Jose F. Fontanari, Maurizio Serva - Nonlinear Group Survival in Kimura's Model for the Evolution of...
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Nonlinear group survival in Kimura’s model for the evolution of altruism J F Fontanari 1 and M Serva 2 ‡ 1 Institu to de F ´ ısica de S˜ ao Carlos, Universidade de S˜ ao Paulo, Caixa Postal 369, 13560-970 S˜ao Carlos SP, Brazil 2 Departamento de Biof ´ ısica e F armacologia, Universidade Federal do Rio Grande do Norte , 59072-970 Natal, RN, Brazil Abstract. Esta bl is hin g the co nditions that guar an te e the spreading or the sustenance of altruistic traits in a population is the main goal of intergroup selection models . Of particular interes t is the balanc e of the parame ters associat ed to group size, migration and group survival against the selective advantage of the non-altruistic individuals. Here we use Kimura’s diffusion model of intergroup selection to determine those conditions in the case the group survival probability is a nonlinear non-decreasing func tion of the proportion of altruists in a group . In the case this function is linear, there are two possible steady states which correspond to the non-altruistic and the altru istic phases. At the discontinuous transitio n line separating these phases there is a non-e rgodic coexistenc e phase . F or a continu ous concav e surviv al funct ion, we find an ergodic coexistence phase that occupies a finite region of the parameter space in between the altruistic and the non-altruistic phases, and is separated from these phases by contin uous transition lines. F or a con vex surviv al function, the coexistenc e phase disappears altogether but a bistable phase appears for which the choice of the initial condition determines whether the evolutionary dynamics leads to the altruistic or the non-altruistic steady state. PACS numbers: 87.10.Ed, 87.23.Kg, 87.23.Cc 1. Introductio n The question of the evolution and maintenance of altruism or, as Wilson [1] put it bluntly “the surrender of personal genetic fitness for the enhancement of personal genetic fitness in others”, has been subject of stern dispute since the 1960s (see, e.g., [ 2]). The central point of this debate is the potential of intergroup selection, whose underlying mechanism is the differen tial p opulation (group) exti nction, to counteract indiv idual selectio n. Of particular historical relevance to this matter was Wynne-Edwards’ suggestion that in order to control population growth animals would limit their own fertility for the sake of group survival [3, 4]. In fact, if the extinc tio n of group s occurs at a rate depending on the ir composition, then suc h ext inctio ns wil l, in pri nciple, favor the exi ste nce of ‡ On lea ve of absence from Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universit` a dell’Aquila, Italy a r X i v : 1 3 0 7 . 6 1 7 7 v 1 [ q b i o . P E ] 2 3 J u l 2 0 1 3
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Transcript of Jose F. Fontanari, Maurizio Serva - Nonlinear Group Survival in Kimura's Model for the Evolution of...
7/27/2019 Jose F. Fontanari, Maurizio Serva - Nonlinear Group Survival in Kimura's Model for the Evolution of Altruism (ArXiv, July 2013, 24th)
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7/27/2019 Jose F. Fontanari, Maurizio Serva - Nonlinear Group Survival in Kimura's Model for the Evolution of Altruism (ArXiv, July 2013, 24th)
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7/27/2019 Jose F. Fontanari, Maurizio Serva - Nonlinear Group Survival in Kimura's Model for the Evolution of Altruism (ArXiv, July 2013, 24th)
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7/27/2019 Jose F. Fontanari, Maurizio Serva - Nonlinear Group Survival in Kimura's Model for the Evolution of Altruism (ArXiv, July 2013, 24th)
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