Drift reversal in asymmetric coevolutionary conflicts: Influence of microscopic processes and population size

The coevolutionary dynamics in finite populations currently is investigated in a wide range of disciplines, as chemical catalysis, biological evolution, social and economic systems. The dynamics of those systems can be formulated within the unifying framework of evolutionary game theory. However it is not a priori clear which mathematical description is appropriate when populations are not infinitely large. Whereas the replicator equation approach describes the infinite population size limit by deterministic differential equations, in finite populations the dynamics is inherently stochastic which can lead to new effects. Recently, an explicit mean-field description in the form of a Fokker-Planck equation was derived for frequency-dependent selection in finite populations based on microscopic processes. In asymmetric conflicts between two populations with a cyclic dominance, a finite-size dependent drift reversal was demonstrated, depending on the underlying microscopic process of the evolutionary update. Cyclic dynamics appears widely in biological coevolution, be it within a homogeneous population, or be it between disjunct populations as female and male. Here explicit analytic address is given and the average drift is calculated for the frequency-dependent Moran process and for different pairwise comparison processes. It is explicitely shown that the drift reversal cannot occur if the process relies on payoff differences between pairs of individuals. Further, also a linear comparison with the average payoff does not lead to a drift towards the internal fixed point. Hence the nonlinear comparison function of the frequency-dependent Moran process, together with its usage of nonlocal information via the average payoff, is the essential part of the mechanism.

💡 Research Summary

The paper investigates how the microscopic rules governing evolutionary updates and the finite size of populations affect the direction of average drift in asymmetric coevolutionary conflicts that exhibit cyclic dominance. The authors place their work within the broader context of evolutionary game theory, noting that while the replicator equation accurately describes infinite populations through deterministic differential equations, real systems are finite and therefore inherently stochastic. Recent advances have provided a mean‑field description of frequency‑dependent selection in finite populations via a Fokker‑Planck equation derived from specific microscopic processes.

The study focuses on a two‑population game where each population has two strategies and the payoff matrix creates a rock‑paper‑scissors‑like cyclic dominance across the populations (e.g., male‑female or predator‑prey interactions). The state of the system is represented by the pair (x, y), the frequencies of a given strategy in each population. To quantify the tendency of the system to move toward or away from the interior mixed‑strategy fixed point, the authors introduce a potential‑like function H(x, y) and compute the average change ⟨ΔH⟩ over one update step. A negative ⟨ΔH⟩ indicates drift toward the fixed point (stability), whereas a positive value signals drift away—a phenomenon they term “drift reversal.”

Two families of update rules are examined. The first is the frequency‑dependent Moran process (FD‑Moran). In this process the reproduction probability of an individual i depends on a nonlinear function of the difference between its payoff π_i and the population‑average payoff ⟨π⟩, typically a logistic (Fermi) function:
 P_i ∝ 1 /