Abstract
We study the discrete model for cooperation as expressed through the dynamics of the family of noninvertible planar maps (x, y) → (x exp(r(1 − x) + sy), y exp(r(1 − y) + sx)), with parameters r, s > 0. We prove that the map is proper in the open positive quadrant and describe its various stretching and folding actions. We determine conditions for a Hopf bifurcation — probably one of a cascade of double, quadruple, ... limit cycles, as a curve is followed in parameter space. For r > s an approximating version of the map is dissipative and permanent in the positive quadrant. We include the results of an extensive computer simulation, including a bifurcation diagram (y vs. r, with s fixed) through which is cut a number of x−y phase-plane plots; (an r−y curve penetrates each plot like a thread through cards). These indicate a complex dynamical evolution for cooperation, from stable cycle to strange attractor. A general conclusion is that the benefit of cooperation can be relatively high average values at the cost of oscillations of high amplitude.
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Krawcewicz, W., Rogers, T.D. Perfect harmony: the discrete dynamics of cooperation. J. Math. Biol. 28, 383–410 (1990). https://doi.org/10.1007/BF00178325
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DOI: https://doi.org/10.1007/BF00178325