Stability of difference equations with interspecific density dependence, competition, and maturation delays

by Geoffrey R. Hosack, Maud El-Hachem and Nicholas J. Beeton

The stability properties of delayed discrete-time model of interspecific competition are examined using directed graphs (figure). For a competitive multi-species model that extends the Beverton-Holt model, return towards the coexistence equilibrium after a local perturbation is guaranteed if intraspecific competition is stronger than interspecific competition. The rate of return depends on the species composition. This property is used to predict an optimised configuration of interspecific competition and rate of return for a system of morphologically indistinguishable species: Although direct observation of the species abundances in the field is not possible, available genotyping methods provide information on species composition. 

Directed graph for three species that connects the nodes of age classes zero (immature recruits) to mature adult stage individuals of age class delta with arcs: survival transitions are shown by red arrows, density dependent recruitment by black arrows, and interspecific interactions by blue arrows. These arcs form cycles that in turn provide sufficient conditions for the stable coexistence of species.