Lotka-Volterra Predator-Prey Population Model with Spatial and Temporal Effects.

Abstract

Mathematical models have received extensive attention, and in particular, the Lotka-Volterra predator-prey population models are of great interests to mathematicians and ecologists. This has given rise to systems which represent more realistic ecological scenarios that appear in the context of species living and interacting in the same habitat.

In this work, a Lotka-Volterra predator-prey population dynamics model which depends on time and location of resources is developed.

Further, a numerical scheme for solving the partial differential equation (PDE) system is developed. The PDE model is both time and space dependent. Therefore finite difference methods (FDMs) are used. That is, both the Explicit and the Implicit schemes are developed, whose stability is investigated using the Von Neumann stability conditions. The implicit scheme results into a system of nonlinear equations. Hence the Newton’s method for solving nonlinear systems of equations is applied.

The effects of random movement (diffusion) and velocity caused by location of resources (advection) are analysed using numerical simulations, obtained by implementing the implicit scheme in MATLAB.

Results show that the explicit scheme is conditionally stable while the implicit scheme is unconditionally stable.

Numerical results reveal that diffusion and velocity caused by advection affect the spatial distribution of the predator-prey population but do not affect the coexistence of the predator-prey populations.

Furthermore the effect of diffusion and advection on the population is seen when a variable initial condition is chosen. Moreover, with or without diffusion and advection, the predator-prey population exhibit an asymptotic behavior when the predator-prey populations converge to the stable fixed point, (1, 1) in the long run. It is therefore reasonable to consider a PDE reaction-diffusion advection model since it allows analyzing a range of dynamics in the predator-prey interaction.