Here we show that a particular one-parameter generalization of the exponential function is suitable to unify most of the popular one-species discrete population dynamics models into a simple formula. A physical interpretation is given to this new introduced parameter in the context of the continuous Richards model, which remains valid for the discrete case. From the discretization of the continuous Richards' model (generalization of the Gompertz and Verhuslt models), one obtains a generalized logistic map and we briefly study its properties. Notice, however that the physical interpretation for the introduced parameter persists valid for the discrete case. Next, we generalize the (scramble competition) θ-Ricker discrete model and analytically calculate the fixed points as well as their stability. In contrast to previous generalizations, from the generalized θ-Ricker model one is able to retrieve either scramble or contest models.
The spatial Prisoner's Dilemma is a prototype model to show the emergence of cooperation in very competitive environments. It considers players, at site of lattices, that can either cooperate or defect when playing the Prisoner's Dilemma with other z players. This model presents a rich phase diagram. Here we consider players in cells of one-dimensional cellular automata. Each player interacts with other z players. This geometry allows us to vary, in a simple manner, the number of neighbors ranging from one up to the lattice size, including self-interaction. This approach has multiple advantages. It is simple to implement numerically and we are able to retrieve all the previous results found in the previously considered lattices, with a faster convergence to stationary values. More remarkable, it permits us to keep track of the spatio-temporal evolution of each player of the automaton. Giving rise to interesting patterns. These patterns allow the interpretation of cooperation/defection clusters as particles, which can be absorbed and collided among themselves. The presented approach represents a new paradigm to study the emergence and maintenance of cooperation in the spatial Prisoner's Dilemma.
The Prisoner's Dilemma (PD) is one of the most popular games of the Game Theory due to the emergence of cooperation among competitive rational players. In this paper, we present the PD played in cells of onedimension cellular automata, where the number of possible neighbors that each cell interacts, z, can vary. This makes possible to retrieve results obtained previously in regular lattices. Exhaustive exploration of the parameters space is presented. We show that the final state of the system is governed mainly by the number of neighbors z and there is a drastic difference if it is even or odd.
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