American Institute of Mathematical Sciences

Advanced
2008, 5(4): 889-903. doi: 10.3934/mbe.2008.5.889

Modeling frequency-dependent selection with an application to cichlid fish

Sheree L. Arpin 1, and  J. M. Cushing 2,

1. 

Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, University of Arizona, Tucson AZ 85721, United States
2. 

Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, University of Arizona, Tucson AZ 85721, United States

Received  January 2008 Revised  April 2008 Published  October 2008

Negative frequency-dependent selection is a well known microevolutionary process that has been documented in a population of Perissodus microlepis, a species of cichlid fish endemic to Lake Tanganyika (Africa). Adult P. microlepis are lepidophages, feeding on the scales of other living fish. As an adaptation for this feeding behavior P. microlepis exhibit lateral asymmetry with respect to jaw morphology: the mouth either opens to the right or left side of the body. Field data illustrate a temporal phenotypic oscillation in the mouth-handedness, and this oscillation is maintained by frequency-dependent selection. Since both genetic and population dynamics occur on the same time scale in this case, we develop a (discrete time) model for P. microlepis populations that accounts for both dynamic processes. We establish conditions on model parameters under which the model predicts extinction and conditions under which there exists a unique positive (survival) equilibrium. We show that at the positive equilibrium there is a 1:1 phenotypic ratio. Using a local stability and bifurcation analysis, we give further conditions under which the positive equilibrium is stable and conditions under which it is unstable. Destabilization results in a bifurcation to a periodic oscillation and occurs when frequency-dependent selection is sufficiently strong. This bifurcation is offered as an explanation of the phenotypic frequency oscillations observed in P. microlepis. An analysis of the bifurcating periodic cycle results in some interesting and unexpected predictions.
Keywords: phenotypic oscillation period doubling bifurcation, microevolution, frequency-dependent selection, population dynamics, population genetics.
Mathematics Subject Classification: Primary: 92D40, 92D25, 92D10; Secondary: 39A11.
Citation: Sheree L. Arpin, J. M. Cushing. Modeling frequency-dependent selection with an application to cichlid fish. Mathematical Biosciences & Engineering, 2008, 5 (4) : 889-903. doi: 10.3934/mbe.2008.5.889
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