Stability and traveling waves of a stage-structured predator–prey model with Holling type-II functional response and harvesting

In this paper, we consider a reaction–diffusion predator–prey model with stage-structure, Holling type-II functional response, nonlocal spatial impact and harvesting. The stability of the equilibria is investigated. Furthermore, by the cross-iteration scheme companied with a pair of admissible upper and lower solutions and Schauder fixed point theorem, we deduce the existence of traveling wave solution which connects the zero solution and the positive constant equilibrium.     

Stage-structured Harvest Models

We formulate a stage-structured population model where the population is divided to two classes, the juveniles and the adults. Then, we include harvest in the model and assume that the harvesting is only on adults. The cases where the harvesting rate is constant, proportional to the amount of adults, or of Holling-II type are studied. While the model dynamics are relatively simple when the harvesting rate is proportional, the model system with a constant or a Holling-II type harvesting rate can have multiple positive equilibria. We explore the existence of all possible equilibria and investigate their stability. We also give numerical examples to confirm our findings.     

Dynamics of a ratio-dependent eco-epidemiological system with prey harvesting

In this article, we study a ratio-dependent eco-epidemiological system where prey population is subjected to harvesting. Mathematical results like positive invariance, boundedness, stability of equilibria, and permanence of the system have been established. The dynamics of zero equilibria have been thoroughly investigated to find out conditions on the system parameters such that trajectories starting from the domain of interest can reach a zero equilibrium following any fixed direction. We have also studied suitable conditions for non-existence of a periodic solution around the interior equilibrium. Computer simulations have been carried out to illustrate different analytical findings.     

A PREDATOR-PREY SYSTEM WITH STAGE STRUCTURE AND HARVESTING FOR PREDATOR

In this paper, we consider a predator-prey system with stage structure and harvesting (where the predator population has two stages, an immature stage and a mature stage with harvesting, and the growth of the prey population is of Lotka-Volterra nature). We obtain the conditions of the globally asymptotic stability for three nonne-gative equilibria of this system.     

Stability and bifurcation analysis for a fractional prey–predator scavenger model

In this study, we consider a fractional prey–predator scavenger model as well as harvesting by a predator and scavenger. We prove the positivity and boundedness of the solutions in this system. The model undergoes a Hopf bifurcation around one of the existing equilibria where the conditions are met for the occurrence of a Hopf bifurcation. The results show that chaos disappears in this biological model. We conclude that the fractional system is more stable compared with the classical case and the stability domain can be extended under fractional order. In addition, a suitable amount of prey harvesting and a fractional order derivative can control the chaotic dynamics and stabilize them. We also present an extended numerical simulation to validate the results.     

A ratio‐dependent predator–prey model with stage structure and optimal harvesting policy

In this paper, a ratio‐dependent predator–prey model with stage structure and harvesting is investigated. Mathematical analyses of the model equations with regard to boundedness of solutions, nature of equilibria, permanence and stability are performed. By constructing appropriate Lyapunov functions, a set of easily verifiable sufficient conditions are obtained for the global asymptotic stability of nonnegative equilibria of the model. The existence possibilities of bioeconomic equilibria have been examined. An optimal harvesting policy is also given by using Pontryagin's maximal principle. Copyright © 2007 John Wiley & Sons, Ltd.     

Harvesting of a phytoplankton–zooplankton model

Considering that some phytoplankton and zooplankton are harvested for food, a phytoplankton–zooplankton model with harvesting is proposed and investigated. First, stability conditions of equilibria and existence conditions of a Hopf-bifurcation are established. Our results indicate that over exploitation would result in the extinction of the population and an appropriate harvesting strategy should ensure the sustainability of the population which is in line with reality. Furthermore, the existence of bionomic equilibria and the optimal harvesting policy are discussed. The present value of revenues is maximized by using Pontryagin’s maximum principle subject to the state equations and the control constraints. We discussed the case of optimal equilibrium solution. It is found that the shadow prices remain constant over time in optimal equilibrium when they satisfy the transversality condition. It is established that the zero discounting leads to the maximization of economic revenue and that an infinite discount rate leads to complete dissipation of economic rent. Finally, some numerical simulations are given to illustrate our results.     

Dynamics of prey threshold harvesting and refuge

The dynamics of a predator-prey model with continuous threshold prey harvesting and prey refuge is studied. One central question is how harvesting and refuge could directly affect the dynamics of the ecosystem, such as the stability properties of some coexistence equilibria and periodic solutions. Theoretical and numerical methods are used to investigate boundedness of solutions, existence of bionomic equilibria, as well as the existence and stability properties of equilibrium points and periodic solutions. Several bifurcations are also studied.     

具有食饵避难的Leslie-Gower捕食系统最优收获分析

研究了一类具有食饵避难的Leslie-Gower捕食与被捕食系统收获模型,利用Hurwitz判据,得到了正平衡点局部渐近稳定,进一步构造了适当的Lyapunov函数,证明了正平衡点的全局渐近稳定性.并且在捕获努力量假说下,对发生食饵避难的两种群同时捕获,考虑了生态经济平衡点的存在性和利用Pontryagin最大值原理对两种群进行最优收获,得到当贴现率为零时,既保持了生态平衡,又使得在渔业开发过程中取得最大经济利益.     

Stability and traveling waves of diffusive predator-prey model with age-structure and nonlocal effect

The paper is concerned with the dynamical behaviors of a stage-structured diffusive predator-prey model with nonlocal effect and harvesting. The linear stability of the equilibria is investigated by using the characteristic equation technique. By constructing a closed convex set bounded by a pair of upper-lower solutions and using Schauder fixed point theorem, the existence of traveling wave solution connecting two steady states is also derived. Finally, a pair of upper-lower solutions is constructed by using inequality technique and characteristic equations.     

