Trade-off geometries and the adaptive dynamics of two co-evolving species

Background: A recently developed geometric method makes it possible to explore how the shape of a trade-off determines the outcome of adaptive evolution in any complex model and without committing to a particular functional form of the trade-off function. Aim: Extend the method to the co-evolution of two species. (The two species may be distantly related such as a predator and its prey, or may be closely related like two strategies produced by evolutionary branching.) Results: Thresholds of the local convexity of the trade-off functions are obtained that guarantee evolutionary and convergence stability when a given species pair is singular. In contrast to the single-species case, the condition for convergence is sufficient but not necessary. Criteria for evolutionary branching generalize from the single-species case. A cross-derivative of the invasion fitness determines whether evolutionary branching is possible; this quantity is independent of the trade-offs and if it is negative at a certain species pair, then the trade-offs can be chosen such that evolutionary branching occurs at this point. Worked example: A simple predator–prey model shows how these results can be used to identify trade-off functions such that evolution leads to an evolutionarily stable species pair or to evolutionary branching in either species.