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Lotka-volterra systems and polymatrix replicators
Publication . Peixe, Telmo Jorge Lucas, 1975-; Duarte, Pedro Miguel Nunes da Rosa Dias
In the 1970’s John M. Smith and George R. Price [22] applied the theory of strategic games developed by John von Neumann and Oskar Morgenstern [42] in the 1940’s to investigate the dynamical processes of biological populations, giving rise to the field of the Evolutionary Game Theory (EGT). Some classes of ordinary differential equations (o.d.e.s) which plays a central role in EGT are the Lotka-Volterra systems (LV), the replicator equation, the bimatrix replicator and the polymatrix replicator. Many properties of the LV systems can be geometrically expressed in terms of its associated graph, constructed from the system’s interaction matrix. For the class of stably dissipative LV systems we prove that the rank of its defining matrix, which is the dimension of the associated invariant foliation, is completely determined by the system’s graph. In this thesis we also study analytic flows defined on polytopes. We present a theory that allows us to analyze the asymptotic dynamics of the flow along the heteroclinic network composed by the flowing-edges and the vertices of the polytope where the flow is defined. In this context, given a flow defined on a polytope, we give sufficient conditions for the existence of normally hyperbolic stable and unstable manifolds for heteroclinic cycles. In polymatrix games population is divided in a finite number of groups, each one with a finite number of strategies. Interactions between individuals of any two groups are allowed, including the same group. The differential equation associated to a polymatrix game, that we designate as polymatrix replicator, is defined in a polytope given by a finite product of simplices. Karl Sigmund and Josef Hofbauer [16] and Wolfgang Jansen [18] give sufficient conditions for permanence in the usual replicators. We generalize these results for polymatrix replicators. Also for polymatrix replicators we extend the concept of stably dissipativeness developed by Ray Redheffer et al. [25–29]. In this context we generalize a theorem of Waldyr Oliva et al. [6] about the Hamiltonian nature of the limit dynamics in “stably dissipative” polymatrix replicators. We present also some examples to illustrate fundamental results and concepts developed along the thesis.
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Fundação para a Ciência e a Tecnologia
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SFRH
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SFRH/BD/72755/2010