The theory of adaptive dynamics derives from considering ecological interactions and phenotypic variation at the level of individuals. Extending classical birth and death processes through mutation, adaptive dynamics models keep track, across time, of the phenotypic composition of populations in which trait values of offspring are allowed to differ from those of their parents.

Throughout this chapter we will adhere to the following notation. Time is denoted by t. The number of species in the considered community is N. The values of quantitative traits in species i are denoted by Xi, be they univariate or multivariate. The abundance of individuals with trait value xi is denoted by ni(xi), while ni denotes the total abundance of individuals in species i. If species i harbors individuals with mi distinct trait values xik, its phenotypic density is given by pi(xi) = i ni(xik)6(xi — xik), where 6 denotes Dirac's delta function. A species with mi = 1 is said to be monomor-phic. For small mi, species i may be characterized as being oligomorphic, when mi is large, it will be called polymorphic. The community's pheno-

typic composition is described by p = (pi,... ,pn). The per capita birth and death rates of individuals with trait value xi in a community with phenotypic composition p are denoted by bi(x'i,p) and di(x'i,p). Reproduction is clonal, mutant individuals arise with probabilities pi(xi) per birth event, and their trait values xi are drawn from distributions Mi(x'i,xi) around parental trait values xi.

If all species in the community are monomorphic, with resident trait values x = (xi,...,xn), and if their ecological dynamics attain an equilibrium attractor, with resident abundances ni(x), the resultant phenotypic composition is denoted by p(x). The per capita birth, death, and growth rates of individuals with trait value xi will then be given by bi(x'i,x) = bi(x'i,p(x)), di(xi,x) = di(x'i,p(x)), and fi(x'i,x) = bi(x'i,x) — di(x'i,x), respectively. In adaptive dynamics theory, the latter quantity is called invasion fitness. For a mutant xi to have a chance of invading a resident community x, its invasion fitness needs to be positive. The notion of invasion fitness fi(x'i,x) makes explicit that the fitness fi of individuals with trait values xi can only be evaluated relative to the environment in which they live, which, in the presence of density- and frequency-dependent selection, depends on x. Invasion fitness can be calculated also for more complicated ecological scenarios, for example, when species exhibit physiological population structure, when they experience non-equilibrium ecological dynamics, or when they are exposed to fluctuating environments (Metz et al. 1992). If a community's ecological dynamics possess several coexisting attractors, invasion fitness will be multivalued. While strictly monomorphic populations will seldom be found in nature, it turns out that the dynamics of polymorphic populations can often be well approximated and understood in terms of the simpler monomorphic cases. For univariate traits, depicting the sign structure of invasion fitness results in so-called pairwise invasibility plots (Matsuda 1985; van Tienderen and de Jong 1986, Metz et al. 1992, 1996; Kisdi and Meszena 1993; Geritz et al. 1997).

Derivatives of invasion fitness help to understand the course and outcome of evolution. The selection pressure gi,(x) = dx.fi(x'i,x)\xi=x>. acting on trait value xi is given by the local slope of the fitness landscape fi(xi,x) at xi = xi. When xi is multivariate, this derivative is a gradient vector. Selection pressures in multi-species communities are characterized by g(x) = (gi(xi),... ,gN(xn)). Trait values x* at which this selection gradient vanishes, g(x*) = 0, are called evolutionarily singular (Metz et al. 1992). Also the signs of the second derivatives of invasion fitness at evolutionarily singular trait values reveal important information. When the mutant Hessian hmm,i(x*) = dX1 fi(xi,x)\X>.=x*,x=x* is negative definite, x* is at a fitness maximum, implying (local) evolutionary stability. When hmm^i(x* ) — hrr^i(x*) is negative definite, where hrr,i(x*) = d? fx ,x)\x.=x*,x=x* denotes the resident Hessian, subsequent invasion steps in the vicinity of xi* will approach xi* , implying (strong) convergence stability.

Based on these considerations, four classes of models are used to investigate the adaptive dynamics of ecological communities at different levels of resolution and generality. Details concerning the derivations of these models are provided in the Appendix and their formal relations are summarized in Fig. 8.2. We now introduce these four model classes in turn.

Individual-based birth-death-mutation processes: polymorphic and stochastic. Under the individual-based model specified above, polymorphic distributions of trait values stochastically drift and diffuse through selection and mutation (Dieckmann 1994; Dieckmann et al. 1995). See Fig. 8.1a for an illustration. Using the specification of the birth, death, and mutation processes provided by the functions bi, di, Mi, and Mi, efficient algorithms for this class of models (Dieckmann 1994) will typically employ Gillespie's minimal process method (Gillespie 1976).

Time, f

Fig. 8.1. Models of adaptive dynamics. Panel a illustrates the individual-based birth-death-mutation process (polymorphic and stochastic), panel b shows an evolutionary random walk (monomorphic and stochastic), panel c represents the gradient-ascent model (monomorphic and deterministic, described by the canonical equation of adaptive dynamics), and panel d depicts an evolutionary reaction-diffusion model (polymorphic and deterministic)

Time, f

Fig. 8.1. Models of adaptive dynamics. Panel a illustrates the individual-based birth-death-mutation process (polymorphic and stochastic), panel b shows an evolutionary random walk (monomorphic and stochastic), panel c represents the gradient-ascent model (monomorphic and deterministic, described by the canonical equation of adaptive dynamics), and panel d depicts an evolutionary reaction-diffusion model (polymorphic and deterministic)

Evolutionary random walks: monomorphic and stochastic. In large populations characterized by low mutation rates, evolution in the individual-based birth-death-mutation process proceeds through sequences of trait substitutions (Metz et al. 1992). During each trait substitution, a mutant with positive invasion fitness quickly invades a resident population, typically ousting the former resident (Geritz et al. 2002). The concatenation of trait substitutions produces the sort of directed random walk depicted in Fig. 8.1b, formally described by the master equation

for the probability density P(x) of observing trait value x, with probabilistic transition rates

r(x',x) = Y, )bi(xi,x)Mi(x'i,Xi)n(x)si(xi,x) JJ 5(xj - Xj) i= 1 j=1,j=i o i/i c

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