Simple Case of Evolutionary Discrete Dynamics for Altruism with Reputation

An important issue in biological and social sciences is understanding how behavioural and cultural traits, inherited from previous generations or produced in response to changes in socio-economic or ecological conditions, are spread in populations. Following the formalism of Rice (Evolutionary Theory: Mathematical and Conceptual Foundations. Sinaurer, Sunderland; 2004), we considered a trait for the altruism, which is vertically transmitted from parents to their descendants. We showed a simple model, revisited from Rice (Evolutionary Theory: Mathematical and Conceptual Foundations. Sinaurer, Sunderland; 2004), to study the evolutionary dynamics resulting from the transmission of this trait in the population. The dynamics depends on several parameters: the value of the fitness of the parents, the probability of successful transmission, the possible changes of social and environmental contest and the initial percentage of altruists (initial data of altruism). In this article, the evolutionary dynamics of altruism was investigated using a particular definition of reputation. Taking this definition into account, the evolutionary dynamics for the trait for the altruism were analysed with reference to the parameters. We generally observed a range of values of the initial data of altruism for which there is a decrease-increase (which we called “accordion effect”) of the frequency of the trait to a stable equilibrium value. Furthermore, there is a threshold level of initial data of altruism (depending on the other parameters) beyond which there is an explosion of the propagation of altruism. To sum up, our model adds knowledge to those scenarios in which the presence of reputation plays an important role to the fixation of a stable equilibrium of the frequency of the trait for the altruism. Further and more sophisticated models are needed to investigate our results.