On a discrete-time model with replicator dynamics in renewable resource exploitation

We consider a discrete-time version of the model proposed by Lamantia and Radi [15 F. Lamantia, and D. Radi, Exploitation of renewable resources with differentiated technologies: an evolutionary analysis, Math. Comput. Simulation 108 (2015), pp. 155–174. doi:10.1016/j.matcom.2013.09.013.[Crossref], [Web of Science ®] , [Google Scholar]] to describe a fishery where a population regulated by a logistic growth function is exploited by a pool of agents that can choose, at each time period, between two different harvesting strategies according to a profit-driven evolutionary selection rule. The resulting discrete dynamical system, represented by a two-dimensional nonlinear map, is characterized by the presence of invariant lines on which the dynamics are governed by one-dimensional restrictions that represent pure, i.e. adopted by all players, strategies. However, interesting dynamics related to interior attractors, where players playing both strategies coexist, are evidenced by analytical as well as numerical methods that reveal local and global bifurcations.