Cyclic Ecological Systems with an Exceptional Species

Abstract

We consider cyclic ecological systems consisting of N species ui, i=1,…,N, where N=3 and N=4. We employ a model in which species ui competes only with members of its own species (intraspecies competition), with a strength normalized to unity, and with species ui+1, where i+1 is taken to mean (imodN)+1 (interspecies competition). The strength of the competition is quantified by an interaction coefficient αi>0.When the interspecies competition is weak for all species (αi<1 for all i), the only stable critical point is a unique coexistence state where all species coexist at nonzero concentrations (one might think that all species in the community form an alliance of the whole). All states involving extinction of at least one species are unstable. For strong competition (αi>1 for all i), coexistence can still be stable under certain circumstances, but the predominant feature is complex dynamics for N=3, involving heteroclinic cycles, spatiotemporal chaos, traveling waves, and modulated traveling waves, and interactions among stable alliances of noncompeting species for N=4. All states other than the coexistence state involving alliances of competing species will be unphysical.We consider the case where one species (say, species j) is either a weak competitor in a community of strong competitors (αj>1, αi<1 for i≠j), or a strong competitor in a community of weak competitors (αj<1, αi>1 for i≠j). We show that in these cases, physical partial alliances of competing states are stable and there is a parameter regime where they represent the only stable states of the system. In all cases, we find that the behavior is dominated by transcritical bifurcations between two states: (i) the coexistence state and (ii) partial alliance states involving competing species. These transcritical bifurcations unify the behavior of three- and four-species systems for weak and strong exceptional species. We consider both homogeneous mixtures (i.e., ODE models) and the displacement of one state by another in both one and two dimensions (spatial dependence - systems of partial differential equations - PDEs). In the latter case, we develop and compare displacement speeds with approximate analytical results and show good agreement between computed and analytic speeds.