Evolutionarily stable strategy, stable state, periodic cycle and chaos in a simple discrete time two-phenotype model

Area of Science:

• Evolutionary Game Theory
• Population Dynamics
• Mathematical Biology

Background:

• Understanding evolutionary stable strategies (ESS) is crucial for predicting population dynamics.
• Previous models often used linear fitness functions, limiting applicability.
• The role of non-linear fitness functions in ESS dynamics requires further investigation.

Purpose of the Study:

• To investigate a discrete time, two-phenotype matrix game model with an exponential fitness function.
• To determine the conditions for ESS in this model.
• To analyze the relationship between static ESS conditions and dynamic population behavior.

Main Methods:

• Developed a discrete time, two-phenotype matrix game model.
• Defined individual fitness as an exponential function of expected payoff.
• Analyzed the static conditions for ESS.
• Examined the dynamic properties of the pure strategy model.

Main Results:

• Static ESS conditions depend solely on the payoff matrix, independent of the specific fitness function form.
• ESS conditions are identical to those in models with linear fitness functions.
• An unstable ESS equilibrium in the pure strategy model corresponds to cyclic or chaotic population states.

Conclusions:

• The choice of fitness function (linear vs. exponential) does not alter the static ESS conditions in this two-phenotype game.
• The stability of ESS equilibria is directly linked to population dynamics, with instability predicting complex behaviors like cycles or chaos.