The Stability of the Critical Points of the Generalized Gause Type Predator-Prey Fishery Models With Proportional Harvesting and Time Delay

Abstract

In the marine ecosystem, the time delay or lag may occur in the predator response function, which measures the rate of capture of prey by a predator. This is because, when the growth of the prey population is null at the time delay period, the predator’s growth is affected by its population and prey population densities only after the time delay period. Therefore, the generalized Gause type predator-prey fishery models with a selective proportional harvesting rate of fish and time lag in the Holling type II predator response function are proposed to simulate and solve the population dynamical problem. From the mathematical analysis of the models, a certain dimension of time delays in the predator response or reaction function can change originally stable non-trivial critical points to unstable ones. This is due to the existence of the Hopf bifurcation that measures the critical values of the time lag, which will affect the stabilities of the non-trivial critical points of the models. Therefore, the effects of increasing and decreasing the values of selective proportional harvesting rate terms of prey and predator on the stabilities of the non-trivial critical points of the fishery models were analysed. Results have shown that, by increasing the values of the total proportion of prey and predator harvesting denoted by qx Ex and qy Ey respectively, within the range 0.3102 ≤ qx Ex ≤ 0.9984 and 0.5049 ≤ qy Ey ≤ 0.5363, the originally unstable non-trivial critical points of the fishery models can be stable.

The Impacts of Scavenging on the Prey-Predator-Scavenger Fishery Model in the Existence of Harvesting and Toxin

“…( 2) always unstable. Based on the research conducted by Wan Hussin et al, [27], large periodic oscillations of both prey and predator populations around their unstable coexistence steady-states occur when the time delays at the predator reaction function which measures the capture rate of prey by a predator exceed their threshold or Hopf bifurcation value. The growth of the predator population is influenced by its population and prey population densities at time delays above the HB point, which makes the coexistence steady-states of the fishery models unstable.…”