Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator–prey model

Abstract

Time delay is an essential ingredient of spatio-temporal predator-prey models since the reproduction of the predator population after predating the prey will not be instantaneous, but is mediated by a constant time lag accounting for the gestation of predators. In this paper we study a predator-prey reaction-diffusion system with time delay, where a stability analysis involving Hopf bifurcations with respect to the delay parameter and simulations produced by a new numerical method reveal how this delay affects the formation of spatial patterns in the distribution of the species. In particular, it turns out that when the carrying capacity of the prey is large and whenever the delay exceeds a critical value, the reaction-diffusion system admits a limit cycle due to the Hopf bifurcation. This limit cycle induces the spatio-temporal pattern. The proposed discretization consists of a finite volume element (FVE) method combined with a Runge-Kutta scheme.

Fractional Dynamics Based-Enhancing Control Scheme of a Delayed Predator-Prey Model

“…In this section, time delay shall be selected as a bifurcation parameter to investigate the problem of bifurcation control for the predator-prey model (5). The existence bifurcation and bifurcation point for the proposed model shall be established.…”

A Modified Immersed Finite Volume Element Method for Elliptic Interface Problems

“…Due to the local conservation property and other attractive properties, the finite volume element (FVE) method is widely used in computational fluid dynamics (see [5,7,13,25,[37][38][39] and the references therein). Although the numerical analysis of the FVE method is more difficult than that of the FE method, a general framework for analysing the FVE methods has been proposed [8,9,11].…”

Identification of the initial population of a nonlinear predator-prey system backwards in time

“…In [20], the authors considered a degenerate predator-prey nonlinear system with homogeneous Dirichlet boundary conditions to describe the local interactions of prey and predator species where there is a direct movement of predators caused by a variation in prey. As far as numerical simulations are concerned, in [3], the authors studied numerical methods for obtaining spatio-temporal patterns described by a predator-prey model with time delay and diffusion.…”