Altruism in populations at the extinction transition

Abstract

We study the evolution of cooperation as a birth-death process in spatially extended populations. The benefit from the altruistic behavior of a cooperator is implemented by decreasing the death rate of its direct neighbours. The cost of cooperation is the increase of a cooperator's death rate proportional to the number of its neighbors. For any benefit-cost ratio above 1, the stable stationary concentrations of agents pass through four regimes as the baseline death rate p increases: (i) defection only, (ii) coexistence, (iii) cooperation only, (iv) extinction. We investigate the transitions between these regimes. Qualitatively identical phase diagrams are obtained for populations on square lattices and in pair approximation. Spatial correlations with nearest neighbours only are thus sufficient for sustained cooperation. arXiv:1906.04202v1 [q-bio.PE]

Deterministic and stochastic cooperation transitions in evolutionary games on networks

“…with a rate π − σ . Note that the dynamics can be seen as a birth-death process, hence suitable for being analyzed as in previous works [12,49]. Taking into account the steps (ii) and (iii) of the evolution given in the previous section, the rates can be written as…”

“…One of the mechanisms known to favor cooperation is the reciprocity induced by the spatial distribution of the players as shown by Nowak and May [8]. When interactions are no longer well-mixed and players are distributed in a spatial/topological structure, cooperators can cluster together and might survive surrounded by defectors, changing the mean-field equilibrium panorama of many games [9][10][11][12].…”

Structured interactions as a stabilizing mechanism for competitive ecological communities

“…Let M (s) be the matrix of time-dependent coefficients of the system (A17). A necessary, but not sufficient, condition for the homogeneous solutions x i (s) to be unstable is that some eigenvalue of M (s) has positive real part for some time s ∈ [0, T ] (see a proof of a similar result in [31]). During these times, even if the trajectory turns out to be linearly stable, its stability is reduced and more susceptible to non-infinitesimal perturbations or noise.…”

“…This state evolves as follows: (i) with a rate r, a randomly chosen individual (say, located at ν) dies, then (ii) two neighbors of the dead individual (thus pertaining to the set P ν of neighbors of ν) are chosen at random and compete to generate the offspring: a winner species is selected according to the probabilities in the dominance matrix H. And (iii) this offspring is immediately located at the vacant node. Following standard procedures (for example see [30,31]) the master equation for the probability p(S, t) of finding the system in a state S at time t can be written as…”