Abstract
Nonlinear dynamics makes use of attactors to describe asymptotic behavior of complex systems. However, in real life can be unattainable, as achieving them might require time much exceeding all relevant timescales of a system. Therefore, there is an increasing interest in quasi-stationary states, where the system rapidly converges to and remains for a long time, before getting into an absorbing (asymptotic) state. Exemplifying in the famous Dawkins‘ Battle of the Sexes game, we demonstrate that quasi-stationary distributions can produce not simply different, but a much more complex behavior, then the asymptotic ones, that is transient self-sustained oscillations of player numbers and the corresponding non-unimodal probability distribution. We find that parameters of the quasi-stationary limit cycle depend on the population size.
Highlights
Nonlinear dynamics offers powerful tools to describe behavior of complex physical, biological, chemical or socio-ecological systems
If a system is dissipative, typically, one or another asymptotic regime, a stationary state, limit cycle or chaotic attractor is reached after a transient process
Employing the concept of the quasi-stationary distribution in absorbing Markov chains [Darroch et al, 1965; Collet et al, 2012], we find that the metastable dynamics is oscillatory
Summary
Nonlinear dynamics offers powerful tools to describe behavior of complex physical, biological, chemical or socio-ecological systems. An asymptotic state may not be reached during a relevant timescale, for example, the coherence time of a quantum system, or a population lifetime [Biroli and Kurchan, 2001; Rose et al, 2016; Assaf and Mobilia, 2012] In this case, one speaks about metastable or quasistationary states, rapidly converged on by a trajectory from generic initial conditions, and resided at for a long time [Rabinovich et al, 2008; Macieszczak et al, 2016]. As a particular complex system we take an evolutionary “Battle of Sexes” game of populations that models the gender conflict over parental care [Dawkins, 1976] In this game males and females have two alternative behavioral strategies, which stem from various expectations for courtship and different assistance in raising an offspring.
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