Michael Kopp and Sergey Gavrilets 2004 "COEVOLUTION IN QUANTITATIVE TRAITS:
AN EXPLICIT GENETIC MODEL"
We develop and analyze an explicit multilocus genetic model of coevolution.
We assume that interactions between two species (mutualists, competitors,
or victim and exploiter) are mediated by a pair of additive quantitative traits
which are also subject to direct stabilizing selection towards intermediate
optima. Using a weak selection approximation, we derive analytical results
for a symmetric case with equal allelic effects and no mutation, and we complement
these results by numerical studies of more general cases. We show that mutualistic
and competitive interactions always result in coevolution towards a stable
equilibrium with no more than one polymorphic locus per species. Victim-exploiter
interactions can lead to a number of different dynamic regimes including evolution
towards stable equilibria, cycles, and chaos. At equilibrium, the victim
is often characterized by a very large genetic variance whereas the exploiter
is polymorphic in no more than one locus. Compared to related one-locus or
quantitative genetic models, the multilocus model exhibits two major new properties:
First, the equilibrium structure is considerably more complex. We derive
detailed conditions for the existence and stability of various classes of
equilibria and demonstrate the possibility of multiple simultaneously stable
states. Second, the genetic variances change dynamically, which in turn significantly
affects the dynamics of the mean trait values. In particular, the dynamics
tend to be destabilized by an increase in the number of loci. Thus, dynamic
details depend on genetic details.