Stability of the replicator equation for a single species with a multi-dimensional continuous trait space

Natural gradient ascent in evolutionary games

We consider evolutionary games with a continuous trait space where the replicator dynamics are restricted to the manifold of multivariate Gaussian distributions. We demonstrate that replicator dynamics are gradient flows with respect to the Fisher information metric. The potential function for these gradient flows is closely related to the mean fitness. Our findings extend previous results on natural gradient ascent in evolutionary games with a finite strategy set. Throughout the paper we pursue an information-geometric point of view on evolutionary games. This sheds a new light on the replicator dynamics as a learning process, realizing the compromise between maximization of the mean fitness and preservation of the diversity.

The fundamental theorem of natural selection in optimization and games

The Fisher's fundamental theorem of natural selection (FTNS) is a matter of longstanding debate in the community of mathematical biologists. Many researchers proposed different clarifications and mathematical reconstructions of the Fisher's original statement. The present study is motivated by our opinion that the controversy can be resolved by examining the Fisher's statement within the framework of two mathematical theories that are inspired by the Darwinian formalism: evolutionary game theory (EGT) and evolutionary optimization (EO). We present four rigorous formulations (some of them previously reported) of FTNS in four different setups that come from EGT and EO. Our study demonstrates that FTNS in its original form is correct only in certain setups. In order to be recognized as a universal law, the Fisher's statement should be: (a) clarified and completed and (b) relaxed by replacing the words "is equal to" with "does not exceed". Moreover, the real meaning of FTNS can be best understood from the information-geometric point of view. Such an approach shows that FTNS imposes an upper geometric bound on information flows in evolutionary systems. In this light, FTNS appears to be a statement about the intrinsic time scale of an evolutionary system. This leads to a novel insight: FTNS is an analogue of the time-energy uncertainty relation in physics. This further emphasizes a close relation with results on speed limits in stochastic thermodynamics.

Bayes and Darwin: How replicator populations implement Bayesian computations

Bayesian learning theory and evolutionary theory both formalize adaptive competition dynamics in possibly high-dimensional, varying, and noisy environments. What do they have in common and how do they differ? In this paper, we discuss structural and dynamical analogies and their limits, both at a computational and an algorithmic-mechanical level. We point out mathematical equivalences between their basic dynamical equations, generalizing the isomorphism between Bayesian update and replicator dynamics. We discuss how these mechanisms provide analogous answers to the challenge of adapting to stochastically changing environments at multiple timescales. We elucidate an algorithmic equivalence between a sampling approximation, particle filters, and the Wright-Fisher model of population genetics. These equivalences suggest that the frequency distribution of types in replicator populations optimally encodes regularities of a stochastic environment to predict future environments, without invoking the known mechanisms of multilevel selection and evolvability. A unified view of the theories of learning and evolution comes in sight.

Red queen dynamics in specific predator-prey systems

The dynamics of a predator-prey system are studied, with a comparison of discrete and continuous strategy spaces. For a [Formula: see text] system, the average strategies used in the discrete and continuous case are shown to be the same. It is further shown that the inclusion of constant prey switching in the discrete case can have a stabilising effect and reduce the number of available predator types through extinction.

CSS, NIS and dynamic stability for two-species behavioral models with continuous trait spaces

Static continuously stable strategy (CSS) and neighborhood invader strategy (NIS) conditions are developed for two-species models of frequency-dependent behavioral evolution when individuals have traits in continuous strategy spaces. These are intuitive stability conditions that predict the eventual outcome of evolution from a dynamic perspective. It is shown how the CSS is related to convergence stability for the canonical equation of adaptive dynamics and the NIS to convergence to a monomorphism for the replicator equation of evolutionary game theory. The CSS and NIS are also shown to be special cases of neighborhood p(*)- superiority for p(*) equal to one half and zero, respectively. The theory is illustrated when each species has a one-dimensional trait space.

Revisiting matrix games: The concept of neighborhood invader strategies

We extend the concept of neighborhood invader strategy (NIS) to finite-dimensional matrix games and compare this concept to the evolutionarily stable strategy (ESS) concept. We show that these two concepts are not equivalent in general. Just as ESS's may not be unique, NIS's may also not be unique. However, if there is an ESS and a NIS then these strategies must be the same. We show that an ESNIS (an ESS and NIS) for any matrix game is unique and that a mixed ESS with full support is a NIS. Thus a mixed ESS with full support is not invadable by any pure or mixed strategy and it can invade any pure or mixed strategy. An ESS which is an ESNIS, therefore, has better chance of being established evolutionarily through dynamic selection.