Perfect harmony: the discrete dynamics of cooperation |
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Authors: | W Krawcewicz T D Rogers |
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Institution: | (1) Department of Mathematics, University of Alberta, T6G 2G1 Edmonton, Alberta, Canada |
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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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Keywords: | Discrete dynamical systems Strange attractor Lotka-Volterra cooperation Permanence Hopf bifurcation |
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