Intermingled basins in a two species system

Abstract

We present simple examples of (competitive) two species systems with complicated dynamic behaviour. From almost all initial conditions one of the two species dies out. But the survivor is unpredictable: The basins of the two chaotic one-species attractors are everywhere dense and intermingled.

Riddled basins in complex physical and biological systems

“…Moreover, there are also experimental evidences of chaotic behavior in the time evolution of the populations of single species of Tribolium [24]. Hence we may regard the extinction of either species as an asymptotic state with a well-defined attractor in the phase space [25]. The observed extreme sensitivity on the initial condition has led to the hypothesis that the basins of these attractors are riddled [26].…”

“…This proof has been extended to the case 0 < κ < 1 by Hofbauer and collaborators [25]. In the present paper we consider a slightly modified version of Kan's model, by choosing…”

“…The sensitivity observed in the Tribolium sp. experiments suggests that a mathematical model describing the problem should exhibit intermingled basins, since the initial population in the experiments correspond to an initial condition which is unavoidably plagued with some uncertainty, and thus the outcome becomes uncertain, even if the accuracy is very large in determining the initial condition [25,26].…”

“…Hofbauer and coworkers [25] have developed a class of twodimensional models in which we consider two species with populations x 1 (n) and x 2 (n) at (discrete) times n = 0, 1, 2, . .…”

Unequal Crossover Dynamics in Discrete and Continuous Time

“…So, at least locally, the contraction property should be preserved. Nevertheless, we do not expand on this here since it seems possible to use a rather different approach [7], which has been used for similar problems in game theory, to establish a slightly weaker type of convergence result for all 0 < q < 1, and probably even on the larger compact set M + 1,m,C of Lemma 2.…”