Cultural and Biological Transmission: A Simple Case of Evolutionary Discrete Dynamics

A remarkable issue in biological and social sciences is understanding how behavioral and cultural traits, inherited from previous generations or produced in response to changes in socioeconomic or ecological conditions, are spread in populations. Transmissions of behavioral traits have been observed in animals like reef fish (Helfman and Schultz, Anim Behav 32 (2):379–384, 1984) and birds (Fisher and Hinde, British Birds 42:347–357, 1949) and also between parents and their offspring (Cavalli-Sforza and Feldman, Cultural Trasmission and Ecolution: A Quantitative Approach, Princeton University Press, Princeton, 1981). Following the formalism of Rice (Rice, Evolutionary Theory: Mathematical and Conceptual Foundations. Sinaurer, 2004), we considered a cultural (behavioral) trait, which is vertically transmitted from parents to their descendants. We showed a simple model, revisited from Rice (Rice, Evolutionary Theory: Mathematical and Conceptual Foundations. Sinaurer, 2004), to study the evolutionary dynamics resulting from the transmission of this trait in the population. The dynamics depends on several parameters. In particular, the value of the fitness of the parents (previous generation), the probability of successful transmission, and the possible changes of social and environmental contest were taken into account. We generally observed the fixation of the frequency of the trait to a stable equilibrium value. Nevertheless, for some particular values of the parameters, the dynamics produced a stable limit cycle or chaotic behavior of the frequency of the trait. Furthermore, we showed that when the equilibrium of the frequency is achieved, the mean population fitness did not necessary reach a local maximum of the fitness. Overall, our model opens to the scenarios in which the best possible value of the fitness was not reached by means of the equilibrium of the frequency of the trait. Applications of our model in different biological contests are possible. Further and more sophisticated models are needed to investigate our results.