Feasibility in MacArthur's Consumer-Resource Model

Abstract

Finding the conditions that ensure the survival of species has occupied ecologists for decades. Theoretically, for mechanistic models such as MacArthurs consumer-resource model, most of the efforts have concentrated on proving the stability of an equilibrium assuming that it is feasible, but overlooking the conditions that ensure its feasibility. Here we address this gap by finding the range of conditions that lead to a feasible equilibrium of MacArthurs consumer-resource model and study how changes in the systems structural and parametric properties affect those ranges. We characterize the relationship between the loss of feasibility and the increase in complexity (measured by the systems richness and connectance) by a power law that can be extended to random competition matrices. Focusing on the pool of consumers, we find that while the feasibility of the entire system decreases with the size of the pool, the expected fraction of feasible consumers increases --safety in consumer numbers. Focusing on the pool of resources, we find that if resources grow linearly, the larger the pool of resources, the lower the feasibility of the system and the expected fraction of feasible consumers --danger in resource numbers. However, if resources grow logistically, this pattern is reversed with a sublinear increase in feasibility, as it has been previously reported in experimental work. This work provides testable predictions for consumer-resource systems and is a gateway to exploring feasibility in other mechanistic models.