The dynamics of adaptation an illuminating example and a Hamilton Jacobi approach

The dynamics of adaptation : an illuminating example and a Hamilton-Jacobi approach Odo Diekmann1, Pierre-Emanuel Jabin2, Stephane Mischler 3 and Benoıt Perthame2 May 19, 2004 Abstract Our starting point is a selection-mutation equation describing the adaptive dynamics of a quan- titative trait under the influence of an ecological feedback loop. Based on the assumption of small (but frequent) mutations we employ asymptotic analysis to derive a Hamilton-Jacobi equation. Well-established and powerful numerical tools for solving the Hamilton-Jacobi equations then al- low us to easily compute the evolution of the trait in a monomorphic population. By adapting the numerical method we can, at the expense of a significantly increased computing time, also capture the branching event in which a monomorphic population turns dimorphic and subsequently follow the evolution of the two traits in the dimorphic population. From the beginning we concentrate on a caricatural yet interesting model for competition for two resources. This provides the perhaps simplest example of branghing and has the great advantage that it can be analysed and understood in detail. Contents 1 Introduction 2 2 Competition for two resources 3 3 The selection-mutation equation and its Hamilton-Jacobi limit 5 4 Trait substitutions, singular points and branching 7 4.1 Invasibility . . . . . . . . . . . . . . . . . .

• mutation


• jacobi equation


• unique steady


• trait


• ds2 dt


• s02 ?


• equation does


• direct numerical