Fitness waves in Population Genetics aim at describing the dynamics of adaptation of a population in an abstract (1-d) trait space. When selection is strong and mutations are rare so that the population is monomorphic (all individuals have the same trait), the dynamics is simple: when a beneficial mutation occurs, the population undergoes a selective sweep and reaches equilibrium before the next mutation occurs.
We consider a stochastic population in a more realistic and challenging polymorphic regime. We show that the scaling limit of the distribution of individuals in the (1-d) trait space converges to the solution of a general PDE. When the population is located around a point in the landscape with positive fitness gradient, this PDE can be seen as a generalization of Schweinsberg’s results in [1] on the dynamics of the travelling wave. In Adaptive Dynamics, the case where the population approaches a so-called evolutionary branching point instead (where in particular the gradient is null) has not been mathematically described. Our PDE formally characterizes the dynamics of the population until the wave splits into two components.
Biologically, by considering both the ecological and the evolutionary dynamics of the population, our results help understanding how their interplay can lead to the emergence of multiple, separated traits even though the population evolves in the same fitness landscape. Mathematically, our population model is simple enough to use standard proof techniques, and introduces a general PDE for travelling waves in evolution, naturally emerging from our stochastic population model.
[1]-Jason Schweinsberg. Rigorous results for a population model with selection
I: evolution of the fitness distribution. Electronic Journal of Probability,
22:1–94, 2017.
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