Adaptive Dynamics in Two Dimensions

1. Adaptive Dynamics in Two Dimensions

2. Properties of Evolutionary Singularities • Evolutionary stabilityIs a singular phenotype immune to invasions by neighboring phenotypes? • Convergence stabilityWhen starting from neighboring phenotypes, do successful invaders lie closer to the singular one? • Invasion potentialIs the singular phenotype capable of invading into all its neighboring types? • Mutual invasibilityCan a pair of neighboring phenotypes on either side of a singular one invade each other?

3.                 Classification of Evolutionary Singularities for One-dimensional Adaptive Traits (1) Evolutionary stability, (2) Convergence stability, (3) Invasion potential, (4) Mutual invasibility.

4. Invasion Fitness, Gradient, and Hessians • Invasion fitness at ecological equilibrium • Selection gradient at evolutionary equilibrium • Hessians of invasion fitness

5. Analytic Conditionsfor One-dimensional Traits • Evolutionary stability • Convergence stability • Invasion potential • Mutual invasibility Important if not monomorphic (or ) Central to monomorphic analysis Not so important for small mutations Important if not monomorphic

6. Constraints on Second Derivativesfor One-dimensional Traits • Constraints and • Degrees of freedom 4 (second derivatives) minus1+1 (constraints) minus1 (scaling convention) gives1 effective degree of freedom

7. Conditions for Evolutionary Branchingfor One-dimensional Traits • In general, four prerequisites for evolutionary branching: • 1. Monomorphic convergence • 2. Invasibility • 3. Mutual invasibility • 4. Dimorphic divergence • Resultant condition for one-dim. evolutionary branching: • Monomorphic convergence and invasibility

8. Constraints on Second Derivativesfor Two-dimensional Traits • Constraints and and and • Degrees of freedom 4x4 (matrix elements of second derivatives) minus4+1+1+3 (constraints) minus1 (scaling convention) gives6 effective degrees of freedom

9. Convergence Stability for Two-dimensional Traits • Absolute convergence (in)stability is symmetric and negative (positive) definite • Strong convergence (in)stability Symmetric component of is negative (positive) definite • Canonical convergence (in)stability is negative (positive) Stability may depend on mutational variance-covariance matrix

10. Evolutionary Stability for Two-dimensional Traits • Evolutionary stability is negative definite • Partial invasibility is indefinite Branching impossible along trait axes, yet feasible in diagonal direction. • Full invasibility is positive definite

11. Mutual Invasibility for Two-dimensional Traits • No mutual invasibility is negative definite • Partial mutual invasibility is indefinite • Full mutual invasibility is positive definite

12. Alignment upon Evolutionary Branchingfor Two-dimensional Traits Consider the fitness landscape in the mutant direction as described by cThmmc: Using the orthogonal matrix c, made up from the normalized eigenvectors of σ-2, the variance-covariance matrix σ-2 is transformed into the identity matrix, thus removing mutational bias from the branching process. Outgoing branches are expected to dynamically align on the direction of steepest upward curvature, that is, on the dominant eigenvector of cThmmc.

13. Conditions for Evolutionary Branchingfor Two-dimensional Traits • Monomorphic convergence • Invasibility • Mutual invasibility with • Alternative sufficient condition symmetric and negative definite and positive definite

14. Dimorphic Invasion Fitness for Two-dimensional Adaptive Traits The infamous non-differentiability –snapshots of an elusive beast:

15. Summary • The structure of evolutionary singularities in higher-dimensional trait spaces is more complex than in one-dimensional adaptive dynamics. There are three reasons for this: • Matrices can generically be indefinite, while vanishing scalar second derivatives are non-generic. • The two mixed Hessians are not identical. • The mutational variance-covariance matrix can affect evolutionary outcomes. • Conditions for primary evolutionary branching can nevertheless be derived. • Tackling the issue of dimorphic divergence and the full classification of singularities in higher-dimensional trait spaces will still benefit from deriving and analyzing the normal form of dimorphic invasion fitness.