Classification of global dynamics of a periodic diffusive consumer-resource model

Abstract

In [40], a biological experiment with yeast was successfully designed to verify a seemingly counterintuitive mathematical phenomenon: in a heterogeneous environment, dispersal can allow a species’ total population to exceed its carrying capacity. A key finding of [40] highlights the critical role of resource dynamics in shaping population dynamics. To shed light on this striking biological insight, a novel consumer-resource model was proposed in [40]. This paper aims to analyze this consumer-resource system, which incorporates resource decay and consumer loss, under conditions where the resource input rate may exhibit spatial heterogeneity and temporal periodicity. We characterize the persistence or extinction of both consumer and resource populations based on their dispersal rates and a relaxation-time parameter. Additionally, we derive the asymptotic profiles of positive periodic solutions as the resource dispersal rate becomes sufficiently small or large. Our results reveal several notable insights: (a) resource decay, even when slight, acts as a decisive factor preventing unlimited growth of resource abundance; (b) the consumer mortality rate is a key determinant of the consumer population’s persistence or extinction; (c) when the consumer mortality rate is moderate, a temporally homogeneous resource input may be more advantageous for the consumer population than a temporally periodic one. We employ a variety of methods to establish our results, including the parabolic comparison principle, the upper-lower solution method for mixed quasi-monotone systems, the theory of asymptotically periodic systems, the uniform persistence theory for infinite-dimensional dynamical systems, and the principal eigenvalue theory.