Exact solutions for diffusive transport on heterogeneous growing domains

Abstract

From the smallest biological systems to the largest cosmological structures, spatial domains undergo expansion and contraction. Within these growing domains, diffusive transport is a common phenomenon. Mathematical models have been widely employed to investigate diffusive processes on growing domains. However, a standard assumption is that the domain growth is spatially uniform. There are many relevant examples where this is not the case, such as the colonization of growing gut tissue by neural crest cells. As such, it is not straightforward to disentangle the individual roles of heterogeneous growth and diffusive transport. Here we present exact solutions to models of diffusive transport on domains undergoing spatially non-uniform growth. The exact solutions are obtained via a combination of transformation, convolution and superposition techniques. We verify the accuracy of these solutions via comparison with simulations of a corresponding lattice-based random walk. We explore various domain evolution functions, including linear growth, exponential growth and contraction, and oscillatory evolution. Provided the domain size remains positive, we find that the derived solutions are valid. The exact solutions reveal the relationship between model parameters, such as the diffusivity and the type and rate of domain evolution, and key statistics, such as the survival and splitting probabilities.