  We propose a numerical approach to study the invasion fitness of a mutant and
to determine evolutionary singular strategies in evolutionary structured models
in which the competitive exclusion principle holds. Our approach is based on a
dual representation, which consists of the modelling of the small size mutant
population by a stochastic model and the computation of its corresponding
deterministic model. The use of the deterministic model greatly facilitates the
numerical determination of the feasibility of invasion as well as the
convergence-stability of the evolutionary singular strategy. Our approach
combines standard adaptive dynamics with the link between the mutant survival
criterion in the stochastic model and the sign of the eigenvalue in the
corresponding deterministic model. We present our method in the context of a
mass-structured individual-based chemostat model. We exploit a previously
derived mathematical relationship between stochastic and deterministic
representations of the mutant population in the chemostat model to derive a
general numerical method for analyzing the invasion fitness in the stochastic
models. Our method can be applied to the broad class of evolutionary models for
which a link between the stochastic and deterministic invasion fitnesses can be
established.
