The dynamics of an age structured predator–prey model with disturbing pulse and time delays

Many of the existing predator–prey models on stage structured populations are some ordinary differential equations (ODE) or models without a disturbing effect of human behavior. In reality, death of the juvenile during its immature stage and catching or poisoning for the prey or predator occur continuously. From this basic standpoint, we formulate a general and robust prey-dependent consumption predator–prey model with periodic harvesting (catching or poisoning) for the prey and stage structure for the predator with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and ecological study. We show that the conditions for global attractivity of the ‘predator-extinction’ (‘predator-eradication’) periodic solution and permanence of the population of the model depend on time delay, so, we call it “profitless”. We also show that constant maturation time delay and impulsive catching or poisoning for the prey can bring great effects on the dynamics of system by numerical analysis. In this paper, the main feature is that we introduce time delay and pulse into the predator–prey (natural enemy–pest) model with age structure, exhibit a new modeling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.     

Predator-prey dynamics with square root functional responses

A predator-prey model is considered in which a modified Lotka-Volterra interaction term is used as the functional response of the predator to the prey. The interaction term is proportional to the square root of the prey population, which appropriately models systems in which the prey exhibits strong herd structure implying that the predator generally interacts with the prey along the outer corridor of the herd. Because of the square root term, the solution behavior near the origin is more subtle and interesting than standard models and makes sense ecologically.     

Global qualitative analysis of a non-smooth Gause predator–prey model with a refuge

The present paper aims to provide a detailed qualitative analysis of a non-smooth Gause predator–prey model. In this model, the saturating functional response function with a discontinuity at a critical prey density was employed to show the effects of a prey refuge on the population dynamic behavior. Analysis of this model revealed rich dynamics including locally (or globally) stable canard cycles, a locally (globally) stable pseudo-equilibrium, unbounded trajectories in which both populations go to infinity or the prey goes to infinity and the predator dies out eventually. The main purpose of the present work is to carry out a completely qualitative analysis for this model. In particular, two sets of sufficient conditions drive both populations to approach infinity and the sufficient and necessary conditions for all of the other main results are presented.     

A modified Leslie–Gower predator–prey model with ratio‐dependent functional response and alternative food for the predator

In this work, a modified Leslie–Gower predator–prey model is analyzed, considering an alternative food for the predator and a ratio‐dependent functional response to express the species interaction. The system is well defined in the entire first quadrant except at the origin ( 0 , 0 ) . Given the importance of the origin ( 0 , 0 ) as it represents the extinction of both populations, it is convenient to provide a continuous extension of the system to the origin. By changing variables and a time rescaling, we obtain a polynomial differential equations system, which is topologically equivalent to the original one, obtaining that the non‐hyperbolic equilibrium point ( 0 , 0 ) in the new system is a repellor for all parameter values. Therefore, our novel model presents a remarkable difference with other models using ratio‐dependent functional response. We establish conditions on the parameter values for the existence of up to two positive equilibrium points; when this happen, one of them is always a hyperbolic saddle point, and the other can be either an attractor or a repellor surrounded by at least one limit cycle. We also show the existence of a separatrix curve dividing the behavior of the trajectories in the phase plane. Moreover, we establish parameter sets for which a homoclinic curve exits, and we show the existence of saddle‐node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation, and homoclinic bifurcation. An important feature in this model is that the prey population can go to extinction; meanwhile, population of predators can survive because of the consumption of alternative food in the absence of prey. In addition, the prey population can attain their carrying capacity level when predators go to extinction. We demonstrate that the solutions are non‐negatives and bounded (dissipativity and permanence of population in many other works). Furthermore, some simulations to reinforce our mathematical results are shown, and we further discuss their ecological meanings. Copyright © 2017 John Wiley & Sons, Ltd.     

Turing pattern formation in a three species model with generalist predator and cross-diffusion

In a natural ecosystem, specialist predators feed almost exclusively on one species of prey. But generalist predators feed on many types of species. Consequently, their dynamics is not coupled to the dynamics of a specific prey population. However, the defense of prey formed by congregating made the predator tend to move in the direction of lower concentration of prey species. This is described by cross-diffusion in a generalist predator–prey model. First, the positive equilibrium solution is globally asymptotically stable for the ODE system and for the reaction–diffusion system without cross-diffusion, respectively, hence it does not belong to the classical Turing instability scheme. But it becomes linearly unstable only when cross-diffusion also plays a role. This implies that cross–diffusion can lead to the occurrence and disappearance of the instability. Our results exhibit some interesting combining effects of cross-diffusion, predations and intra-species interactions. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions.     

