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.

     

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.     

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.

     

Predator cannibalism can give rise to regular spatial pattern in a predator–prey system

One of the central issues in ecology is the study of spatial pattern in the distribution of organisms. Thus, in this paper, spatial pattern of a predator–prey system with predator cannibalism is considered. By mathematical analysis, we obtain the condition for emerging Turing pattern formation. Furthermore, numerical simulations reveal that large variety of different spatiotemporal dynamics emerge as the consequence of the interaction of Holling type II with predator cannibalism. The obtained results show predator cannibalism has great influence on the spatial pattern formation. In other words, the regular pattern is induced by predator cannibalism. Moreover, we find that although the environment is heterogeneous, the system still exhibits Turing pattern, which means the pattern is self-organized. It may help us better understand the dynamics of predator–prey interaction in a real environment.     

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.     

Optimal harvesting and complex dynamics in a delayed eco-epidemiological model with weak Allee effects

In this article, an eco-epidemiological system with weak Allee effect and harvesting in prey population is discussed by a system of delay differential equations. The delay parameter regarding the time lag corresponds to the predator gestation period. Mathematical features such as uniform persistence, permanence, stability, Hopf bifurcation at the interior equilibrium point of the system is analyzed and verified by numerical simulations. Bistability between different equilibrium points is properly discussed. The chaotic behaviors of the system are recognized through bifurcation diagram, Poincare section and maximum Lyapunov exponent. Our simulation results suggest that for increasing the delay parameter, the system undergoes chaotic oscillation via period doubling. We also observe a quasi-periodicity route to chaos and complex dynamics with respect to Allee parameter; such behavior can be subdued by the strength of the Allee effect and harvesting effort through period-halving bifurcation. To find out the optimal harvesting policy for the time delay model, we consider the profit earned by harvesting of both the prey populations. The effect of Allee and gestation delay on optimal harvesting policy is also discussed.     

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.

     

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.     

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.     

A Free Boundary Problem for the Predator–Prey Model with Double Free Boundaries

In this paper we investigate a free boundary problem for the classical Lotka–Volterra type predator–prey model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species and are described by Stefan-like condition. We prove a spreading–vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as \(t\rightarrow \infty \) at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The long time behavior of solution and criteria for spreading and vanishing are also obtained.     

Neimark–Sacker bifurcation and chaos control in discrete-time predator–prey model with parasites

In this paper, we discuss the qualitative behavior of a four-dimensional discrete-time predator–prey model with parasites. We investigate existence and uniqueness of positive steady state and find parametric conditions for local asymptotic stability of positive equilibrium point of given system. It is also proved that the system undergoes Neimark–Sacker bifurcation (NSB) at positive equilibrium point with the help of an explicit criterion for NSB. The system shows chaotic dynamics at increasing values of bifurcation parameter. Chaos control is also discussed through implementation of hybrid control strategy, which is based on feedback control methodology and parameter perturbation. Finally, numerical simulations are conducted to illustrate theoretical results.     

Bifurcations and stability of a discrete singular bioeconomic system

The paper analyzes the stability and bifurcations of a discrete singular bioeconomic system in the closed first quadrant $R_{+}^{3}$ . First, applying the Poincaré scheme to a differential-algebraic predator–prey system where the economic interest of harvesting is taken into account, a discrete singular bioeconomic system is proposed. Then, local stability and the existing conditions of the flip bifurcation and Neimark–Sacker bifurcation around the interior equilibria of the proposed model are discussed by using the normal form of the discrete singular bioeconomic system, the center manifold theorem and the bifurcation theory, when choosing the step size δ as the parameter of the bifurcation. Finally, the results are illustrated and the complex dynamical behaviors are exhibited by computer numerical simulations.     

Dynamical behaviour of a delayed three species predator–prey model with cooperation among the prey species

Nonlinear Dynamics - In this paper we have discussed about the dynamics of three species (two preys and one predator) delayed predator–prey model with cooperation among the preys against...     

Bifurcation analysis of 4-axle rail vehicle models in a curved track

The dynamics of a diffusive predator–prey model with time delay and Michaelis–Menten-type harvesting subject to Neumann boundary condition is considered. Turing instability and Hopf bifurcation at positive equilibrium for the system without delay are investigated. Time delay-induced instability and Hopf bifurcation are also discussed. By the theory of normal form and center manifold, conditions for determining the bifurcation direction and the stability of bifurcating periodic solution are derived. Some numerical simulations are carried out for illustrating the theoretical results.     

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.     

Optimal predator control policy and weak Allee effect in a delayed prey–predator system

Nonlinear Dynamics - In this article, a system of delay differential equations to represent the predator–prey dynamics with weak Allee effect in the growth of predator population is...     

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.     

Forced Convection Flow of Power-Law Fluids Over a Flat Plate Embedded in a Darcy-Brinkman Porous Medium

The steady forced convection flow of a power-law fluid over a horizontal plate embedded in a saturated Darcy-Brinkman porous medium is considered. The flow is driven by a constant pressure gradient. In addition to the convective inertia, also the “porous Forchheimer inertia” effects are taken into account. The pertinent boundary value problem is investigated analytically, as well as numerically by a finite difference method. It is found that far away from the leading edge, the velocity boundary layer always approaches an asymptotic state with identically vanishing transverse component. This holds for pseudoplastic (0 < n < 1), Newtonian (n = 1), and dilatant (n > 1) fluids as well. The asymptotic solution is given for several particular values of the power-law index n in an exact analytical form. The main flow characteristics of physical and engineering interest are discussed in the paper in some detail.     

Predicting solutions of the Lotka‐Volterra equation using hybrid deep network

Prediction of Lotka-Volterra equations has always been a complex problem due to their dynamic properties. In this paper, we present an algorithm for predicting the Lotka-Volterra equation and investigate the prediction for both the original system and the system driven by noise. This demonstrates that deep learning can be applied in dynamics of population. This is the first study that uses deep learning algorithms to predict Lotka-Volterra equations. Several numerical examples are presented to illustrate the performances of the proposed algorithm, including Predator nonlinear breeding and prey competition systems, one prey and two predator competition systems, and their respective systems. All the results suggest that the proposed algorithm is feasible and effective for predicting Lotka-Volterra equations. Furthermore, the influence of the optimizer on the algorithm is discussed in detail. These results indicate that the performance of the machine learning technique can be improved by constructing the neural networks appropriately.     

Modelling and analysis of a multiple delayed exploited ecosystem towards coexistence perspective

This paper describes a multiple delayed modified Leslie–Gower type predator–prey system with a strong Allee effect in prey population growth. Non-selective effort is used to harvest the population. The dynamical characteristics of the delay induced system are rigorously studied using mathematical tools. The existence of coexistence equilibria is ensured, and the dynamic behavior of the system is investigated around coexistence equilibria. Uniform strong persistence and permanence of the system are discussed in order to ensure long-term survival of the species. The stability of the delay preserved system is investigated. Sufficient conditions are derived for local and global stability of the system. The existence of Hopf bifurcation phenomenon is examined around interior equilibria of the system. Subsequently, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. Finally, numerical simulations are carried out to validate the analytical findings.