On the trap of extinction and its elimination

We investigate the dynamics of a system of three interacting populations in presence of extinction and substitution: each population whose number of individuals drops under some threshold value becomes extinct, and it is substituted by another population with different fitness and different coefficients of interaction with the other populations. We study the influence of extinction on the system states, which in the absence of extinction can be fixed points, limit cycles or chaotic attractors of Shilnikov kind. The extinction can destabilize each of these states. We observe two possible kinds of evolution in the destabilized system: (i) it can remain forever in the trap of extinction, i.e., the extinctions and substitutions of populations continue for indefinitely long time or (ii) it can avoid the trap of extinction by means of the substitution, i.e., the fitness and the coefficients of the interactions between the species move the system attractor away from the zone of the threshold values, the extinction stops, and the system settles on a new attractor. The obtained results are discussed from the point of view of products competing for the preference of buyers that can change their opinion.