Optimal control of effort of a stage structured prey–predator fishery model with harvesting

This paper describes a prey–predator fishery model with stage structure for prey. The adult prey and predator populations are harvested in the proposed system. The dynamic behavior of the model system is discussed. It is observed that singularity induced bifurcation phenomenon is appeared when variation of the economic interest of harvesting is taken into account. We have incorporated state feedback controller to stabilize the model system in the case of positive economic interest. Fishing effort used to harvest the adult prey and predator populations is used as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent earned from the resource. Pontryagin’s maximum principle is used to characterize the optimal control. The optimal system is derived and then solved numerically using an iterative method with Runge–Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem.     

A numerical study of the formation of spatial patterns in twospotted spider mites

The aim of this paper is to study the formation of spatial patterns in a predator–prey system with Tetranychus urticae as prey and Phytoseiulus persimilis as predator. Logistic Lotka–Volterra predator–prey equations are solved numerically with two different response functions, two initial conditions and one data set. The spatial patterns are generated by introducing diffusion-driven instability in the predator–prey system. Among all parameters involved in predator–prey equations, only the predator interference parameter is varied to generate diffusion-driven instability leading to spatial patterns of population density. Spatial patterns are further generated with the inclusion of prey-taxis in the predator–prey system. Routh–Hurwitz’s conditions for stability are used to create instability with prey-taxis in the system. It is shown that it is possible to generate spatial patterns with zero flux boundary conditions even in a smaller domain with a suitable value of the predator interference parameter or prey-taxis.     

Dynamics of a Stochastic Predator–Prey Model with Stage Structure for Predator and Holling Type II Functional Response

In this paper, we develop and study a stochastic predator–prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results.     

Waves analysis and spatiotemporal pattern formation of an ecosystem model

In this paper, we consider a predator–prey model given by a reaction–diffusion system. This model incorporates Holling-type-II (Michaelis–Menten) and modified Leslie-Gower functional responses. We show the existence of qualitatively different types of system behaviors realized for various parameter values. Our model is investigated with methods of the qualitative theory and the theory of bifurcations. We generalize the traveling waves existence method for populations dynamics with positive derivative densities, to the predator–prey system in which growth densities may change sign. Parallel to this is a discussion and an analysis of alternative model outcomes such as complex pattern formation and spatio-temporal chaos behavior.     

Chaos in delay-induced Leslie–Gower prey–predator–parasite model and its control through prey harvesting

In this paper we analyze a delay-induced predator–prey–parasite model with prey harvesting, where the predator–prey interaction is represented by Leslie–Gower type model with type II functional response. Infection is assumed to spread horizontally from one infected prey to another susceptible prey following mass action law. Spreading of disease is not instantaneous but mediated by a time lag to take into account the time required for incubation process. Both the susceptible and infected preys are subjected to linear harvesting. The analysis is accomplished in two phases. First we analyze the delay-induced predator–prey–parasite system in absence of harvesting and proved the local & global dynamics of different (six) equilibrium points. It is proved that the delay has no influence on the stability of different equilibrium points except the interior one. Delay may cause instability in an otherwise stable interior equilibrium point of the system and larger delay may even produce chaos if the infection rate is also high. In the second phase, we explored the dynamics of the delay-induced harvested system. It is shown that harvesting of prey population can suppress the abrupt fluctuations in the population densities and can stabilize the system when it exceeds some threshold value.     

A cannibalistic eco-epidemiological model with disease in predator population

In this present article, we propose and analyze a cannibalistic predator–prey model with disease in the predator population. We consider two important factors for the dynamics of predator population. The first one is governed through cannibalistic interaction, and the second one is governed through the disease in the predator population via cannibalism. The local stability analysis of the model system around the biologically feasible equilibria are investigated. We perform global dynamics of the model using Lyapunov functions. We analyze and compare the community structure of the system in terms of ecological and disease basic reproduction numbers. The existence of Hopf bifurcation around the interior steady state is investigated. We also derive the sufficient conditions for the permanence and impermanence of the system. The study reveals that the cannibalism acts as a self-regulatory mechanism and controls the disease transmission among the predators by stabilizing the predator–prey oscillations.     

