We present a dynamical systems analysis of a decision-making mechanism inspired by collective choice in house-hunting honeybee swarms, revealing the crucial role of cross-inhibitory ‘stop-signalling’ in improving the decision-making capabilities. We show that strength of cross-inhibition is a decision-parameter influencing how decisions depend both on the difference in value and on the mean value of the alternatives; this is in contrast to many previous mechanistic models of decision-making, which are typically sensitive to decision accuracy rather than the value of the option chosen. The strength of cross-inhibition determines when deadlock over similarly valued alternatives is maintained or broken, as a function of the mean value; thus, changes in cross-inhibition strength allow adaptive time-dependent decision-making strategies. Cross-inhibition also tunes the minimum difference between alternatives required for reliable discrimination, in a manner similar to Weber's law of just-noticeable difference. Finally, cross-inhibition tunes the speed-accuracy trade-off realised when differences in the values of the alternatives are sufficiently large to matter. We propose that the model, and the significant role of the values of the alternatives, may describe other decision-making systems, including intracellular regulatory circuits, and simple neural circuits, and may provide guidance in the design of decision-making algorithms for artificial systems, particularly those functioning without centralised control.

Abstract. The replicator-mutator equations from evolutionary dynamics serve as a model for the evolution of language, behavioral dynamics in social networks, and decision-making dynamics in networked multi-agent systems. Analysis of the stable equilibria of these dynamics has been a focus in the literature, where symmetry in fitness functions is typically assumed. We explore asymmetry in fitness and show that the replicator-mutator equations exhibit Hopf bifurcations and limit cycles. We prove conditions for the existence of stable limit cycles arising from multiple distinct Hopf bifurcations of the dynamics in the case of circulant fitness matrices. In the noncirculant case we illustrate how stable limit cycles of the dynamics are coupled to embedded directed cycles in the payoff graph. These cycles correspond to oscillations of grammar dominance in language evolution and to oscillations in behavioral preferences in social networks; for decision-making systems, these limit cycles correspond to sustained oscillations in decisions across the group.Key words. replicator-mutator dynamics, limit cycle bifurcations, social networks AMS subject classifications. 91A22, 37G15, 91D301. Introduction. Evolutionary dynamics [32,19,7,31] are, broadly speaking, an effort to cast the basic tenets of Darwinian natural selection (replication, competition, strategy dependent fitness, mutation) in a mathematical framework that can be simulated, interpreted, and often rigorously analyzed. John Maynard Smith's pioneering work [25] made formal connections between classical game theory and evolutionary dynamics. Particularly important was Maynard Smith's definition of evolutionarily stable strategies (ESS's), which are equilibria of an evolutionary dynamical system that are uninvadable by other competing strategies in the environment, and hence stable in an evolutionary sense. From a game theoretic perspective, ESS's are a subset of the Nash equilibria of a game: they satisfy both the Nash best reply condition and evolutionary uninvadability.The replicator dynamics [27] are the simplest model of evolutionary dynamics for a large population comprised of N sub-populations, each subscribed to a different competing strategy. These differential equations model the game theoretic interactions among the sub-populations and determine how each sub-population changes in size as a consequence of these interactions. Lyapunov stable equilibria of the replicator dynamics are Nash equilibria of the corresponding game [34]. Further, all ESS's of the replicator dynamics are asymptotically stable [34].Although the replicator dynamics have proved to be a powerful tool in analyzing a variety of classical games from an evolutionary perspective, they do not model mutation, a key ingredient of selection theory. Mutation can be included by adding the possibility that individuals spontaneously change from one strategy to another. This yields the replicator-mutator dynamics [2,21], which have played a prominent role in evolutionary theory and contain as limiti...

The evolution of leadership in migratory populations depends not only on costs and benefits of leadership investments but also on the opportunities for individuals to rely on cues from others through social interactions. We derive an analytically tractable adaptive dynamic network model of collective migration with fast timescale migration dynamics and slow timescale adaptive dynamics of individual leadership investment and social interaction. For large populations, our analysis of bifurcations with respect to investment cost explains the observed hysteretic effect associated with recovery of migration in fragmented environments. Further, we show a minimum connectivity threshold above which there is evolutionary branching into leader and follower populations. For small populations, we show how the topology of the underlying social interaction network influences the emergence and location of leaders in the adaptive system. Our model and analysis can describe other adaptive network dynamics involving collective tracking or collective learning of a noisy, unknown signal, and likewise can inform the design of robotic networks where agents use decentralized strategies that balance direct environmental measurements with agent interactions.

Abstract-The replicator-mutator equations from evolutionary dynamics serve as a model for the evolution of language, behavioral dynamics in social networks, and decision-making dynamics in networked multi-agent systems. Analysis of the stable equilibria of these dynamics has been a focus in the literature, where symmetry in fitness functions is typically assumed. We explore asymmetry in fitness and show that the replicator-mutator equations exhibit Hopf bifurcations and limit cycles. We prove conditions for the existence of stable limit cycles for the dynamics in the case of circulant fitness matrices, and illustrate their existence in the noncirculant case. For decision-making networks, these limit cycles correspond to sustained oscillations in decisions across the group. I. INTRODUCTIONIn this paper we study the replicator-mutator equations from evolutionary dynamics; these serve as a model for the evolution of language [1], for behavior selection in social networks [2], [3], and also for decision-making dynamics in networked multi-agent systems [4]. Our main result is to prove a Hopf bifurcation for these dynamics with certain network interconnection topologies; this implies the existence of limit cycle behavior. For decision-making networks, the limit cycles correspond to sustained oscillations in decisions across the group.Evolutionary dynamics [5], [6], [7] are, broadly speaking, an effort to cast the basic tenets of Darwinian natural selection (replication, competition, strategy dependent fitness, mutation) in a mathematical framework. The set of replicator equations [8] are the simplest model of evolutionary dynamics for a population divided among a finite set of N competing strategies (N ≥ 2). The differential equations model the game theoretic interactions among the sub-populations, each subscribed to a different competing strategy, and determine how the different sub-populations change in size as a consequence of these interactions.Although the replicator dynamics have proved to be a powerful tool in analyzing a variety of classical games from an evolutionary perspective, they do not model mutation, a key ingredient of selection theory. Mutation can be included by adding the possibility that individuals spontaneously change from one strategy to another. This yields the replicatormutator dynamics [9], which have played a prominent role in evolutionary theory and have recently been employed to model social and multi-agent network dynamics [2]. The replicator-mutator dynamics can be interpreted from a

Pursuit and evasion strategies are used in both biological and engineered settings; common examples include predator-prey interactions among animals, dogfighting aircraft, car chases, and missile pursuit with target evasion. In this paper, we consider an evolutionary game between three strategies of pursuit (classical, constant bearing, motion camouflage) and three strategies of evasion (classical, random, optical-flow based). Pursuer and evader agents are modeled as self-propelled steered particles with constant speed and strategydependent heading control. We use Monte-Carlo simulations and theoretical analysis to show convergence of the evolutionary dynamics to a pure strategy Nash equilibrium of classical pursuit vs. classical evasion. Here, evolutionary dynamics serve as a powerful tool in determining equilibria in complicated game-theoretic interactions. We extend our work to consider a novel pursuit and evasion based collective motion scheme, motivated by collective pursuit and evasion in locusts. We present simulations of collective dynamics and point to several avenues for future work.