Pursuit and evasion strategies are used in both biological and engineered settings; common examples include predator-prey interactions among animals, dog fighting 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 camouage) 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 strategy-dependent 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.