Eco-evolutionary Dynamics of Individual-Based Food Webs - Advances in Ecological Research - page 230

Ef kj ¼

m þ m

1
2

1
2

f G 1 ðÞ j 1 f G 1 ðÞ j f G 2 ðÞ j 1 f G 2 ðÞ j

ð m Þþ ð m Þþ

1 1

ð 6 Þ

Substituting f G 1 ðÞ j ¼ ð 1 þ q G 1 ðÞ j = 2 Þ and f G 2 ðÞ j ¼ ð 1 þ q G 2 ðÞ j = 2 Þ from Eq. (3)

gives:

Ef kj

q G 1 ðÞ j
2

q G 1 ðÞ j
2

q G 1 ðÞ j
2

1
2

1 þ 1 þ 1

¼ þ m þ m m

1
2

1 þ q G 2 ðÞ j

2

1 þ q G 2 ðÞ j

2

1 q G 2 ðÞ j

2

þ þ m þ m m ; ð 7 Þ

and after simplification, we obtain:
Ef kj ¼

þ þ

q G 1 ðÞ j
2

q G 2 ðÞ j
2

1
2

1
2 m

1
2

1
2 m

q G 1 ðÞ j q G 2 ðÞ j þ ; ð 8 Þ

Ef kj ¼

m þ

1
2

q G 1 ðÞ j
2 þ

q G 2 ðÞ j
2

q G 1 ðÞ j q G 2 ðÞ j 1 m ; ð 9 Þ

and substituting in Eq. (2) :
Eq kj ¼ 2Ef kj

m þ

q G 1 ðÞ j
2

q G 2 ðÞ j
2

q G 1 ðÞ j q G 2 ðÞ j

¼ m þ ;

1 1 1

ð 10 Þ

and after simplification, we obtain:

<

0 1

1
2 m

@ A q G 1 ðÞ j

Eq kj ½ ¼ ð þ q G 2 ðÞ j Þ;

ð Þ

11

:

Eq kk ½ ¼ 1 ;

and the expected mean genetic similarity at equilibrium is Q* ¼ 1/( yþ 1),

where ( Higgs and Derrida, 1992; Meli´n et al., 2010 ). In summary,

each new offspring has a genome inherited from its two parents with repro-
duction producing similar individuals and mutation acting so that offspring
differ from their parents and from all the individuals in the population.

y¼ 4J m

2. Integrating Population Genetics in the Neutral Metacommunity
Model

In the previous section, we derived the expected genetic similarity among all
the individuals in the initial population of size J starting from all individuals
with an extremely large and similar genome. The dynamics in the original