Mathematical analysis of an ecological system using a non-monotonic functional response: effects of gestation delay and predator harvesting

This article has formulated a mathematical model for the dynamics of an ecological system consisting of one prey (nutrient) and two predators (two fish populations). We have assumed Holling Type I and generalized Holling Type IV (non-monotonic) functional responses for first and second predators, respectively, and harvesting of both predators. We have considered constant recruitment of nutrients (prey), and a certain fraction of nutrients is assumed to be unused/decomposed during interaction with the predators. The local stability behavior and the existence of the Hopf bifurcation of coexisting steady-state for both the delayed and non-delayed systems are analyzed. Using the normal form theory and the center manifold theorem, we have discussed the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution. Finally, the results of theoretical analysis are verified via numerical simulations. We have established that time delay and harvesting parameters play a significant role in the system's dynamics. Model dynamics are sensitive to the initial size of the population. As the rate of harvesting increases, the steady state's stability shifts from unstable to stable, consequently bifurcation occurs when the harvesting rate changes. We have found that the Hopf bifurcation of both delayed and the non-delayed system is of sub-critical type.