Replicator Equations, Response Functions And Entropy Measures In Science: Mathematical Background
Free (open access)
Y. A. Pykh & I. G. Malkina-Pykh
Replicator dynamics is an evolutionary strategy well established in different disciplines of biological sciences. It describes the evolution of self-reproducing entities called replicators in various independent models of, e.g., genetics, ecology, prebiotic evolution, and sociobiology. Besides this, replicator selection has been applied to problem solving in combinatorial optimization and to learning in neural networks and also in fluid mechanics, game and laser theory. So, the replicator systems arise in an extraordinary variety of modeling situations. In this report we’ll consider the new class of generalized replicator equations with nonlinear response functions and construct the Energy Lyapunov function and entropy measures for this system. Keywords: replicator equations, Lyapunov functions, entropy, distance measure. 1 An overview and introduction A replicator is a fundamental unit in evolutionary processes, representing a population type, and characterized by two attributes: ()i p t, its proportion in the population at time t , and its non-negative fitness at time t . A replicator's fitness is a measure of its significance to the future evolution of the population. The proportions of replicators in a population change as a result of their mutual interactions, and their relative fitnesses. The founding fathers of evolutionary genetics, Fisher, Haldane and Wright used mathematical models to generate a synthesis between Mendelian genetics and Darwinian evolution. Kimura’s theory of neutral evolution, Hamilton’s kin
replicator equations, Lyapunov functions, entropy, distance measure.