Adaptive Dynamics in Two Dimensions

Properties of Evolutionary Singularities n Evolutionary stability Is a singular phenotype immune to invasions by neighboring phenotypes? n Convergence stability When starting from neighboring phenotypes, do successful invaders lie closer to the singular one? n Invasion potential Is the singular phenotype capable of invading into all its neighboring types? n Mutual invasibility Can a pair of neighboring phenotypes on either side of a singular one invade each other?

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

Invasion Fitness, Gradient, and Hessians n Invasion fitness n Selection gradient n Hessians of invasion fitness at ecological equilibrium at evolutionary equilibrium

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

Constraints on Second Derivatives for One-dimensional Traits n Constraints n Degrees of freedom 4 (second derivatives) minus 1+1 (constraints) minus 1 (scaling convention) gives 1 effective degree of freedom and

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

Constraints on Second Derivatives for Two-dimensional Traits n Constraints n Degrees of freedom 4x4 (matrix elements of second derivatives) minus (constraints) minus 1 (scaling convention) gives 6 effective degrees of freedom and

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

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

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

Alignment upon Evolutionary Branching for Two-dimensional Traits Consider the fitness landscape in the mutant direction as described by c T h mm c : Outgoing branches are expected to dynamically align on the direction of steepest upward curvature, that is, on the dominant eigenvector of c T h mm c. 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.

Conditions for Evolutionary Branching for Two-dimensional Traits n Monomorphic convergence n Invasibility n Mutual invasibility with n Alternative sufficient condition symmetric and negative definite andpositive definite

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

Summary n 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: u Matrices can generically be indefinite, while vanishing scalar second derivatives are non-generic. u The two mixed Hessians are not identical. u The mutational variance-covariance matrix can affect evolutionary outcomes. n Conditions for primary evolutionary branching can nevertheless be derived. n 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.