Box 12.3 (continued)

when- V- r v is tin totjl number ot individuals competing loi the breeding site of a mutant that has ri-it dispersed. and .V.., is the total number of individuals competing for the breeding site of a mutant that has disposed.

i.uhidutinn V. -j,,,., Ret..«use only one indi\ breeds on eaili bite, a site containing nondispersing mutant offspring must have been occupied by a single mutant individual in the pievmus generation. I'hu\ a focal patch containing a mutant nondisperser will contain two types ot individuals: mutant types that have not dispersed and (ii■ lesident types that have immigrated into the patch. We can igiioie the possibility that a mutant immigrates in liom another patch, because the mutant is rare and mutants migrating out ot other patches arc e.xceedinglv unlike ly to end up in a patch that happens to contain othei mutant individuals.

We have already calculated Ihe number ot mutant types that do not disperse away from the local patch. n\l - i/.,i. Next, we must calculate the nuuibei of lesident types that migrate into the loi al patch. I et S" be the total number of sites and /' be the proportion ot these sites occupied by mutants. There are thus S 11 - />> residents, who produce a total of n .S" 11 - p) offspring, of which a fraction d disperse. Of these migrants, only a traction il c) survive, and only a fraction l'.s land on the foul patch. I lius, the number of resident types that immigrate to Unpad h is n s ■ 1 [>) d 11 £ j/s -«il ¡n d (1 ;) Bec ause we assume that the mutant type is vciv rare, this number is approximated ri J (1 - u. to leading ordei in p. Overall, the total number ot offspring competing tor the breeding site ot .1 non-dispersing mutant is thus

Ciilculalinj \ A focal patch containing a dispersing mutant offspnrig will contain three types nl individuals: ii) the mutant itself.«ii■ resident types that ha\e not dispersed, and liii) resident tvpes that have dispeised I mm other sites. We assume that onlv the one mutant would land on the site, because ihe mutant is very rare. The number ot resident types that have not dispersed will equal n 11 J), hnallv, the number of resident types thai have dispersed limn othei siles to the site is the- same .is in the previous paragraph: approximately nd >1 i'i. Overall, we have V,.,,.,..j - I - nil J* ■ n d (I ci. If the number ot ollspring pmdiued per parent, n. is very large (<■ n.. when ,i plant produces a laige iniriibei of sc\dsi. we can make the further approximation that N^^ng ~ n (I - d) + n d (1 c).

Substituting lhe-e expressions into M2 -5.11 yields the mutant leproduaive factor:

n 1 </,„"■ \ n.M »1 d- ■ ti d • 1 .

Pividinv: through bv /; gives equation 112.7> m the main tc.xt.

Unlike (12.4c), equation (12.7) cannot be rearranged in any way that allows us to rewrite the invasion condition A (dm,d) > 1 as f{dm) > f(d), with mutant parameters on one side and resident parameters on the other. Consequently, there is no function that is strictly maximized over evolutionary time. Rather, the ability of a mutant allele to invade depends on both its own characteristics and those of the resident allele, as described by A(dm,d)- Nevertheless, we can use the mutant reproductive factor (12.7) to predict how dispersal rate evolves by determining which mutations can invade which resident populations (i.e., by determining when A(dm,d) > 1).

Although mathematically equivalent, it is easier to see if A(dm,d) - 1 is positive than it is to see if k{dm,d) > 1, because we can focus on the sign of the result. Subtracting one from (12.7) and factoring yields k{dm,d) ~ 1

(1 - c)(dm - d){ 1 - cd - dm) (1 - cd)(( 1 ~ dm) + d(1 - c)Y

We attempted to write each term in equation (12.8) as a quantity whose sign was obviously positive, because we can ignore positive factors when identifying the sign of\(dm,d) - 1. For example, we wrote ((1 - dm) + d (1 - c)) rather than (1 + d - dm - c). All terms in A (dm,d) ~ 1 are necessarily positive except (dm -d)( 1 - cd - dm) because the dispersal probability d and the probability of dying while dispersing, c, are less than one. Therefore, when (dm - d)(l - c d - dm) > 0, we know that Hdmtd) - 1 > 0, implying that the mutant will spread.

The quantity (dm - d)( 1 - c d - dm) is positive when both terms in parentheses are positive (dm > d and 1 - c d > dm) or when both terms are negative (dm < d and 1 - c d < dm). In other words, the mutant allele will spread only if its dispersal probability lies between d and 1 - c d. Figure 12.2 illustrates four o o

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