Function-valued adaptive dynamics and optimal control theory
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In this article we further develop the theory of adaptive dynamics of function-valued traits. Previous work has concentrated on models for which invasion fitness can be written as an integral in which the integrand for each argument value is a function of the strategy value at that argument value only. For this type of models of direct effect, singular strategies can be found using the calculus of variations, with singular strategies needing to satisfy Euler’s equation with environmental feedback. In a broader, more mechanistically oriented class of models, the function-valued strategy affects a process described by differential equations, and fitness can be expressed as an integral in which the integrand for each argument value depends both on the strategy and on process variables at that argument value. In general, the calculus of variations cannot help analyzing this much broader class of models. Here we explain how to find singular strategies in this class of process-mediated models using optimal control theory. In particular, we show that singular strategies need to satisfy Pontryagin’s maximum principle with environmental feedback. We demonstrate the utility of this approach by studying the evolution of strategies determining seasonal flowering schedules.
KeywordsAdaptive dynamics Function-valued traits Theory of optimal control Pontryagin’s maximum principle Environmental feedback
Mathematics Subject Classification92D15 49-00
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- Gilchrist M, Sulsky D, Pringle A (2006) Identifying fitness and optimal life-history strategies for an asexual filamentous fungus. Evolution 60: 970–979Google Scholar
- Heino M, Parvinen K, Dieckmann U (2008) Evolution of foraging strategies on resource gradients. Evol Ecol Res 10: 1131–1156Google Scholar
- Jaffra’ezic F, Pletcher SD (2000) Statistical models for estimating the genetic basis of repeated measures and other function-valued traits. Genetics 156: 913–922Google Scholar
- Maynard Smith J (1976) Evolution and the theory of games. Am Sci 64: 41–45Google Scholar
- Metz JAJ, Geritz SAH, Meszéna G, Jacobs FJA, van Heerwaarden JS (1996) Adaptive dynamics, a geometrical study of the consequenses of nearly faithful reproduction. In: van Strien SJ, Verduyn Lunel SM (eds) Stochastic and spatial structures of dynamical systems. North-Holland, Amsterdam, pp 183–231Google Scholar