Fractured alliances in a four-species cyclic ecological system

Abstract

We consider two Lotka–Volterra type models for ecological communities exhibiting a modified form of cyclic competition. The first governs a four-species ecological system. When each species competes only with one other species cyclically, then, it is known that, for strong competition the system evolves to one of two alliances of non-competing species.

We consider the case when there is internal competition and predation within one of the alliances. This leads to an embedded three-species rock–paper–scissors ($RPS$) community, which can be dynamically unstable — leading to attracting heteroclinic cycles within the four-species system. We show that, even for vanishingly small fracturing, other outcomes are also possible. We identify all possible physical steady states, their stabilities and bifurcations which can occur between them (eight bifurcations in all).

For our second model, we consider the case of two ecological communities within a heterogeneous environment. The communities are coupled, allowing information exchange between them. We prove that for strong coupling the two communities will form a unified state corresponding to one with averaged ecological parameters. For the case of dynamically unstable communities (i.e., heteroclinic cycles), we develop a method to characterize the averaged heteroclinic cycle based on the rate of expansion of trajectory visits to the appropriate saddle. Information exchange can allow small ecological heterogeneities to lead to very major changes in the steady state. In particular, information exchange can quench heteroclinic cycles within both communities or conversely can allow for heteroclinic cycles where none would occur for each community in isolation.