Domain decomposition methods for a class of spatially heterogeneous delayed reaction–diffusion equations

Abstract

We derive and study a class of delayed reaction–diffusion equations with spatial heterogeneity, which models the population of a single species with different habitats for mature and immature individuals. We introduce new solid cones, obtain spectral bounds of several spatial heterogeneous operators, and establish limiting non-negativeness property for the whole space and the eventual comparison principle for bounded domains. As a result, we develop new domain decomposition methods so that one can compare solutions with those to associated equations from a suitable bounded spatial domain to the whole space. Then by employing domain decomposition methods and dynamical system approaches, we obtain threshold results under the supremum norm. These results are greatly different from the existing ones of other evolution equations in unbounded domains or the whole space. The main results are applied to two examples with the Ricker birth function and with the Mackey–Glass birth function. It reveals that the size of the immature habitat can affect the reproduction and spread of the population.