Periodic Oscillations and Transient Dynamics Caused by Saturating Density-Dependent Benefits and Costs in Obligate Mutualisms.

It is generally recognized that oscillatory dynamics of mutualism systems arise from external factors such as environmental fluctuations and additional interspecific interactions. However, we here theoretically demonstrate that the saturating density dependence of mutualistic benefits and costs can lead to the periodic oscillations of obligate mutualism systems. This suggests that the dynamic complexity of mutualisms can also arise intrinsically. Our model differentiates benefits in mutualistic interactions from costs and assumes they respectively influence the reproduction rate and mortality of populations. In the symmetric case, where the model structure and parameters are the same for both species, this model shows multiple equilibria and oscillatory dynamics. The difference between benefit and cost may be the primary determinant of these phenomena. The system exhibits damped or periodic oscillations when this difference is intermediate. The two species can stably coexist when benefits significantly outweigh costs, whereas the system faces extinction when costs become relatively high. Asymmetry in benefit and cost between mutualists dramatically changes the system's dynamical regimes. Essentially, these oscillations of mutualism are caused by the transitions of the system between mutualism and antagonism. In addition, our model reveals the transient dynamics of the mutualism system (a phenomenon of regime shift without parameter change), including saddle crawl-bys (moving slowly by saddles) and ghost attractors (slow change in system state near the attractors). Our findings highlight the crucial role of nonlinear benefits and costs in the dynamical complexity of mutualisms.