The evolutionary stability of quantitative traits depends on whether a population can resist invasion by any mutant. While uninvadability is well understood in well-mixed populations, it is much less so in subdivided populations when multiple traits evolve jointly. Here, we investigate whether a spatially subdivided population at a monomorphic equilibrium for multiple traits can withstand invasion by any mutant, or is subject to diversifying selection. Our model also explores the among traits correlations arising from diversifying selection and how they depend on relatedness due to limited dispersal. We find that selection favours a positive (negative) correlation between two traits, when the selective effects of one trait on relatedness is positively (negatively) correlated to the indirect fitness effects of the other trait. We study the evolution of traits for which this matters: dispersal that decreases relatedness, and helping that has positive indirect fitness effects. We find that when dispersal cost is low and the benefits of helping accelerate faster than its costs, selection leads to the coexistence of mobile defectors and sessile helpers. Otherwise, the population evolves to a monomorphic state with intermediate helping and dispersal. Overall, our results highlight the importance of population subdivision for evolutionary stability and correlations among traits.
We propose a faster algorithm for individual based simulations for adaptive dynamics based on a simple modification to the standard Gillespie Algorithm for simulating stochastic birth-death processes. We provide an analytical explanation that shows that simulations based on the modified algorithm, in the deterministic limit, lead to the same equations of adaptive dynamics as well as same conditions for evolutionary branching as those obtained from the standard Gillespie algorithm. Based on this algorithm, we provide an intuitive and simple interpretation of the canonical equation of adaptive dynamics. With the help of examples we compare the performance of this algorithm to the standard Gillespie algorithm and demonstrate its efficiency. We also study an example using this algorithm to study evolutionary dynamics in a multi-dimensional phenotypic space and study the question of predictability of evolution.