Frequency-dependent growth in class-structured populations: continuous dynamics in the limit of weak selection

Area of Science:

• Population Dynamics
• Mathematical Biology
• Evolutionary Theory

Background:

• Discrete-time models are often used to study population dynamics.
• Approximating these discrete models with continuous ones can simplify analysis and provide insights into long-term behavior.
• Understanding frequency-dependent growth is crucial in evolutionary biology.

Purpose of the Study:

• To establish a continuous model that accurately approximates a discrete-time, class-structured population model.
• To analyze the dynamics of frequency-dependent growth in haploid populations with finite class structure.
• To derive the limiting behavior of type frequencies under weak selection.

Main Methods:

• Consideration of infinite haploid populations with finite class structure.
• Analysis in the limit of weak selection, using the inverse of selection intensity as the unit of time.
• Derivation of generalized fecundity parameters with perturbations dependent on class and type frequencies.
• Mathematical analysis of the dynamics as the number of time steps (N) approaches infinity.

Main Results:

• The type frequencies, in the limit, obey the replicator equation.
• Instantaneous growth rates depend on equilibrium class frequencies and reproductive values.
• The model provides an approximation for discrete population dynamics.

Conclusions:

• A continuous model effectively approximates discrete, class-structured population dynamics under weak selection.
• The derived replicator equation offers insights into evolutionary game theory applications.
• The findings are supported by simulation results.