Numerical solution of the spatial-temporal model of population distribution in heterogeneous domain

In this work we consider temporal-spatial models of one species population distribution in heterogeneous domain. Mathematical models of such problems are described by a nonlinear parabolic equation in the two- dimensional domain D: 𝜕𝑢/𝜕t −∇ (𝜀 (x) ∇ u) = 𝑟(x) (1 − 𝑢) 𝑢, x ∈ 𝐷, where u is the function of 2D coordinates x and time t, r is the rate of reproduction and is the diffusion. This equation has certain initial and boundary conditions pending on the specifics of the problem. Here we have used a finite element method to solve this problem in the complex geometry with heterogeneous inclusions and implicit-explicit time approximation. Numerical implementation is performed by using python and finite-element library FEniCS. We have done numerical investigation of the influence of the geometry and heterogeneous properties on the solution.