Bifurcation and Control in a Singular Phytoplankton-Zooplankton-Fish Model with Nonlinear Fish Harvesting and Taxation

In this paper, a delayed differential algebraic phytoplankton-zooplankton-fish model with taxation and nonlinear fish harvesting is proposed. In the absence of time delay, the existence of singularity induced bifurcation is discussed by regarding economic interest as bifurcation parameter. A state feedback controller is designed to eliminate singularity induced bifurcation. Based on Liu’s criterion, Hopf bifurcation occurs at the interior equilibrium when taxation is taken as bifurcation parameter and is more than its corresponding critical value. In the presence of time delay, by analyzing the associated characteristic transcendental equation, the interior equilibrium loses local stability when time delay crosses its critical value. What’s more, the direction of Hopf bifurcation and stability of the bifurcating periodic solutions are investigated based on normal form theory and center manifold theorem, and nonlinear state feedback controller is designed to eliminate Hopf bifurcation. Furthermore, Pontryagin’s maximum principle has been used to obtain optimal tax policy to maximize the benefit as well as the conservation of the ecosystem. Finally, some numerical simulations are given to demonstrate our theoretical analysis.

The Modelling and Control of a Singular Biological Economic System in a Polluted Environment

A singular biological economic model with harvesting and stage structure is presented. The local stability of equilibriums of the system is investigated when the economic profit parameter is zero, and the conditions of the singularity induced bifurcation occurring at the positive equilibrium are obtained by the singular systems theory and bifurcation theory. In order to eliminate the singularity induced bifurcation, a state feedback controller is designed by controlling the fishing effort. At last, an application example is given to illustrate the validity of the theoretical results.

Complex Dynamics in a Singular Delayed Bioeconomic Model with and without Stochastic Fluctuation

A singular delayed biological economic predator-prey system with and without stochastic fluctuation is proposed. The conditions of singularity induced bifurcation are gained, and a state feedback controller is designed to eliminate such bifurcation. Furthermore, saddle-node bifurcation is also showed. Next, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of roots of the corresponding characteristic equation, and the hybrid control strategy is used to control the occurrence of Hopf bifurcation. In addition, some explicit formulas determining the spectral densities of the populations and harvest effort are given when the system is considered with stochastic fluctuation. Finally, numerical simulations are illustrated to verify the theoretical results.

Bifurcation and Feedback Control of an Exploited Prey-Predator System

This paper makes an attempt to highlight a differential algebraic model in order to investigate the dynamical behavior of a prey-predator system due to the variation of economic interest of harvesting. In this regard, it is observed that the model exhibits a singularity induced bifurcation when economic profit is zero. For the purpose of stabilizing the proposed model at the positive equilibrium, a state feedback controller is therefore designed. Finally, some numerical simulations are carried out to show the consistency with theoretical analysis and to illustrate the effectiveness of the proposed controller.

The Bifurcation and Control of a Single-Species Fish Population Logistic Model with the Invasion of Alien Species

The objective of this paper is to study systematically the bifurcation and control of a single-species fish population logistic model with the invasion of alien species based on the theory of singular system and bifurcation. It regardsSpartina anglicaas an invasive species, which invades the fisheries and aquaculture. Firstly, the stabilities of equilibria in this model are discussed. Moreover, the sufficient conditions for existence of the trans-critical bifurcation and the singularity induced bifurcation are obtained. Secondly, the state feedback controller is designed to eliminate the unexpected singularity induced bifurcation by combining harvested effort with the purification capacity. It obviously inhibits the switch of population and makes the system stable. Finally, the numerical simulation is proposed to show the practical significance of the bifurcation and control from the biological point of view.