Dynamic Analysis of a Phytoplankton-Fish Model with Biological and Artificial Control

Abstract

We investigate a nonlinear model of the interaction between phytoplankton and fish, which uses a pair of semicontinuous systems with biological and artificial control. First, the existence of an order-1 periodic solution to the system is analyzed using a Poincaré map and a geometric method. The stability conditions of the order-1 periodic solution are obtained by a theoretical mathematical analysis. Furthermore, based on previous analysis, we investigate the bifurcation in the order-1 periodic solution and prove that the order-1 periodic solution breaks up an order-1 periodic solution at least. In addition, the transcritical bifurcation of the system is described. Finally, we provide a series of numerical results that illustrate the feasibility of the theoretical results. Based on the theoretical and numerical results, we analyzed the feasibility of biological and artificial control, which showed that biological and artificial methods can control phytoplankton blooms. These results are expected to be useful for the study of phytoplankton dynamics in aquatic ecosystems.

Effects of pesticide dose on Holling II predator–prey model with feedback control

“…Since many problems of human interventions originated from real world need to be mathematically modelled and solved, impulsive semi-dynamical systems, which are described by impulsive differential equations, are often being proposed to model control strategies of closed-loop type, rather than fixed time pulsed model which are used to solve problems of open-loop techniques [17,36,37]. For predator-prey systems with closed-loop control (or state-dependent feedback control), many factors of human actions were investigated including culling, resource limitation, insect resistance, biological and chemical control, media report and so on [16,23,[28][29][30]34,38]. For simplicity, the killing rates or reductions resulting from these elements are usually considered as constants.…”

Dynamic Complexity of a Phytoplankton-Fish Model with the Impulsive Feedback Control by means of Poincaré Map

“…People would be faced with increasing shortages of fish resources. erefore, it is imperative to formulate effective fishing strategies, maintain the ecological balance of fishery resources, and protect the ecological environment [1][2][3][4].…”