Strategy diversity stabilizes mutualism through investment cycles, phase polymorphism, and spatial bubbles.

There is continuing interest in understanding factors that facilitate the evolution and stability of cooperation within and between species. Such interactions will often involve plasticity in investment behavior, in response to the interacting partner's investments. Our aim here is to investigate the evolution and stability of reciprocal investment behavior in interspecific interactions, a key phenomenon strongly supported by experimental observations. In particular, we present a comprehensive analysis of a continuous reciprocal investment game between mutualists, both in well-mixed and spatially structured populations, and we demonstrate a series of novel mechanisms for maintaining interspecific mutualism. We demonstrate that mutualistic partners invariably follow investment cycles, during which mutualism first increases, before both partners eventually reduce their investments to zero, so that these cycles always conclude with full defection. We show that the key mechanism for stabilizing mutualism is phase polymorphism along the investment cycle. Although mutualistic partners perpetually change their strategies, the community-level distribution of investment levels becomes stationary. In spatially structured populations, the maintenance of polymorphism is further facilitated by dynamic mosaic structures, in which mutualistic partners form expanding and collapsing spatial bubbles or clusters. Additionally, we reveal strategy-diversity thresholds, both for well-mixed and spatially structured mutualistic communities, and discuss factors for meeting these thresholds, and thus maintaining mutualism. Our results demonstrate that interspecific mutualism, when considered as plastic investment behavior, can be unstable, and, in agreement with empirical observations, may involve a polymorphism of investment levels, varying both in space and in time. Identifying the mechanisms maintaining such polymorphism, and hence mutualism in natural communities, provides a significant step towards understanding the coevolution and population dynamics of mutualistic interactions.