Numerical Methods for Moving-Habitat Models in One and Two Spatial Dimensions

Abstract

Temperature isoclines are shifting with global warming. To survive, species with thermal niches must shift their geographical ranges to stay within the bounds of their suitable habitat, or acclimate to a new environment. Mathematical models that study range shifts are called moving-habitat models. The literature is rich and includes modelling with reaction-diffusion equations. Much of this literature represents space by the real line, with a handful studying 2-dimensional domains that are unbounded in at least one direction. The suitable habitat is represented by the set over which the demographics (reaction term) has a positive net growth rate. In some cases, this is a bounded set, in others, it is not. The unsuitable habitat is represented by the set over which the net growth rate is negative. The environmental shift is captured by an imposed shift of the suitable habitat. Individuals respond to their environment via their movement behaviour and many display habitat-dependent dispersal rates and a habitat bias. Such behaviour corresponds to a jump in density across the interface of suitable and unsuitable habitat. The questions motivating moving-habitat models are: when can a species track its shifting habitat and what is the impact of an environmental shift on a persisting species. Without closed form solutions, researchers rely on numerical methods to study the latter, and depending on the movement of the interface, the former may require numerical tools as well. We construct and analyse two numerical methods, a finite difference (FD) scheme and a finite element (FE) method in 1- and 2-dimensional space, respectively. The FD scheme can capture arbitrary movement of the boundary, and the FE method rather general shapes for the suitable habitat. The difficulty arises in capturing the jump across a shifting interface. We construct a reference frame in which the interfaces are fixed in time. We capture the jump in density with a clever placing of the nodes in the FD scheme, and through a Lagrange multiplier in the FE method. With biological applications, we demonstrate the power of our solvers in advancing research for moving-habitat models.