Attractors and Spatial Patterns in Hypercycles with Negative Interactions

Abstract

This study reports on the effect of adding negative interaction terms to the hypercycle equation. It is shown that there is a simple parameter condition at which the behaviour of the hypercycle switches from dominant catalysis to dominant suppression. In the suppression!dominated hypercycles the main attractor turns out to be different for cycles consisting of an even or odd number of species. In "odd" cycles there is typically a limit cycle attractor, whereas in "even" cycles there are two alternative stable attractors each containing half of the species. In a spatial domain, odd cycles create spiral waves. Even cycles create a "voting pattern", i.e. initial fluctuations are quickly frozen into patches of the alternative attractors and subsequently, very slowly, small patches will disappear and only one of the two attractors remains. In large cycles (both even and odd) there are additional limit cycle attractors[ In a spatial domain these limit cycles fail to form stable spiral waves, but they can form stable rotating waves around an obstacle. However, these waves are outcompeted by the dominant spatial pattern of the system[ In competition between even and odd cycles, the patches of even cycles are generally stronger than the spiral waves of odd cycles. If the growth parameters of the species vary a little, a patch will no longer contain only half of the species but will instead attract "predator" species from the other patch type. In such a system one of the patch types will slowly disappear and the final dynamics resembles that of a predator-prey system with multiple trophic levels. The conclusion is that adding negative interactions to a hypercycle tends to cause the cycle to break and thereafter the system attains an ecosystem type of dynamics.