving the stability is also estimated. Further, the criterion for existence of Hopf-type small amplitude periodic oscillations of the predator and prey biomass is derived. Numerical results indicate that the delay does not affect the stability of the system in the process but makes all populations oscillate more intensively. In addition, the results show that the recovery makes the levels of the infected prey and the predator become lower but makes the sound prey higher in limit time.     

$$\mathcal {H}_\infty $$ H ∞ state-feedback control for continuous-time Markovian jump fuzzy systems using a fuzzy weighting-dependent Lyapunov function

Nonlinear Dynamics - In this paper, we investigate a ratio-dependent prey–predator model with state-dependent impulsive harvesting where the prey growth rate is subject to a strong Allee...     

Stability and Hopf bifurcation for a delayed predator–prey model with stage structure for prey and Ivlev-type functional response

Nonlinear Dynamics - In this paper, we mainly investigate a delayed predator–prey model with stage structure for prey and Ivlev-type functional response. Four assumptions about this model are...     

A Beddington–DeAngelis type one-predator two-prey competitive system with help

Nonlinear Dynamics - In this paper, we present and study a two competing prey and one predator system where during predation both the teams of prey help each other and the rate of predation on both...     

Turing patterns induced by self-diffusion in a predator–prey model with schooling behavior in predator and prey

Nonlinear Dynamics - This article considers a reaction–diffusion predator–prey model with schooling behavior both in predator and prey species and subject to the homogeneous Neumann...     

Global dynamics of a Holling Type-III two prey–one predator discrete model with optimal harvest strategy

Nonlinear Dynamics - This paper deals with a discrete-time two prey–one predator system with Holling Type-III functional response, along with inter-specific competition between the prey and...     

Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey

We investigate a stage-structured delayed predator-prey model with impulsive stocking on prey and continuous harvesting on predator. According to the fact of biological resource management, we improve the assumption of a predator-prey model with stage structure for predator population that each individual predator has the same ability to capture prey. It is assumed that the immature and mature individuals of the predator population are divided by a fixed age, and immature predator population does not have the ability to attach prey. Sufficient conditions are obtained, which guarantee the global attractivity of predator-extinction periodic solution and the permanence of the system. Our results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system, and provide tactical basis for the biological resource management. Numerical analysis is presented to illuminate the dynamics of the system.     

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.     

Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey

We investigate a stage-structured delayed predator-prey model with impulsive stocking on prey and continuous harvesting on predator. According to the fact of biological resource management, we improve the assumption of a predator-prey model with stage structure for predator population that each individual predator has the same ability to capture prey. It is assumed that the immature and mature individuals of the predator population are divided by a fixed age, and immature predator population does not have the ability to attach prey. Sufficient conditions are obtained, which guarantee the global attractivity of predator-extinction periodic solution and the permanence of the system. Our results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system, and provide tactical basis for the biological resource management. Numerical analysis is presented to illuminate the dynamics of the system.     

Global analysis of a Holling type II predator–prey model with a constant prey refuge

A global analysis of a Holling type II predator–prey model with a constant prey refuge is presented. Although this model has been much studied, the threshold condition for the global stability of the unique interior equilibrium and the uniqueness of its limit cycle have not been obtained to date, so far as we are aware. Here we provide a global qualitative analysis to determine the global dynamics of the model. In particular, a combination of the Bendixson–Dulac theorem and the Lyapunov function method was employed to judge the global stability of the equilibrium. The uniqueness theorem of a limit cycle for the Lineard system was used to show the existence and uniqueness of the limit cycle of the model. Further, the effects of prey refuges and parameter space on the threshold condition are discussed in the light of sensitivity analyses. Additional interesting topics based on the discontinuous (or Filippov) Gause predator–prey model are addressed in the discussion.     

The study of a ratio-dependent predator–prey model with stage structure in the prey

In this paper, a ratio-dependent predator–prey model with stage structure in the prey is constructed and investigated. In the first part of this paper, some sufficient conditions for the existence and stability of three equilibriums are obtained. In the second part, we consider the effect of impulsive release of predator on the original system. A sufficient condition for the global asymptotical stability of the prey-eradication periodic solution is obtained. We also get the condition, under which the prey would never be eradicated, i.e., the impulsive system is permanent. At last, we give a brief discussion.

     

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.     

