Population growth modifies the optimal equilibrium between a stationary population and its resource, producing instead a line of equilibria, characterized by fluctuating population size, resource quantity, harvest per head, and birth rates. The Pontryagin procedure allows the analytical expression of the Nash equilibria for two populations sharing a common resource and capable of growth. An alternative procedure, which avoids solving differential equations and inherently includes state constraints, involves building the capture-viability kernel of an auxiliary system. For two populations, all Nash equilibria under state constraints are obtained as the intersection of the boundaries of two capture-viability kernels. The two methods, Pontryagin and viability, yield concordant results. Viability is more flexible and avoids solving differential equations for each initial condition.
Journal of Optimization Theory and Applications – Springer Journals
Published: Jul 11, 2017
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