Dynamics of a consumer–resource reaction–diffusion model: Homogeneous versus heterogeneous environments

We study the dynamics of a consumer–resource reaction–diffusion model, proposed recently by Zhang et al. (Ecol Lett 20(9):1118–1128, 2017), in both homogeneous and heterogeneous environments. For homogeneous environments we establish the global stability of constant steady states. For heterogeneous environments we study the existence and stability of positive steady states and the persistence of time-dependent solutions. Our results illustrate that for heterogeneous environments there are some parameter regions in which the resources are only partially limited in space, a unique feature which does not occur in homogeneous environments. Such difference between homogeneous and heterogeneous environments seems to be closely connected with a recent finding by Zhang et al. (2017), which says that in consumer–resource models, homogeneously distributed resources could support higher population abundance than heterogeneously distributed resources. This is opposite to the prediction by Lou (J Differ Equ 223(2):400–426, 2006. https://doi.org/10.1016/j.jde.2005.05.010) for logistic-type models. For both small and high yield rates, we also show that when a consumer exists in a region with a heterogeneously distributed input of exploitable renewed limiting resources, the total population abundance at equilibrium can reach a greater abundance when it diffuses than when it does not. In contrast, such phenomenon may fail for intermediate yield rates.