Harvesting of a prey–predator fishery in the presence of toxicity

In this model we discuss the bioeconomic harvesting of a prey–predator fishery in which both the species are infected by some toxicants released by some other species. Here both the species are harvested where we use the usual catch-per-unit-effort hypothesis. The dynamical behaviour of the exploited system is examined. The possibility of existence of a bionomic equilibrium is considered. The optimal harvesting policy is studied by using Pontryagin’s maximal principle. Some numerical examples and the corresponding solution curves are studied to illustrate the results of the model. Finally, the existence of limit cycle is discussed.     

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.     

HARVESTING A TWO-PATCH PREDATOR-PREY METAPOPULATION

ABSTRACT. A mathematical model for a two-patch predator-prey metapoplation is developed as a generalization of single-species metapopulation harvesting theory. We find optimal harvesting strategies using dynamic programming and La-grange multipliers. If predator economic efficiency is relatively high, then we should protect a relative source prey subpopulation in two different ways: directly, with a higher escapement of the relative source prey subpopulation, and indirectly, with a lower escapement of the predator living in the same patch as the relative source prey subpopulation. Numerical examples show that if the growth of the predator is relatively low and there is no difference between prey and predator prices, then it may be optimal to harvest the predator to extinction. While, if the predator is more valuable compared to the prey, then it may be optimal to leave the relative exporter prey subpopulation unharvested. We also discuss how a ‘negative’ harvest might be optimal. A negative harvest might be considered a seeding strategy.     

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.     

A mathematical model for chemical defense mechanism of two competing species

In this paper, a non-linear mathematical model is proposed and analyzed to study the phenomenon of a chemical defense mechanism involving two competing species, where each species produces a toxicant affecting the other. It is shown that if the emission rate coefficient of toxicant, produced by one species increases, the equilibrium density of the other species decreases and its magnitude is lower than its original carrying capacity. It is found that the usual principle of competitive exclusion (coexistence) in the absence of toxicant may change in the case under consideration. It is also observed that increases in the values of production rates of toxicants by the competing species and depletion rates of environmental toxicants due to its assimilation by the species has a destabilizing effect, and decrease in the washout rates of environmental toxicants has a destabilizing effect on the dynamics of the system. In the case of allelopathy, where only one species produces a toxicant affecting the other species, it is shown that the affected species is driven to extinction for large production rate of this toxicant.     

A Regularity Condition of 3d Axisymmetric Navier-Stokes Equations

This paper is concerned with a Variable-territory model with limited self-limitation of predator. By the bifurcation theorem, regular perturbation theorem and numerical simulation, the conditions of existence, stability and convergence of positive solutions are established. This work shows that prey and predator can be controlled by parameters in Variable-territory model, such as intrinsic growth rate, death rate, handling time and the self-limitation of predator.     

食饵具有常投放的一类稀疏效应捕食系统

本文讨论食饵具有常投放的一类稀疏效应的一般捕食系统 (1) ,利用对此系统 (1)在第一象限内轨线的拓朴分类 ,分别给出了捕食者种群必将灭绝或持续生存的条件 .     

Challenges of living in the harsh environments: A mathematical modeling study

We propose and analyze a mathematical model, which mimics community dynamics of plants and animals in harsh environments. The mathematical model exploits type IV functional responses whose idiosyncrasies have been recognized only in recent years. The interaction of the middle predator with the top predator is cast into Leslie-Gower scheme. Linear and non-linear stability analyses are performed to get an idea of the stability behavior of the model food chain. It turns out that carrying capacity of the prey and the immunity parameter of the middle predator are two crucial parameters governing the model. Availability of alternative food options to the generalist predator also plays a key role in deciding the model dynamics.Simulation runs performed on this model provide insight into population dynamics of monkeys of macaque family found in northern Japan. These monkeys are social animals which reproduce sexually. The characteristic feature of the model dynamics is that the generalist predator (macaque monkeys) is able to avoid impending extinction frequently and recovers at a rate which falsify threats from exogenous external forces; extreme weather conditions, etc.     

STABILITY ANALYSIS OF A TRITROPHIC FOOD CHAIN MODEL WITH AN ADAPTIVE PARAMETER FOR THE PREDATOR

Abstract The study of three‐species communities have become the focus of considerable attention, and because the studies of ecological communities start with their food web, we consider a tritrophic food chain model comprised of the prey, the predator, and the super‐predator. The classical assumption of the domino effect is supplemented with an adaptive parameter for the predator (in the absence of prey). Thus, the model exhibits an equilibrium with the predator‐top‐predator steady state, which is a saddle point. Dynamical behaviors such as boundedness, existence of periodic orbits, persistence, as well as stability are analyzed. The long‐term coexistence of the three interacting species is addressed, and the stability analysis of the model shows that the biologically most relevant equilibrium point is globally asymptotically stable whenever it satisfies a certain criterion. Practical implications are explored and related to real populations.     

A two-patch prey-predator model with food-gathering activity

In insect ecosystem, the dynamics of prey and predator is regulated by complex interactions between them. Insect pests are spatially aggregated in patches forming a spatial pattern in the environment. An efficient predator dynamically changes its strategies and time for its random search movements to concentrate on higher resource patches based on the benefit of assessment. This food-gathering activity of both prey and predator plays a major role in stabilizing the system by influencing the per unit food consumption. Extending Holling time-budget argument by migration, here we formulate a two patch prey-predator model and show that how several foraging parameters such as handling time, dispersal rate can have important consequences in stability of prey-predator system. Specifically, the ratio between timings that a predator remains mobile in searching and handling their food, is the most important one and simulation on this suggests that the stabilizing effect continues to operate when the dispersal process is modeled more realistically. Thus we conclude that the migration submodel is an important constituent of a spatial predator-prey system. These results are shown to have important implications for possible biological control.     

