Modeling frequency-dependent selection with an application to cichlid fish - Sheree L. Arpin; J. M. Cushing (Mathematical Biosciences and Engineering)
Negative frequency-dependent selection is a wellknown microevolutionary process that has been documented in a population ofPerissodus microlepis, a species of cichlid fish endemic to Lake Tanganyika(Africa). Adult P. microlepis are lepidophages, feeding on the scales of otherliving fish. As an adaptation for this feeding behavior P. microlepis exhibitlateral asymmetry with respect to jaw morphology: the mouth either opens tothe right or left side of the body. Field data illustrate a temporalphenotypic oscillation in the mouth-handedness, and this oscillation ismaintained by frequency-dependent selection. Since both genetic and populationdynamics occur on the same time scale in this case, we develop a (discretetime) model for P. microlepis populations that accounts for both dynamicprocesses. We establish conditions on model parameters under which the modelpredicts extinction and conditions under which there exists a unique positive(survival) equilibrium. We show that at the positive equilibrium there is a1:1 phenotypic ratio. Using a local stability and bifurcation analysis, wegive further conditions under which the positive equilibrium is stable andconditions under which it is unstable. Destabilization results in abifurcation to a periodic oscillation and occurs when frequency-dependentselection is sufficiently strong. This bifurcation is offered as anexplanation of the phenotypic frequency oscillations observed in P.microlepis. An analysis of the bifurcating periodic cycle results in someinteresting and unexpected predictions.