Modeling frequency-dependent selection with an application to cichlid fish
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Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, University of Arizona, Tucson AZ 85721
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2.
Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, University of Arizona, Tucson AZ 85721
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Received:
01 January 2008
Accepted:
29 June 2018
Published:
01 October 2008
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MSC :
Primary: 92D40, 92D25, 92D10; Secondary: 39A11
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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.
Citation: Sheree L. Arpin, J. M. Cushing. Modeling frequency-dependent selection with an application to cichlid fish[J]. Mathematical Biosciences and Engineering, 2008, 5(4): 889-903. doi: 10.3934/mbe.2008.5.889
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Abstract
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.
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