一类捕食者存在疾病的捕食系统传染病模型

建立并分析一类捕食者存在疾病的捕食系统传染病模型,模型中不考虑疾病对捕获率的影响.通过极限系统理论、Lyapunov稳定性理论分析和Bendixson判据,给出了各类平衡点存在及其全局稳定的条件,并得到了捕食者绝灭和疾病成为地方病的充分必要条件.     

A foraging problem: Sit-and-wait versus active predation

The literature on foraging shows that some predators use a combination of ambush and active search to locate a prey. Let us suppose that a prey must go every day to some determined places to feed, and to another place, 0, to drink. A predator can stay at zone 0 waiting for the prey (sit-and-wait strategy) or it can move between the different places where the prey will go to eat (search strategy). If predator and prey meet each other in the same place, prey will be caught with a probability depending on the place. We study this problem in different situations, modelling them as two-person zero-sum games. We solve them in closed form, giving optimal strategies for prey and for predator and the value of the games.     

A Holling II functional response food chain model with impulsive perturbations

In this paper, we investigate a three trophic level food chain system with Holling II functional responses and periodic constant impulsive perturbations of top predator. Conditions for extinction of predator as a pest are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics (for example: period doubling, period halfing, chaos crisis) in the positive octant. The dynamics behavior is found to be very sensitive to the parameter values and initial value.     

TRAVELING WAVE SOLUTIONS FOR A PREDATOR-PREY MODEL WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE

In this paper, we study a class of predator-prey models with Beddington-De Angelis functional response. And the predator equation has singularity in zero prey population, where a smoothing auxiliary function is introduced to overcome it. Our aim is to see if the predator and prey can eventually survive when an alien predator enters the habitat of an existing prey by employing traveling wave solutions, based on the upper and lower solutions and Schauder's fixed point theorem. In addition, the non-existence of traveling wave solutions is discussed by the comparison principle. At the same time, some simulations are carried out to further verify the results.     

A mathematical model for a spatial predator–prey interaction

A predator–prey model is proposed within the general scheme of extended thermodynamics. An additional equation of balance is needed to account for the flux of the number density of the predator as an independent field variable. Copyright © 2002 John Wiley & Sons, Ltd.     

A discrete-time model with non-monotonic functional response and strong Allee effect in prey

In this paper, we investigate the impact of strong Allee effect on the stability of a discrete-time predator–prey model with a non-monotonic functional response. The dynamics of discrete-time predator–prey models with strong Allee effect is studied earlier. But, the mathematical investigations of predator–prey dynamics in discrete-time set up with Holling type-IV functional response and strong Allee effect in prey are lacking. The proposed model supports the coexistence of two steady states, and the mathematical features of the model are analyzed based on local stability and bifurcation theory. By considering the Allee parameter as the bifurcation parameter, we provide sufficient conditions for the flip and the Neimark–Sacker bifurcations. We observe that Allee parameter plays a significant role in the dynamics of the system.     

一个群体防卫捕-食系统的定性分析

该文对一个群体防卫捕一食系统进行了较全面的定性分析．讨论了分界线的相对位置，得到了极限环的存在性、唯一性以及分界线环的存在性，首次证明了群体防卫捕一食系统可以至少存在两个或三个极限环．     

Predator invasion in predator–prey model with prey-taxis in spatially heterogeneous environment

We consider a predator–prey model with prey-taxis and Holling-type II functional responses in a spatially heterogeneous environment to analyze the effects of prey-taxis and the heterogeneity of an environment on predator invasion. To achieve our goal, we investigate the stability of semi-trivial solution in which the predator is absent. It is known that both the predator diffusion and the death rate contribute to the predator invasion in a heterogeneous habitat when there is no prey-taxis. In this paper, we show that predator invasion is affected by the prey-taxis and diffusions of the prey-taxis model for a certain range of predator death rates in a heterogeneous environment. Furthermore, in cases where predator invasion by predator diffusion does not occur in a particular death rate range of the predator, predator invasion can occur by prey-taxis in a spatially heterogeneous habitat. In addition, we compare this phenomenon to the corresponding predator–prey model with ratio-dependent functional responses. It is observed that none of the predator’s diffusion and prey-taxis affect the predator’s invasion, and that only the predator’s death rate contributes to predator invasion for the model with ratio-dependent functional responses.     

POSITIVE STEADY STATES OF A DIFFUSIVE PREDATOR-PREY SYSTEM WITH PREDATOR CANNIBALISM

The purpose of this paper is to investigate positive steady states of a diffusive predator-prey system with predator cannibalism under homogeneous Neumann boundary conditions. With the help of implicit function theorem and energy integral method, nonexistence of non-constant positive steady states of the system is obtained, whereas coexistence of non-constant positive steady states is derived from topological degree theory. The results indicate that if dispersal rate of the predator or prey is sufficiently large, there is no nonconstant positive steady states. However, under some appropriate hypotheses, if the dispersal rate of the predator is larger than some positive constant, for certain ranges of dispersal rates of the prey, there exists at least one non-constant positive steady state.     

PERMANENCE OF A NONLINEAR PERIODIC PREDATOR-PREY SYSTEM WITH PREY DISPERSAL AND PREDATOR DENSITY-INDEPENDENCE

In this paper,a set of suffcient conditions which ensure the permanence of a nonlinear periodic predator-prey system with prey dispersal and predator density-independence are obtained,where the prey species can disperse among n patches,while the density-independent predator is confined to one of the patches and cannot disperse. Our results generalize some known results.