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

This is a Preprint and has not been peer reviewed. This is version 1 of this Preprint. You must log in to post a comment. There are no comments or no comments have been made public for this article. This is a Preprint and has not been peer reviewed. This is version 1 of this Preprint. Add a Comment You must log in to post a comment. Comments There are no comments or no comments have been made public for this article. 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. https://doi.org/10.32942/X2HH1T Ecology and Evolutionary Biology, Population Biology population dynamics, population ecology, Epidemiology, Diffusion-Reaction Partial Differential Equation Published: 2025-09-24 10:24 Last Updated: 2025-09-24 10:24 CC BY Attribution 4.0 International Conflict of interest statement: There is no conflict of interest to declare. Data and Code Availability Statement: Data will be made available upon request.. Language: English