A COMMENSAL SYMBIOSIS MODEL WITH HOLLING TYPE FUNCTIONAL RESPONSE AND NON-SELECTIVE HARVESTING IN A PARTIAL CLOSURE

A two species commensal symbiosis model with Holling type functional response and non-selective harvesting in a partial closure is considered. Local and global stability property of the equilibria are investigated. Depending on the the area available for capture, we show that the system maybe extinct or one of the species will be driven to extinction, while the rest one is permanent,or both of the species coexist in a stable state. The dynamic behaviors of the system is complicated and sensitive to the fraction of the harvesting area.     

Analysis of predator-prey models with continuous threshold harvesting

We formulate an alternative form of threshold harvesting and investigate its properties within the framework of a predator-prey model. Our formulation accounts for economic constraints and is defined as a continuous harvesting function of the predator species. Theoretical and numerical analysis indicate that our formulation, which we call continuous threshold policy (CTP), can help improve undesirable behavior of the predator-prey ecosystem. Our study includes uniform boundedness of solutions, stability of equilibria and periodic orbits, bifurcations, and heteroclinic orbits.     

DYNAMICAL ANALYSIS OF A SINGLE-SPECIES POPULATION MODEL WITH MIGRATIONS AND HARVEST BETWEEN PATCHES

A single-species population model with migrations and harvest between the protected patch and the unprotected patch is formulated and investigated in this paper. We study the local stability and the global stability of the equilibria. The research points out, under some suitable conditions, the singlespecies population model admits a unique positive equilibrium, which is globally asymptotically stable. We also derive that the trivial solution is globally asymptotically stable when the harvesting rate exceeds the threshold. Further,we discuss the practical effects of the protection zones and the harvest. The main results indicate that the protective zones indeed eliminate the extinction of the species under some cases, and the theoretical threshold of harvest to the practical management of the endangered species is provided as well. To end this contribution and to check the validity of the main results, numerical simulations are separately carried out to illustrate these results.     

Stability preserving non-standard finite difference scheme for a harvesting Leslie–Gower predator–prey model

A non-standard finite difference scheme for a harvesting Leslie–Gower equations is constructed. It is shown that the obtained difference system has the same dynamics as the original continuous system, such as positivity of solutions, equilibria and their local stability properties, irrespective of the size of numerical time step. To illustrate the analytical results, we present some numerical simulations.     

A STAGE-STRUCTURED AND HARVESTING PREDATOR-PREY SYSTEM

A predator-prey system with independent harvesting in either species and BeddingtonDeAngelis functional response is investigated. By analyzing characteristic equations and using an iterative technique,we obtain a set of easily verifiable sufficient conditions,which ensure the local and global stability of the nonnegative equilibria of the system. It is also shown that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities. Numerical simulations are carried out t...     

Global asymptotic stability of a two species competitive system with stage structure and harvesting

Introduction' There have recently appeared in the literature several mathematical models of stagestructured population growth, i. e., models which take into account the faCt that individuals in a population may belong to one of two classes, the immatures and the matureslllZI.Cannibalism has been observed in a great variety of species, including a number of fish species.Cannibalism models of various types have also been investigatedI3"l. In these models, the ageto maturity is represented by a…     

Optimal Harvesting and Stability for a Predator-prey System with Stage Structure

The dynamics of a predator-prey system, where prey population has two stages, an immature stage and a mature stage with harvesting, the growth of predator population is of Lotka-Volterra nature, are modelled by a system of retarded functional differential equations. We obtain conditions for global asymptotic stability of three nonnegative equilibria and a threshold of harvesting for the mature prey population. The effect of delay on the population at positive equilibrium and the optimal harvesting of the mature prey population are also considered.     

Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting

In this paper, we investigate the dynamics of a ratio dependent predator-prey model with quadratic harvesting. We examine the existence of the positive equilibria, the related dynamical behaviors of the model, as well as the boundedness and permanence property of the system. We also study the global stability of the interior equilibrium without time delay. Finally some bifurcation analysis is carried out for the system with delay and the results are illustrated numerically.     

A bioeconomic model of a ratio-dependent predator-prey system and optimal harvesting

This paper deals with the problem of a ratio-dependent prey-predator model with combined harvesting. The existence of steady states and their stability are studied using eigenvalue analysis. Boundedness of the exploited system is examined. We derive conditions for persistence and global stability of the system. The possibility of existence of bionomic equilibria has been considered. The problem of optimal harvest policy is then solved by using Pontryagin’s maximal principle.     

Stable coexistence mediated by specialist harvesting in a two zooplankton–phytoplankton system

This paper deals with a predator–prey model with specialist harvesting, representing a two predators (Zooplankton) and one resource (Phytoplankton) system. First, the existence and stability of equilibria is analyzed both from local and global point of view. Our results indicate that a specialist harvesting which is discriminate may mediate the coexistence of the two zooplankton species which competitively exclude each other in absence harvesting. Although in most cases increasing harvesting reduces the two zooplankton species numbers, when harvesting leads to coexistence, it may also lead to increase the two zooplankton species numbers. Furthermore, to protect fish population from over exploitation a control instrument tax is imposed. The problem of optimal taxation policy is then solved by using Pontryagin’s maximal principle. It is established that the zero discounting leads to the maximization of the net economic revenue to the society and an infinite discount rate leads to complete dissipation of the net economic revenue to the society. Finally, the impact of harvesting is mentioned along with numerical results to provide some support to the analytical findings.