Attractors and Spatial Patterns in Hypercycles with Negative Interactions

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

Equal Pay for all Prisoners / The Logic of Contrition

This report deals with two questions concerning the emergence of cooperative strategies in repeated games. The first part is concerned with the Perfect Folk Theorem and presents a vast class of equilibrium solutions based on Markovian strategies. Simple strategies, called equalizers, are introduced and discussed: if players adopt such strategies, the same payoff results for every opponent. The second part analyzes strategies implemented by finite automata. Such strategies are relevant in an evolutionary context; an important instance is called Contrite Tit For Tat. In populations of players adopting such strategies, Contrite Tit For Tat survives very well -- at least as long as errors are restricted to mistakes in implementation ("the trembling hand"). However, this cooperative strategy cannot persist if mistakes in perception are included as well

Phase transition and selection in a four-species cyclic Lotka-Volterra model

We study a four species ecological system with cyclic dominance whose individuals are distributed on a square lattice. Randomly chosen individuals migrate to one of the neighboring sites if it is empty or invade this site if occupied by their prey. The cyclic dominance maintains the coexistence of all the four species if the concentration of vacant sites is lower than a threshold value. Above the treshold, a symmetry breaking ordering occurs via growing domains containing only two neutral species inside. These two neutral species can protect each other from the external invaders (predators) and extend their common territory. According to our Monte Carlo simulations the observed phase transition is equivalent to those found in spreading models with two equivalent absorbing states although the present model has continuous sets of absorbing states with different portions of the two neutral species. The selection mechanism yielding symmetric phases is related to the domain growth process whith wide boundaries where the four species coexist.Comment: 4 pages, 5 figure

Cellular Automaton Modeling of Pattern Formation

Book review Andreas Deutsch and Sabine Dormann, Cellular Automaton Modeling of Biological Pattern Formation, Characterization, Applications, and Analysis, Birkhäuser (2005) ISBN 0-8176-4281-1 331pp.