Compactly Supported Solutions of Reaction–Diffusion Models of Biological Spread

Abstract

Lie group analysis is one of the most useful techniques for analyzing the analytic structure of the solutions of differential equations. Here, reaction–diffusion (RD) modelling of biological invasion is used to illustrate this fact in terms of identifying the conditions that the diffusion and reaction terms must satisfy for their solutions to have compact support. Biological invasion, such as the spread of viruses on the leaves of plants and the invasive spread of animals and weeds into new environments, has a well-defined progressing compactly supported spatial \(\mathbb {R}^2\) structure. There are two distinct ways in which such progressing compact structure can be modelled mathematically; namely, cellular automata modelling and reaction–diffusion (RD) equation modelling. The goal in this paper is to review the extensive literature on RD equations to investigate the extent to which RD equations are known to have compactly supported solutions. Though the existence of compactly supported solutions of nonlinear diffusion equations, without reaction, is well documented, the conditions that the reaction terms should satisfy in conjunction with such nonlinear diffusion equations, for the compact support to be retained, has not been examined in specific detail. A possible partial connection relates to the results of Arrigo, Hill, Goard and Broadbridge, who examined, under various symmetry analysis assumptions, situations where the diffusion and reaction terms are connected by explicit relationships. However, it was not investigated whether the reaction terms generated by these relationships are such that the compact support of the solutions is maintained. Here, results from a computational analysis for the addition of different reaction terms to power law diffusion are presented and discussed. It appears that whether or not the reaction term is zero, as a function of its argument at zero, is an important consideration. In addition, it is confirmed algebraically and graphically that the shapes of compactly supported solutions are strongly controlled by the choice of the reaction term.