Adaptive dynamics in logistic branching populations

The recent biological theory of adaptive dynamics [6,9] proposes a description of the long term evolution of an asexual population by putting emphasis on the ecological interactions between individuals, in contrast with classical population genetics models which focus on the genetic structure of the population. The basic models are individual-based models in which the population dynamics is precisely described and includes birth, death, competition and mutation. The basic idea of the theory of adaptive dynamics is to try to get insights about the interplay between ecology and evolution by studying the invasion of a single mutant type appearing in a simplified (monotype stable) resident population. The evolution of the population can then be described as a sequence of mutant invasions in the population. If the resident type goes extinct when the mutant type invades (we say that the mutant type fixates), the evolution is described by the so-called ‘trait substitution sequence’ (TSS) [10]. This appraoch has revealed powerful to predict the qualitative behaviour of complicated evolutionary dynamics. In particular, it allows to determine the (local) direction of evolution in the space of phenotypic traits (or simply traits) from the individual ecological