Numerical convergence of a Telegraph Predator-Prey System

TL;DR

Numerical convergence of a Telegraph Predator-Prey system is studied and it was verified that the mesh refinement and the model parameters, reactive constants, diffusion coefficients and delay constants, determine the stability/instability conditions of the discretized equations.

Abstract

The numerical convergence of a Telegraph Predator-Prey system is studied. This system of partial differential equations (PDEs) can describe various biological systems with reactive, diffusive and delay effects. Initially, our problem is mathematically modeled. Then, the PDEs system is discretized using the Finite Difference method, obtaining a system of equations in the explicit form in time and implicit form in space. The consistency of the Telegraph Predator-Prey system discretization was verified. Next, the von Neumann stability conditions were calculated for a Predator-Prey system with reactive terms and for a Telegraph system with delay. For our Telegraph Predator-Prey system, through numerical experiments, it was verified tat the mesh refinement and the model parameters (reactive constants, diffusion coefficient and delay term) determine the stability/instability conditions of the model. Keywords: Telegraph-Diffusive-Reactive System. Maxwell-Cattaneo Delay. Discretization Consistency. Von Neumann Stability. Numerical Experimentation.

Figures (14)

• Figure 1: The three types of functional responses identified by Holling.
• Figure 2: Discrete one-dimensional mesh used for the Telegraph Predator-Prey system.
• Figure 3: Region of stability/instability for the Telegraph Predator-Prey system for refinements $\Delta x = 0.1$ and $\Delta t = 0.002$.
• Figure 4: Region of stability/instability for the Telegraph Predator-Prey system for refinements $\Delta x = 0.1$ and $\Delta t = 0.001$.
• Figure 5: Region of stability/instability for the Telegraph Predator-Prey system for refinements $\Delta x = 0.1$ and $\Delta t = 0.0015255$.