Numerical Existence and Stability of Steady State Solutions to a Diffusive Lotka-Volterra Predator Prey Model

The study examines temporal trends in population abundance and distribution of a predator and prey species using a spatial diffusion model. Using Chebyshev Spectral Methods in MATLAB®, we were able to develop a model composed of two partial differential equations to simulate and predict population growth and range expansion for the two species. The foundation of the modeling program is a diffusive Lotka-Volterra predator-prey model, which is space and time-dependent. The program distributes the solution across a Chebyshev grid and searches for steady state equilibrium populations. Once an equilibrium solution is discovered, we assess the stability and long-term behavior of the population. We next evaluate the potential impacts of environmental drivers on future growth and expansion: specifically, we test how population growth and behavior is affected by variation in the species’ diffusion, reproduction, and mortality rates. Introduction The Lotka-Volterra predator prey model has been a staple in the arena of population dynamics since its introduction by Alfred J. Lotka in 1910 (Bacaër 2011). The system of n differential equations simulates interactions between predator and prey species. The traditional, non-diffusive model is space-independent and operates solely as a function of time. It simulates a stationary population over a length of time and is defined by the set of coupled ordinary differential equations.