Non-autonomous dynamics of a semi-Kolmogorov population model with periodic forcing

In this paper we study a semi-Kolmogorov type of population model, arising from a predator–prey system with indirect effects. In particular we are interested in investigating the population dynamics when the indirect effects are time dependent and periodic. We first prove the existence of a global pullback attractor. We then estimate the fractal dimension of the attractor, which is done for a subclass by using Leonov’s theorem and constructing a proper Lyapunov function. To have more insights about the dynamical behavior of the system we also study the coexistence of the three species. Numerical examples are provided to illustrate all the theoretical results.     

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.     

Consequences of refuge and diffusion in a spatiotemporal predator–prey model

In this investigation, we offer and examine a predator–prey interacting model with prey refuge in proportion to both the species and Beddington–DeAngelis functional response. We first prove the well-posedness of the temporal and spatiotemporal models which are restricted in a positive invariant region. Then for the temporal model, we analyse its temporal dynamics including uniform boundedness, permanence, stability of all feasible non-negative equilibria and show that refugia can induce periodic oscillation via Hopf bifurcation around the unique positive equilibrium; for the spatiotemporal model, we not only investigate its permanence, stability of non-negative constant steady states and Turing instability but also study the existence and non-existence of non-constant positive steady states by Leray–Schauder degree theory. The key observation is that the coefficient of refuge cooperates a significant part in modifying the dynamics of the current system and mediates the population permanence, stability of coexisting equilibrium and even the Turing instability parameter space. Finally, general numerical simulation consequences are given to illustrate the validity of the theoretical results. Through numerical simulations, one observes that the model dynamics shows prey refugia and self-diffusion control spatiotemporal pattern growth to spots, stripe–spot mixtures and stripes reproduction. The outcomes assign that the dynamics of the model with prey refuge is not simple, but rich and complex. Additionally, numerical simulations show that the other model parameters have an important effect on species’ spatially inhomogeneous distribution, which results in the formation of spots pattern, mixture of spots and stripes pattern, mixture of spots, stripes and rings pattern and anti-spot pattern. This may improve the model dynamics of the prey refuge on the reaction–diffusion predator–prey system.     

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.     

Necessary optimality conditions for three species reaction–diffusion system

An optimal control problem is studied for an ecosystem composed by one predator and two prey populations. Its dynamics is modelled by a reaction–diffusion system of Volterra type. Two control variables are introduced in the system; their meaning is the mixture rates between predator and each prey population. The goal of this paper is to maximize the total density of the three populations at a fixed time moment. The existence of the optimal control is established and necessary optimality conditions are found with the aid of a maximum principle.     

A discrete prey–predator model preserving the dynamics of a structurally unstable Lotka–Volterra model

Leslie's method to construct a discrete two dimensional dynamical system dynamically consistent with the Lotka–Volterra type of competing two species ordinary differential equations is applied in a newly extended manner for the Lotka–Volterra prey–predator system which is structurally unstable. We show that, independently of the time step size, the derived discrete prey–predator system is dynamically consistent with the continuous counterpart, keeping the nature of neutrally stable periodic orbit. Further, we show that the extended method to construct the discrete prey–predator system can provide a dynamically consistent model also for the logistic Lotka–Volterra one.     

Symmetrization in neumann problems

This paper is concerned with a model of a predator–prey system, where both populations disperse among n patches forming their habitat. Criteria are given tor both survival and extinction of the predator population. In case the predator survives, conditions are derived which guarantee a globally asymptotically stable positive equilibrium     

Spreading speeds for time heterogeneous prey–predator systems with diffusion

We investigate the large time behavior for two components reaction–diffusion systems of prey–predator type in a time varying environment. Here we assume that these variations in time exhibit an averaging property, which will be called mean value in this work. This framework includes in particular time periodicity, almost periodicity and unique ergodicity. We describe the spreading behavior of the prey and the predator, wherein the two populations are able to co-invade the empty space. Our analysis is based the parabolic strong maximum principle for scalar equation and on the derivation of local pointwise estimates that are used to compare the solutions of the prey–predator problem with those of a KPP scalar equation on suitable spatio-temporal domains.     

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.