Travelling wave solutions in a negative nonlinear diffusion-reaction model

Interpreting how nonlinear diffusion affects the fate of bistable populations using a discrete modelling framework

Understanding whether a population will survive or become extinct is a central question in population biology. One way of exploring this question is to study population dynamics using reaction–diffusion equations, where migration is usually represented as a linear diffusion term, and birth–death is represented with a nonlinear source term. While linear diffusion is most commonly employed to study migration, there are several limitations of this approach, such as the inability of linear diffusion-based models to predict a well-defined population front. One way to overcome this is to generalize the constant diffusivity, D, to a nonlinear diffusivity function D(C), where C>0 is the population density. While the choice of D(C) affects long-term survival or extinction of a bistable population, working solely in a continuum framework makes it difficult to understand how the choice of D(C) affects survival or extinction. We address this question by working with a discrete simulation model that is easy to interpret. This approach provides clear insight into how the choice of D(C) either encourages or suppresses population extinction relative to the classical linear diffusion model.

Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation

In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 105 different leapfrog-hopscotch algorithm combinations and narrow this selection down to five during subsequent tests. We demonstrate the performance of these top five methods in the case of large systems with random parameters and discontinuous initial conditions, by comparing them with other methods. We verify the methods by reproducing an analytical solution using a non-equidistant mesh. Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution. The new methods are then applied to the nonlinear Fisher equation. Finally, we analytically prove that the order of accuracy of the methods is two, and present evidence that they are unconditionally stable.