Ol ai Q
(Dieckmann 1994; Dieckmann et al. 1995; Dieckmann and Law 1996). Here S again denotes Dirac's delta function, and si(x'i,x) = max(0, fi(x'i,x))/bi(x'i,x) is the probability with which the mutant xi survives accidental extinction through demographic stochasticity while still being rare in the large population of resident individuals (e.g., Athreya and Ney 1972). If also the resident population is small enough to be subject to accidental extinction, si(x'i,x) = (1 - e-2 fi(x'i,x))/(1 - e-2fi(x'i,x)Tli(x)) with fi(x'i,x) = fi(x'i,x))/[bi(x'i,x) + di(x'i,x)] provides a more accurate approximation (e.g., Crow and Kimura 1970). The resulting evolutionary random walk models are again typically implemented using Gillespie's minimal process method (Dieckmann 1994).
Gradient-ascent models: monomorphic and deterministic. If mutation steps are not only rare but also small, the dynamics of evolutionary random walks are well approximated by smooth trajectories, as shown in Fig. 8.1c. These trajectories represent the evolutionary random walk's expected path and can be approximated by the canonical equation of adaptive dynamics (Dieckmann 1994; Dieckmann et al. 1995; Dieckmann and Law 1996), which, in its simplest form, is given by dt for i = 1,.. .,N, where d Xi = 1 ni(xi)ui(x)a2(xi)gi(x)
) J^ (xi xi) (xi xi) Mi(xi,xi)dxi is the variance-covariance matrix of the symmetric mutation distribution M2 around trait value xi. Implementations of this third class of models typically rely on simple Euler integration or on the fourth-order Runge-Kutta method (e.g., Press et al. 1992).
Reaction-diffusion models: polymorphic and stochastic. In large populations characterized by high mutation rates, stochastic elements in the dynamics of the phenotypic distributions become negligible. This enables formal descriptions of reaction-diffusion type (e. g., Kimura 1965; Bürger 1998). Specifically, the reaction-diffusion approximation of the birth-death-mutation process described above is given by d 1 d2 dtpi(xi) = fi(xi ,p)pi(xi) + 2 ai(xi) * dx Vi(xi )bi(xi ,p)pi(xi)
for i = 1,.. .,N, where a2(xi) is the variance-covariance matrix of the symmetric and homogeneous mutation distribution M2, and where * denotes the elementwise multiplication of two matrices followed by summation over all resultant matrix elements. An illustration of reaction-diffusion dynamics is shown in Fig. 8.1d. Models of this fourth class are best implemented using so-called implicit integration methods (e. g., Crank 1975). It ought to be highlighted, however, that the infinitely extended tails that the distributions p2 instantaneously acquire in this framework can give rise to artifactual dynamics that offer no good match to the actual dynamics of the underling birth-death-mutation processes in finite populations. The derivation of finite-size corrections to the traditional reaction-diffusion limit overcomes these shortcomings (Dieckmann, unpublished).
At the expense of ignoring genetic intricacies, models of adaptive dynamics are geared to analyzing the evolutionary implications of complex ecological settings. In particular, such models can be used to study all types of density-and frequency-dependent selection, and are equally well geared to describing single-species evolution and multi-species coevolution. As explained above, the four model classes specified in this section are part of a single conceptual and mathematical framework, which implies that switching back and forth between alternative descriptions of any evolutionary dynamics driven by births, deaths, and mutations - as mandated by particular problems in evolutionary ecology - will be entirely straightforward.
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