Dynamics of a Diffusive Predator–Prey Model: The Effect of Conversion Rate

A general diffusive predator–prey model is investigated in this paper. We prove the global attractivity of constant equilibria when the conversion rate is small, and the non-existence of non-constant positive steady states when the conversion rate is large. The results are applied to several predator–prey models and give some ranges of parameters where complex pattern formation cannot occur.     

Emergence of diverse dynamical responses in a fractional-order slow–fast pest–predator model

To explore the impact of pest-control strategy on integrated pest management, a three-dimensional (3D) fractional- order slow–fast prey–predator model is introduced in this article. The prey community (assumed as pest) represents fast dynamics and two predators exhibit slow dynamical variables in the three-species interacting prey–predator model. In addition, common enemies of that pest are assumed as predators of two different species. Pest community causes serious damage to the economy. Fractional-order systems can better describe the real scenarios than classical-order dynamical systems, as they show previous history-dependent properties. We establish the ability of a fractional-order model with Caputo’s fractional derivative to capture the dynamics of this prey–predator system and analyze its qualitative properties. To investigate the importance of fractional-order dynamics on the behavior of the pest, we perform the local stability analysis of possible equilibrium points, using certain assumptions for different sets of parameters and reveal that the fractional-order exponent has an impact on the stability and the existence of Hopf bifurcations in the prey–predator model. Next, we discuss the existence, uniqueness and boundedness of the fractional-order system. We also observe diverse oscillatory behavior of different amplitude modulations including mixed mode oscillations (MMOs) for the fractional-order prey–predator model. Higher amplitude pest periods are interspersed with the outbreaks of small pest concentration. With the decrease of fractional-order exponent, small pest concentration increases with decaying long pest periods. We further notice that the reduced-order model is biologically significant and sensitive to the fractional-order exponent. Additionally, the dynamics captures adaptation that occurs over multiple timescales and we find consistent differences in the characteristics of the model for various fractional exponents.

     

The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and?harvesting prey at different fixed moments

In this paper, a delayed pest control model with stage-structure for pests by introducing a constant periodic pesticide input and harvesting prey (Crops) at two different fixed moments is proposed and analyzed. We assume only the pests are affected by pesticide. We prove that the conditions for global asymptotically attractive ??predator-extinction?? periodic solution and permanence of the population of the model depend on time delay, pulse pesticide input, and pulse harvesting prey. By numerical analysis, we also show that constant maturation time delay, pulse pesticide input, and pulse harvesting prey can bring obvious effects on the dynamics of system, which also corroborates our theoretical results. We believe that the results will provide reliable tactic basis for the practical pest management. One of the features of present paper is to investigate the high-dimensional delayed system with impulsive effects at different fixed impulsive moments.     

A simplified model system for <Emphasis Type="Italic">Toxoplasma gondii</Emphasis> spread within a heterogeneous environment

This study is dedicated to building and analyzing the spatial spread of Toxoplasma gondii through a heterogeneous predator–prey system. The spatial domain is made of N patches hosting various population species, some of them being prey, others being predators. Predators offer strong heterogeneities with respect to local sustainable resources yielding variable growth rates, from exponential decay to logistic regulation. T. gondii life cycle goes through several stages, starting in the environment where oocysts are released from cat feces, reaching prey within which asexual reproduction yields cysts and then predators wherein sexual reproduction takes place. The resulting model system is complex to handle. We consider some relevant toy models with three patches, two resident predator species and Lotka–Volterra functional responses to predation. We provide the existence and local stability of a persistent stationary state for the underlying predator–prey model systems. The reproduction number R ₀ is computed in the quasistationary case; it simplifies when slow–fast dynamics are considered. Numerical experiments illustrate our analysis.     

Global stability and bifurcation analysis of a delay induced prey-predator system with stage structure

This paper describes a delay induced prey–predator system with stage structure for prey. The dynamical characteristics of the system are rigorously studied using mathematical tools. The coexistence equilibria of the system is determined and the dynamic behavior of the system is investigated around coexistence equilibria. Sufficient conditions are derived for the global stability of the system. The optimal harvesting problem is formulated and solved in order to achieve the sustainability of the system, keeping the ecological balance, and maximize the monetary social benefit. Maturation time delay of prey is incorporated and the existence of Hopf bifurcation phenomenon is examined at the coexistence equilibria. It is shown that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities. Moreover, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.     

Stochastic analysis of a predator–prey model with modified Leslie–Gower and Holling type II schemes

Nonlinear Dynamics - This article studies a predator–prey model with modified Leslie–Gower and Holling type II schemes under white-noise disturbances. The sensitivity analysis of the...