Abstract: Minimal mathematical models are used to understand complex phenomena in the physical, biological, and social sciences. This modeling philosophy never claims, nor even attempts, to fully capture the mechanisms underlying the phenomena, and instead offers insights and predictions not otherwise possible. Here, we build and explore minimal dynamical systems models to understand three complex animal and human systems. First, we incorporate the assumptions of Zahavi's handicap principle into a mathematical model of ornament evolution and show that this existing hypothesis is sufficient to explain the previously puzzling observation of bimodally distributed ornament sizes in a variety of species. Second, we propose a 'return-to-setpoint' model of chronic pain dynamics in sickle cell disease patients with the goal of offering personalized, data-driven recommendations for treating chronic pain. Third, we present a conceptual model of restaurant competition that predicts the existence of a critical gratuity rate threshold at which restaurant owners will disallow tipping to maximize their profits. Because of their simplicity, these models of complex human and animal systems offer new connections between existing ideas, give optimized solutions with limited data, and provide qualitative predictions of future events.