Fitness potentials and qualitative properties of the Wright-Fisher dynamics

Abstract

We present a mechanistic formalism for the study of evolutionary dynamics models based on the diffusion approximation described by the Kimura Equation. In this formalism, the central component is the fitness potential, from which we obtain an expression for the amount of work necessary for a given type to reach fixation. In particular, within this interpretation, we develop a graphical analysis - similar to the one used in classical mechanics - providing the basic tool for a simple heuristic that describes both the short and long term dynamics. As a by-product, we provide a new definition of an evolutionary stable state in finite populations that includes the case of mixed populations. We finish by showing that our theory - rigorous for two types evolution without mutations- is also consistent with the multi-type case, and with the inclusion of rare mutations.

From Fixation Probabilities to d-player Games: An Inverse Problem in Evolutionary Dynamics

“…Indeed, we will show that almost any arbitrary interaction can be modelled by d-player game theory, provided we do not impose any restriction on the game size. As already observed Wu et al, 2013;Pena et al, 2014;Souza, 2016, 2017;Czuppon and Gokhale, 2018;Chalub and Souza, 2018), d-player games with d ≥ 3 can exhibit quite distinct behaviour from the 2-player ones. In the multi-type case, we expect that a similar approach might work, but much remains to be done-indeed, multi-type evolution is an important source of open mathematical problems.…”

Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective

“…[54,92,93,56], see also the discussion on Kimura's Maximum Principle and the (Svirezhev-)Shahshahani metric in [13]. We note also that, for finite populations, short-term and long-term information on the dynamics has been recently obtained from the free energy [18].…”

“…for a given potential V (x) (the gradient −V ′ representing the fitness difference between the focal type, A, and its opponent B). The parameter κ > 0 is the inverse of the selection strength, see [18] for a detailed analysis of each parameter in Equation (4). In fact, if N (∆t) 1/2 → κ −1 , then the vector p obtained from Equation (2), given a certain initial condition, converges in an appropriate sense to a measure p, where p is the solution of a certain degenerate parabolic partial differential equation of drift-diffusion type known as the Kimura equation…”

“…In fact, it is not difficult to prove that Lemma 5. Let M ∈ A N +1,k be an admissible and micro-reversible transition kernel and G φ be defined by (18) for a given differentiable convex function φ : R + → R. Then…”

“…It is also natural to extend definition (18) to the full dynamics, i.e., by abuse of notation, using the variable p instead of q, which are equivalent in view of Remark 3, we write…”