Pursuit and an evolutionary game

Abstract

Pursuit is a familiar mechanical activity that humans and animals engage in-athletes chasing balls, predators seeking prey and insects manoeuvring in aerial territorial battles. In this paper, we discuss and compare strategies for pursuit, the occurrence in nature of a strategy known as motion camouflage, and some evolutionary arguments to support claims of prevalence of this strategy, as opposed to alternatives. We discuss feedback laws for a pursuer to realize motion camouflage, as well as two alternative strategies. We then set up a discrete-time evolutionary game to model competition among these strategies. This leads to a dynamics in the probability simplex in three dimensions, which captures the mean-field aspects of the evolutionary game. The analysis of this dynamics as an ascent equation solving a linear programming problem is consistent with observed behaviour in Monte Carlo experiments, and lends support to an evolutionary basis for prevalence of motion camouflage.

Symmetry and reduction in collectives: cyclic pursuit strategies

“…The resulting shape dynamics is cast in a natural redundant parametrization of shape space tailored to the setting of SE (2). This parametrization fits well with anticipated singularities in feedback laws, such as one associated with motion camouflage (MC) or motion parallax nulling [1].…”

“…ν i > ν j ), and finite evader maneuverability (i.e. |u j | bounded) [2]. In §3b, we prove asymptotic convergence to the CB pursuit manifold provided…”

“…Thus, we treat the planar self-steering particles as evolving on a matrix Lie group G = SE (2) according to the dynamicsġ…”

“…In §3, we explicitly state the dyadic pursuit strategy we call constant bearing (CB)-of central interest in this paper-using a suitable contrast function to fix a constraint on joint state space. We also recall from Wei et al [2] a feedback law to execute the CB strategy. For a particular choice of parameter, the CB strategy and feedback law specialize to the case of classical pursuit (CP), a topic with a rich mathematical heritage and continuing interest [3].…”

“…It is noteworthy that certain key ideas are not dependent on this specialization, and one can profit from the perspective of interacting particles on a Lie group G (e.g. SE (2), the rigid motion group in the plane) formulated below (in §2). In this broader setting, control laws (and hence collective dynamics) are required to admit a continuous symmetry group (e.g.…”