Coexistence of mutualists and non-mutualists in a dual-lattice model

Evolution and maintenance of mutualism have been one of the major questions in evolutionary ecology, because it is often susceptible of invasion of non-mutualistic strategy. Some previous studies using dual-lattice model suggest that spatial structures of habitat can prevent non-mutualism from prevailing over mutualism, while the detail of the dynamics is not fully revealed. Here we explore population dynamics of the two strategies (mutualism and non-mutualism) in two species engaged in Prisoner's Dilemma game on a dual-lattice space, especially focusing on whether mutualists and non-mutualists can coexist in long-term dynamics. The habitat consists of two layers, each of which a population of species inhabits, and interspecific interaction is restricted between two corresponding sites of the layers. Each individual of the both species is either a mutualist or a non-mutualist and only the former pay cost c for benefit of the partner b. The payoff of the game affects the individuals' fecundity, while the mortality is constant. Reproduction is restricted to neighboring vacant sites of the focal individuals. Our computer simulations of the model show that even if b/c ratio remains constant, mutualists become dominant in both species over wider ranges of basic reproduction rate (reproduction rate without interspecific interaction) as b and c increase. If basic reproduction rates are asymmetric between the species or basic reproduction rates were sufficiently large, mutualists and non-mutualists can coexist in one or both species, while their population sizes often fluctuate. Transition of the final state between mutualism and non-mutualism happens rather discontinuously, then total population sizes change drastically at the transition. Moreover, we also find paradoxical cases of unilateral exploitation, i.e. one species consists of mutualists and other species non-mutualists. Additional simulations reveal that accidental extinction of the non-mutualists of one species can result in extinction of mutualist of the other species.