Invasions in a Four-Species Cyclically Competing Ecological Community

Abstract: We consider ecological communities consisting of 4 species engaging in cyclic competition. One can visualize the competition scheme by considering the four species, call them u, v, w, and z, as located on a clock, at 12:00 o’clock, 3 o’clock, 6 o’clock and 9 o’clock, respectively, with each species having a competitive advantage over its counterclockwise neighbor, i.e., species v (3 o’clock) wins over species u (12 o’clock) and similarly for the other competition pairs. When the competition is strong such communities are known to be dynamically stable, with two stable alliances formed by species which do not directly compete, i.e., u − w and v − z alliances , the only possible long-time outcomes of the competition. However, in nature the competitive interactions will not be the same for each species, furthermore there can be internal competition (and possibly predation) within each alliance. We consider how to determine which alliance is stronger depending on parameters of the competition. Said another way, which alliance will be able to displace the other alliance (in colloquial terms invade its territory). This problem is a version of a classical mathematical problem that transcends ecology – given two stable states of a system how to determine which state is dominant. This problem can be reduced to analysis of a system of four coupled boundary problems on an infinite interval. This system cannot be solved exactly. We will develop mathematical methods to address this problem. We will primarily employ asymptotic analysis – generally formulating the problem as a singular system, with one or more small parameters, and developing methods to approximate the solution in appropriate parameter regimes. Our analysis will enable us to determine the steady state outcome of the competition scheme, i.e., the surviving alliance.