## Long-term evolution of polygenic traits under frequency-dependent intraspecific competition

- Date: 01/24/2007

Kristan Schneider (University of Vienna)

University of British Columbia

We analytically investigate the long-term evolution of a continuously

varying quantitative character in a diploid population that is

determined additively by a finite number of loci. The trait is under a

mixture of frequency-dependent disruptive selection induced by

intraspecific competition and frequency-independent stabilizing

selection. Moreover, the trait is restricted to a finite range by

constraints on the particular loci. Our investigations are based on

explicit analytical on the short-term dynamics under the assumption of

linkage equilibrium. We show that the population always reaches a

long-term equilibrium (LTE), i.e., an equilibrium that is resistant

against perturbations of mutations of sufficiently small effect. In

general, several LTEs can coexist. They can be calculated explicitly,

and we provide necessary and sufficient conditions for their existence.

In the case that more than one LTE exists, we exemplify numerically

that the evolutionary outcome depends crucially on the initial genetic

architecture, on the joint distribution of mutational effects across

loci, and on the particular realization of the mutation process.

Therefore, long-term evolution cannot be predicted from the ecology

alone. We further show that a partial order exists for the LTEs. The

set of LTEs has a `largest' element, an LTE, which is reached during

long-term evolution if the effects of the occurring mutant alleles are

sufficiently large.

MITACS Math Biology Seminar 2007