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 current paper investigates a prey-predator-scavenger fishery model in which both prey and predator species release harmful toxins during predation interaction. The harvesting of scavengers is introduced due to their economic value and serves as one of the important protein sources for humans. Both predator and scavenger fish species consume prey as their food. Scavengers devour the carrion or carcasses of predators which die of natural causes and are infected by the toxins of their preys. The non-negativity of solutions from the fishery model has been derived to ensure biologically meaningful. The model's equilibria are investigated along with its local stability characteristics. We investigate the threshold conditions triggering the bifurcations to occur in the steady-states concerning the scavenging activities by the scavengers. By formulating a suitable Lyapunov function, the global stability analysis of the non-trivial or coexistence steady-state is implemented. The dynamical behaviours of the fishery model, as well as the persistence and extirpation properties, have been analysed using the bifurcation analysis. From the results, it is found that the scavenging parameter in the fishery model with the presence of harvesting and toxin can drive the fish population towards extinction state which is unstable and unfavourable in nature.