A mathematical model of population dynamics revisited with fear factor, maturation delay, and spatial coefficients

Abstract

This study concentrates on the dynamics of a prey-predator model incorporating the fear effect in the birth and death rate of prey, whose physiological changes may give rise to undirect predation. In the presence and absence of time delay, single parameter numerical continuation with respect to two parameters, that are (i) fear level in the death rate of prey and (ii) conversion efficiency by which new predators are introduced through prey consumption in the system, is performed. Basic results on extinction and delay-driven Hopf bifurcation criteria are investigated. Then, the model is extended to involve the spatial dynamics with and without time delay. Theoretical results for orientation and stability of Hopf bifurcation in spatial system are provided by applying the normal form recipe and also the center manifold theory. Classical reaction-diffusion-type models, incorporating self-diffusion, can induce regular (periodic) and irregular (chaotic) oscillations in space. However, space periodic oscillations are not common in prey-predator interactions. Here, it is shown that the dynamics of only diffusion involved model is periodically arranged in space and time. However, introducing a very small value of time delay in predator maturation, spatial dynamics with chaos is initiated as a result of the joint effect of time delay and diffusion. This reassures that time delay has a crucial role in population dynamics incorporated with the role of indirect predation and gives some useful intuition into underlying species interactions. Theoretical results of the model in this paper are supported with numerical experiments.