D

ut J

for i — 1,... ,N (e. g., van Kampen 1981). After inserting r(x , x) as derived above, this yields d dt

Xi Hi(Xi)bi(xi,x)rii(x) j si(xi:x)(Xi )MMi(xi,xi)dxi .

By expanding si(x'i, x) — max(0 , fi(xi, x))/bi(x'i, x) around xi to first order in xi, we obtain si(x'i,x) — max(0, (xi — xi)gi(x))/bi(xi,x) with gi,(x) —

dXfi(x'i,x) ; notice here that fi(xi,x) — 0. This means that in the xi-integral above only half of the total xi-range contributes, while for the other half the integrand is 0. If mutation distributions Mi are symmetric -

Mi(xi + Axi, xi) — Mi(xi — Axi, xi) for all i, xi,, and Axi - we obtain d^xi — ^Pi(xi)ni(x) j (xi — xi)T(xi — xi)Mi(x'i,xi)dx' gi(x) .

The integral is the variance-covariance matrix of the mutation distribution Mi around trait value xi, denoted by o~2(xi,). Hence we recover the canonical equation of adaptive dynamics (Dieckmann 1994; Dieckmann and Law 1996), d^xi — 1 Pi(xi )ni (x)af(xi )gi(x) for i — 1,... ,N. When mutational steps xi ^ xi are not small, higher-order correction terms can be derived: these improve the accuracy of the canonical equation and also cover non-symmetric mutation distributions (Dieckmann 1994; Dieckmann and Law 1996).

Polymorphic Deterministic Model. When mutation probabilities are high, evolution is no longer mutation-limited, so that the two classes of models introduced above - both being derived from the analysis of invasions into essentially monomorphic populations - cannot offer quantitatively accurate approximations of the underlying individual-based birth-death-mutation processes. Provided that population sizes are sufficiently large, it instead becomes appropriate to investigate the average distibution-valued dynamics of many realizations of the birth-death-mutation process, p(x) = j [p'(x) — p(x)]r(p',p)dp' .

Inserting the transition rates r(p' ,p) specified above for the individual-based evolutionary model, we can infer (by collapsing the integrals using the sifting properties of the Dirac delta function and of the generalized delta function)

-dtpi(x) = r+ (xi ,p) — ri (xi,p) for i = l,... ,N. Inserting r+ (xi,p) and r-(xi,p) from above, this gives

+ j pi(x'i)bi(x'i,p)pi(x'i)Mi(x'i,xi)dx'i — di(xi,p)pi(xi) .

Further analysis is simplified by assuming that the mutation distributions Mi are not only symmetric but also homogeneous - Mi(x'i + Axi,xi + Axj) = Mi(x'i,xi) for all i, xi, xi, and Axi. Expanding pi(x'i)bi(x'i,p)pi(x'i) up to second order in x' around xi, pi(x'i)bi(x'i ,p)pi(x'i) = pi(xi)bi(xi,p)pi(xi) + (xi — xi) Pi(xi)bi(xi,p)pi(x{) + 2 (xi — xi)T[ dx Pi(xi )bi(xi,p)pi(xi)](xi — xi), then yields d l d2

dtpi(x) = fi(xi,p)pi(xi) + 2a2(xi) * dx2 Pi(xi)bi(xi,p)pi(xi) , with fi(xi,p) = bi(xi,p) — di(xi,p), a2(xH) = ¡(xi — xH)T(xi — xi)Mi(xi,xi) dxi, and with * denoting the elementwise multiplication of two matrices followed by summation over all resultant matrix elements. This result also provides a good approximation when mutation distributions are heterogeneous, as long as a2(xi), rather than being strictly independent of xi, varies very slowly with xi on the scale given by its elements.

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