Dynamics and Optimal Taxation Control in a Bioeconomic Model with Stage Structure and Gestation Delay

A prey-predator model with gestation delay, stage structure for predator, and selective harvesting effort on mature predator is proposed, where taxation is considered as a control instrument to protect the population resource in prey-predator biosystem from overexploitation. It shows that interior equilibrium is locally asymptotically stable when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. An optimal harvesting policy can be obtained by virtue of Pontryagin's Maximum Principle without considering gestation delay; on the other hand, the interior equilibrium of model system loses as gestation delay increases through critical certain threshold, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of model system is also observed. Finally, numerical simulations are carried out to show consistency with theoretical analysis

Theoretical Study of Pest Control Using Stage Structured Natural Enemies with Maturation Delay: A Crop-Pest-Natural Enemy Model

In the natural world, there are many insect species whose individual members have a life history that takes them through two stages, immature and mature. Moreover, the rates of survival, development, and reproduction almost always depend on age, size, or development stage. Keeping this in mind, in this paper, a three species crop-pest-natural enemy food chain model with two stages for natural enemies is investigated. Using characteristic equations, a set of sufficient conditions for local asymptotic stability of all the feasible equilibria is obtained. Moreover, using approach as in (Beretta and Kuang, 2002), the possibility of the existence of a Hopf bifurcation for the interior equilibrium with respect to maturation delay is explored, which shows that the maturation delay plays an important role in the dynamical behavior of three species system. Also obtain some threshold values of maturation delay for the stability-switching of the particular system. In succession, using the normal form theory and center manifold argument, we derive the explicit formulas which determine the stability and direction of bifurcating periodic solutions. Finally, a numerical simulation for supporting the theoretical analysis is given.Comment: 28 pages, 9 figure

Dynamics of a stage–structure Rosenzweig–MacArthur model with linear harvesting in prey and cannibalism in predator

A kind of stage-structure Rosenzweig–MacArthur model with linear harvesting in prey and cannibalism in predator is investigated in this paper. By analyzing the model, local stability of all possible equilibrium points is discussed. Moreover, the model undergoes a Hopf–bifurcation around the interior equilibrium point. Numerical simulations are carried out to illustrate our main results

Controllability of an eco-epidemiological system with disease transmission delay: A theoretical study

This paper deals with the qualitative analysis of a disease transmission delay induced prey preda-tor system in which disease spreads among the predator species only. The growth of the preda-tors’ susceptible and infected subpopulations is assumed as modified Leslie–Gower type. Suffi-cient conditions for the persistence, permanence, existence and stability of equilibrium points are obtained. Global asymptotic stability of the system is investigated around the coexisting equilib-rium using a geometric approach. The existence of Hopf bifurcation phenomenon is also exam-ined with respect to some important parameters of the system. The criterion for disease a trans-mission delay the induced Hopf bifurcation phenomenon is obtained and subsequently, we use a normal form method and the center manifold theorem to examine the nature of the Hopf bifurca-tion. It is clearly observed that competition among predators can drive the system to a stable from an unstable state. Also the infection and competition among predator population enhance the availability of prey for harvesting when their values are high. Finally, some numerical simu-lations are carried out to illustrate the analytical results

A Fractional-Order Predator-Prey Model with Age Structure on Predator and Nonlinear Harvesting on Prey

In this manuscript, the dynamics of a fractional-order predator-prey model with age structure on predator and nonlinear harvesting on prey are studied. The Caputo fractional-order derivative is used as the operator of the model by considering its capability to explain the present state as the impact of all of the previous conditions. Three biological equilibrium points are successfully identified including their existing properties. The local dynamical behaviors around each equilibrium point are investigated by utilizing the Matignon condition along with the linearization process. The numerical simulations are demonstrated not only to show the local stability which confirms all of the previous analytical results but also to show the existence of periodic signal as the impact of the occurrence of Hopf bifurcation

Hopf bifurcation and optimal control in a diffusive predator-prey system with time delay and prey harvesting

In this paper, we investigated the dynamics of a diffusive delayed predator-prey system with Holling type II functional response and nozero constant prey harvesting on no-flux boundary condition. At first, we obtain the existence and the stability of the equilibria by analyzing the distribution of the roots of associated characteristic equation. Using the time delay as the bifurcation parameter and the harvesting term as the control parameter, we get the existence and the stability of Hopf bifurcation at the positive constant steady state. Applying the normal form theory and the center manifold argument for partial functional differential equations, we derive an explicit formula for determining the direction and the stability of Hopf bifurcation. Finally, an optimal control problem has been considered

A Review of Chaos Control Strategies for Tri-trophic Food Chain Ecological Systems

The existence of chaos in ecological models is quite obvious due to the presence of nonlinear terms. Controlling chaotic phenomena in ecological systems remains a difficult task due to their unpredictability, and thus chaos control is one of the main objectives for constructing mathematical models in ecology today. Our aim in this paper is to review chaos control strategies for the tri-trophic food chain models by using various ecological factors. The factors include additional food, prey refuge, the Allee effect, the fear effect, and harvesting. We establish the essential conditions for the existence of ecologically feasible equilibrium points in the food chain ecological systems and their local stability. This paper provides a unified overview of recent research on the chaos control of ecological systems. The theoretical results suggest a way to control populations of species in ecological systems for fishing and pest management in farming. Numerical examples are performed to justify and compare the theoretical findings through phase portraits and bifurcation diagrams