On Rare Mutation, Chaos and Darwin’s Theory

According to the Holy Quran, the evolution of life is the result of divine will and divine guidance. One of the key elements of natural selection is that offsprings sometimes vary from their parents. For instance, a yellow butterfly may have a black offspring. This kind of variation and mutation, though rare, tends to occur in all species. Over a long period of time, the cumulative effect of such mutations leads to an entirely new species.Evolution describes the change in genetic material of a population of organisms through successive generations. In this paper, we consider the evolution operator, namely the quadratic stochastic operator which naturally occurs to describe the transmission of a trait from parents to their offsprings. This operator describes evolutionary dynamics of the considered population and its long run behaviour plays an important role in many applied problems. We consider Mendelian inheritance for a single gene with three alleles and assume that prior to the formation of the new generation each gene has the possibility to mutate, that is, to change into a gene of the other kind. We then show that the rare mutation can transform a regular dynamical system into a chaotic one. As a corollary, we have that the evolutionary operator, with rare mutation of alleles, contradict Darwin’s evolution theory on genetic level.