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Nonlinear frequency-dependent selection at a single locus with two alleles and two phenotypes.

T Yi¹, R Cressman, B Brooks

¹Center for Animal Ecology and Conservational Biology, Institute of Zoology, Academia Sinica, 19 Zhongguancun Lu, Haidian, Beijing, 100080, P.R. China.

Journal of Mathematical Biology

|November 7, 1999

Summary

This study explores frequency-dependent selection in diploid models, revealing that stable equilibria can coexist in nonlinear models. Period-doubling bifurcations arise with strong selection for diverse phenotype interactions.

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Area of Science:

Evolutionary biology
Population genetics
Mathematical modeling

Background:

Understanding frequency-dependent selection is crucial for evolutionary dynamics.
Diploid models offer a more realistic framework for genetic interactions.
Linear fitness models have limitations in capturing complex population dynamics.

Purpose of the Study:

To investigate discrete frequency dynamics in two-phenotype diploid models.
To compare phenotypic and genotypic equilibria stability under nonlinear fitness.
To identify conditions leading to period-doubling bifurcations.

Main Methods:

Developed mathematical models for diploid populations with two phenotypes.
Defined and analyzed phenotypic and genotypic equilibria.

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Utilized matrix game theory to model genotypic fitness as an exponential function of expected payoff.

Investigated stability and bifurcations under frequency-dependent selection.

Main Results:

Demonstrated the coexistence of stable phenotypic and genotypic equilibria within the same nonlinear model.
Showcased that period-doubling bifurcations emerge under specific selection pressures.
Highlighted the distinct dynamics compared to linear fitness models.

Conclusions:

Nonlinear fitness functions in diploid models can lead to complex evolutionary dynamics.
The coexistence of stable equilibria suggests potential for diverse evolutionary outcomes.
Period-doubling bifurcations indicate transitions to chaotic dynamics under strong selection for phenotype interactions.