Species Invasion in a Two-Dimensional Space with Irregularly Shaped Patches

Accounting for spatial heterogeneity in the evolution of a species population in a given space is of much importance in population ecology, epidemiology and related fields in biosciences. Past literature has presented such analysis in the presence of regions with distinct diffusion/growth properties, often referred to as patches. However, most of the past work is limited to one-dimensional space, whereas in practice, population evolution occurs in two dimensions, and realistic patches may have irregular shapes. This work addresses this limitation by deriving an exact analytical solution for a linear diffusion-reaction population growth problem in two-dimensional space containing an arbitrary number of irregularly shaped patches. The spatial variation in diffusion/growth coefficients is represented using Heaviside functions, and an exact expression for the transient coefficient functions in the series solution is derived. A threshold condition for establishment of the population at large time is derived. Results from this work are shown to reduce to well-known results for simpler problems under limiting conditions. Based on the technique, extinction and establishment regions in the parameter space are identified. A number of illustrative problems containing patches of irregular shapes, such as heart-shaped and leaf-shaped patches are solved in order to demonstrate the versatility of the technique. This work contributes a novel mathematical tool for solving population dynamics problems in realistic conditions, including irregular patch shapes, with potential applications in a number of problems in ecology and epidemiology.