Nonlinear averaging in 2D: intraspecific variation in two interacting species and consequences for population dynamics

Recent empirical and theoretical work shows that intraspecific trait variation (ITV) is prevalent and ecologically important, and should thus be taken into account in ecological models. What is lacking, however, is a comprehensive understanding of the effects of ITV in each of two interacting species (two-dimensional ITV). We address this gap for the cases where interspecific individual-by-individual interactions are affected by the trait values of both participants. Using nonlinear averaging in two dimensions, we show for several interaction functions how ITV affects average predation rates or competition coefficients. We develop an intuition for the direction and magnitude of this effect by using a Taylor approximation based on the local curvatures of the interaction function, the trait means, and the trait variances and covariances. We then incorporate the estimated average interaction parameters into simple competition and predator-prey models to derive the expected population-dynamic consequences. We show that two-dimensional ITV can have both quantitative effects on abundances, but also qualitative effects, e.g. stabilize or destabilize coexistence. Our approach can straightforwardly be applied to other interaction functions and dynamical systems and thus provides a valuable tool for understanding the joint effect of trait variation in two interacting species.