Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics

Abstract

Abstract. The measure dynamics approach to modelling single-species coevolution with a one-dimensional trait space is developed and compared to more traditional methods of adaptive dynamics and the Maximum Principle. It is assumed that individual fitness results from pairwise interactions together with a background fitness that depends only on total population size. When fitness functions are quadratic in the real variables parameterizing the one-dimensional traits of interacting individuals, the following results are derived. It is shown that among monomorphisms (i.e. measures supported on a single trait value), the CSS (Continuously Stable Strategy) characterize those that are Lyapunov stable and attract all initial measures supported in an interval containing this trait value. In the cases where adap- *

Adaptive dynamics for physiologically structured population models

“…How quasi-monomorphicity (and by extension, quasi-N -morphicity) is maintained under some not-too-restrictive assumptions is being addressed more recently (e.g. Cressman and Hofbauer 2005;Meszéna et al 2005). More specifically, the latter authors show for general ODE population models with small differences between the types, that away from evolutionary singularities the dynamics of the relative frequencies p i follows (up to the lowest order of approximation) the familiar population genetics equations for the density independent case.…”

Stability of the replicator equation for a single species with a multi-dimensional continuous trait space

“…If these terms are of the form o(α 2 + β 2 ) (i.e. they go to zero faster than α 2 + β 2 ), they will not affect the instability of m = 0 under (13).…”

“…to a Dirac delta distribution in measure theoretic terminology). In fact, Cressman and Hofbauer (2005) have shown the relevance of the CSS concept (and the closely related concept of a neighborhood invader strategy (NIS) of Apaloo (1997)) for stability of monomorphisms in the measure dynamics of a one-dimensional continuous trait space. Specifically, a non CSS monomorphism is unstable in the measure dynamic and, conversely, a CSS is dynamically stable if the initial distribution of individual behaviors is close to the CSS and satisfies an additional technical requirement concerning the strategies present in the population (i.e.…”

“…m = 0 is called convergence stable (with respect to C(m)) if it is asymptotically stable under (13).…”

Stability and instability of the unbeatable strategy in dynamic processes

“…We consider a slightly stronger concept than the unbeatable strategy: x * ∈ is a GSS if x * · Ax > x · Ax for any x = x * (Cressman and Hofbauer 2005). 4 Proposition 1 (iv) implies that any GSS is unbeatable.…”