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.
Highlights
Phytoplankton plays an important role in ecology and the climate because it participates in the global carbon cycle as the base of the food chain [1]
The results indicated that silver carp and bighead carp controlled the blue algae blooms effectively, and the effective biomass required to contain the blooms was determined to be 46–50 g/m3
Phytoplankton blooms are observed frequently, but the growth rate of fishes being less than the mortality rate of fishes is usually rare in real life
Summary
Phytoplankton plays an important role in ecology and the climate because it participates in the global carbon cycle as the base of the food chain [1]. We used the theory of impulsive differential equations [12,13,14] to develop a phytoplankton-fish model to investigate the feasibility of biological and artificial methods. Many researchers have studied some ecological systems with impulsive differential equations [15,16,17,18,19,20,21,22], including the use of biological and chemical controls. It is known that some filterfeeding fish feed on phytoplankton, so the population density of phytoplankton can control the rate of fish production In this system, the loss of fish occurs via death and natural predation by higher trophic levels in the food web. According to other studies [28, 29], some phytoplankton blooms may be controlled within a short period of time (
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