Total population for a resource-limited single consumer model

In the past several decades, much attention has been focused on the effects of dispersal on total populations of species. In Zhang (EL 20:1118–1128, 2017), a rigorous biological experiment was performed to confirm the mathematical conclusion: Dispersal tends to enhance populations under a suitable hypothesis. In addition, mathematical models keeping track of resource dynamics in population growth were also proposed in Zhang (EL 20:1118–1128, 2017) to understand this remarkable phenomenon. In these models, the self-regulated quantity “loss rate" of the population seems, in general, difficult to measure experimentally. Our main goal in this paper is to study the effects of relations between the loss rate and the resources, the role of dispersal, and the impact of their interactions on total populations. We compare the total population for small and large diffusion under various correlations between loss rate and the resources. Biological evidence seems to support some specific correlations between the loss rate and the resources.