Dynamics Analysis and Optimality in Selective Harvesting Predator-Prey Model With Modified Leslie-Gower and Holling-Type II

In this work, we consider the optimal harvesting and stability problems of a prey-predator model with modified Leslie-Gower and Holling-type II functional response. The model is governed by a system of three differential equations which describe the interactions between prey, predator and harvesting effort. Boundedness and existence of solutions for this system are showed. The existence and local stability of the possible steady states are analyzed and the conditions of global stability of the interior equilibrium are established by using the Lyapunov function, we prove also the occurrence of Hopf bifurcation at this point. By using the Pontryagin’s maximal principle, we formulate and we solve the problem of the optimal harvest policy. In the end, some numerical simulations are given to support our theoretical results.

Optimal harvesting of a prey–predator model with variable carrying capacity

This work deals with a prey–predator model in an environment where the carrying capacities are assumed to be variable with time and one species feeds upon the other. Independent harvesting efforts are applied in either species and asymmetrical intraguild predation occurs. A common resource is consumed by two competing species and at the same time predator also consumes the prey. At first we discuss the model under constant carrying capacity and make the conclusion that no limit cycle exists in this case. Then we discuss the model without intraspecific competition. Our main concern is to cover the above mentioned two cases together, i.e. the model with variable carrying capacity and intraspecific competition. We determine the steady states and examine the dynamical behavior. We also analyze the local and global stability of the interior equilibrium by Routh–Hurwitz criterion and a suitable Lyapunov function respectively. A Hopf bifurcation occurs with respect to a parameter which is the ratio of predator’s and prey’s intrinsic growth rate. The possibility of bionomic equilibrium has been considered. The optimal harvest policy is formulated and solved with Pontryagin’s maximum principle. Some numerical simulations are given to explain most of the analytical results.

Harvesting on Facultative Mutualist Prey Species in Presence of a Predator

<p><em>This paper proposes a model with two preys of facultative mutualist type and one predator. Linear predation functions are considered and preys are only considered to be harvested. The stability of the model is analyzed theoretically and numerically in this paper. The optimal harvest policy is studied and the solution is derived in the interior equilibrium case using Pontryagin’s maximum principle. Finally, some numerical simulations are discussed.</em></p>

Stability and Selective Harvesting of a Phytoplankton-Zooplankton System

Considering that some zooplankton can be harvested for food in some bodies of water, a phytoplankton-zooplankton model with continuous harvesting of zooplankton only is proposed and investigated. By using environmental carrying capacity as a parameter, possible dynamic behaviors, such as stability, global stability, Hopf bifurcation, and transcritical bifurcations, are analyzed. The optimal harvesting policy is disposed by imposing a tax per unit biomass of zooplankton. The problem of determining the optimal harvest policy is solved by using Pontryagin's maximum principle subject to the state equations and the control constraints, and the impact of tax is also discussed. Finally, some numerical simulations are performed to justify analytical findings.

NONCONSTANT PREY HARVESTING IN RATIO-DEPENDENT PREDATOR–PREY SYSTEM INCORPORATING A CONSTANT PREY REFUGE

This paper deals with the problem of nonconstant harvesting of prey in a ratio-dependent predator–prey system incorporating a constant prey refuge. Here we use the reasonable catch-rate function instead of usual catch-per-unit-effort hypothesis. The existence, as well as the stability of possible equilibria, is carried out. Bionomic equilibrium of the system is determined and optimal harvest policy is studied with the help of Pontryagin's maximum principle. The key results developed in this paper are illustrated using numerical simulations. Our results indicate that dynamic behavior of the system very much depends on the prey refuge parameter and increasing amount of refuge could increase prey density and may lead to the extinction of predator population density.

A Dynamic Model for Fishery Resource with Reserve Area and Taxation

The present paper deals with a dynamic reaction model of a fishery. The dynamics of a fishery resource system in an aquatic environment consists of two zones: a free fishing zone and a reserve zone. To protect fish population from over exploitation, a control instrument tax is imposed. The existence of its steady states and their stability are studied. The optimal harvest policy is discussed next with the help of Pontryagin's maximum principle. Our theoretical results are confirmed by numerical simulation.

A bioeconomic model of nonselective harvesting of two competing fish species

This paper deals with the combined bioeconomic harvesting of two competing fish species, each of which obeys the Gompertz law of growth. The catch-rate functions are chosen so as to reflect saturation effects with respect to stock abundance as well as harvesting effort. The stability of the dynamical system is discussed and the existence of a bionomic equilibrium is examined. The optimal harvest policy is studied with the help of Pontryagin's maimum principle. The results are illustrated with the help of a numerical example.

Harvesting in a two-prey one-predator fishery: a bioeconomic model

A multispecies harvesting model with interference is proposed. The model is based on Lotka-Volterra dynamics with two competing species which are affected not only by harvesting but also by the presence of a predator, the third species. In order to understand the dynamics of this complicated system, we choose to model the simplest possible predator response function in which the feeding rate of the predator increases linearly with prey density. We derive the conditions for global stability of the system using a Lyapunov function. The possibility of existence of a bioeconomic equilibrium is discussed. The optimal harvest policy is studied and the solution is derived in the equilibrium case using Pontryagin's maximal principle. Finally, some numerical examples are discussed.

Bioeconomic Modelling of a Single Species Fishery with Gompertz Law of Growth

A single species fishery model has been developed using the Gompertz law of population growth and the CPUE (Catch-per-unit-effort) hypothesis. The dynamical and the bionomic steady states were determined and their natures were examined from the biological as well as economic view points. The optimal harvest policy is discussed by taking the fishing effort as a dynamic control variable. The results are compared with those of the Schaefer model [10].