A new integrated measure of selection: when both demography and selection vary over time Carol Horvitz 1, Tim Coulson 2, Shripad Tuljapurkar 3, Douglas Schemske 4 1 University of Miami, Coral Gables, FL 2 Imperial College, Silwood Park, London, UK 3 Stanford University, Stanford, CA 4 Michigan State University, East Lansing, MI Theoretical Ecology Department Lund University, Lund, Sweden, 29 April 2008

Institute for Theoretical and Mathematical Ecology University of Miami Coral Gables, FL USA Mathematics Steve Cantrell Chris Cosner Shigui Ruan Biology Don De Angelis Carol Horvitz Matthew Potts Marine Science Jerry Ault Don Olson

floral tube length and birth date How can we integrate variable selection across years? *for structured populations and overlapping generations

Preview: Integrated selection on Calathea floral tube length El niño driven Stasis Tree-fall Dry season severity Environmental driver Selection _____________________________________________

Preview: Integrated selection on red deer birth date NAO driven yr cycle IID and equal Quality correlated Environmental driver Selection _____________________________________________

Preview: Integrated selection Environment-specific elasticity X Environment-specific selection gradient summed across all relevant life history and environmental paths Horvitz, Coulson, Tuljapurkar, Schemske (in prep)

a small tropical Mexican herb and a large Scottish mammal Fitness components and stochastic growth rate Selection gradients vary Demographic transitions vary Environmental states are dynamic Environmental driver matters

a small tropical Mexican herb and a large Scottish mammal Floral tube length (pollinator related) 3 yrs of selection gradients Fruit production Demographic projection matrices for 4 yrs Local, regional and global environmental dynamics Birth date (seasonal advantage) 26 yrs of selection gradients Recruitment and survival for two classes Demographic projection matrices for 26 yrs Local, regional and global environmental dynamics

Phenotypic selection theory Relative fitness regressed against quantitative trait value The slope of the regression = selection gradient for the trait (Lande and Arnold 1983 Evolution) fitness something quantitative

Candidate parameters for measuring fitness Fitness components Reproduction (stage-specific) Survival (stage-specific) Growth (stage-specific) Population growth rate

Candidate parameters for measuring fitness Fitness components Reproduction (stage-environment-specific) Survival (stage-environment-specific) Growth (stage-environment-specific)  Stochastic growth rate

Schemske and Horvitz 1989 Evolution ** Fitness component vs floral tube length (relative, mean-standardized) Standardized selection Years => Environments mature fruits

Years => Environments Fitness component vs birth date (relative, mean-standardized) Coulson et al Evolution

Demographic transitions and fitness in a constant world N(t+1) = A N(t) A is a population projection matrix Transitions and contributions between stages, a ij = fitness components λ = population growth rate

N(t+1) = X(t) N(t) X(t) is a random variable A 1, A 2, A 3 …A K, K environments Transitions and contributions in each environment, a ijβ λ S = stochastic growth rate Tuljapurkar 1982, 1990 Demographic transitions and fitness in a variable world

In a variable world : sequences, frequencies and new sensitivities Environmental dynamics sequences along sample paths an expected long run stationary distribution λ s is sensitive to perturbations of means, variances, and transitions in particular environmental states E δ, E μ, E β and others… Tuljapurkar et al Am Nat Horvitz et al Ecology

Environmental dynamics: Scaling up using climate data Calathea : “Dry season ” driver Red Deer: “NAO” driver

Sample years in context of historical record Monthly rainfall during the dry season only

Sample years in context of historical record Monthly rainfall during the dry season only Annual Deviations from Mean NAO Year, starting with 1864

Hypothetical environmental drivers markov chain models El niño Stasis Dry season Tree-falls

Hypothetical environmental drivers markov chain models Quality correlatedNAO IID and equal 26-yr cycle

Hypothetical environmental drivers markov chain models Quality correlatedNAO IID and equal 26-yr cycle

… … El Niño … … Dry season … … Tree-falls … … Stasis Sequences for hypothetical environmental drivers

… … NAO … … Quality correlated … … IID and equal … … 26-yr cycle

Environment-specific elasticity of λ s for each driver (to seed production) El niño Stasis Dry season Tree-falls Years => Environments Environment-specific Elasticity Stage class

Environment-specific elasticity of λ s with NAO driver Survival Recruitment Environment-specific Elasticity Age class Years => Environments NAO

Survival Recruitment Environment-specific Elasticity Age class Years => Environments Quality correlated Environment-specific elasticity of λ s for different driver…

Integrated elasticity stage-specific elasticity, e ij = change in λ due to a change in one element of the matrix X selection gradient = change in one element of the matrix due to a change in the trait value (van Tienderen 2000 Ecology, Coulson et al Evolution)

Integrated stochastic elasticity Integrated selection Environment-specific elasticity, e ijβ = change in λ S due to a change in one matrix element in one state of the environment X selection gradient = change in one matrix element in one state of the environment due to a change in the trait value Horvitz, Coulson, Tuljapurkar, Schemske (in prep)

Calathea Each matrix is 8x8 4 environments (let’s look at 1) Selection gradient only on top row All reproductive stages have same value

1 example (there are 4 per driver) “dry season” driver, envt × = (elementwise multiplication)

Red deer Each matrix is 20 x environments environment-specific elasticity males : zero females : recruitment and survival each age females are in 11 x 11 matrix, top left

1 example (there are 26 per driver) “NAO” driver, envt 5 × = (elementwise multiplication)

Integrated selection by environmental state and TOTAL El niño Total= Stasis Total Tree- falls Total= Dry season Total= Years => Environments

Integrated selection by environmental state and TOTAL 26-yr cycle Total = IID and equal Total= Quality correlated Total = NAO Total = Years => Environments

Note: These are ALL negative Integrated selection by transition rate

Integrated selection by stage and type

Conclusions New parameter Integrates selection across the life cycle and across changing environments Uses λ s and its sensitivities (by environment) The force of selection on a trait depends upon environmental dynamics Historical climate data combined with a few years of demographic observations: plausible long run patterns

Thanks Per Lundberg NSF OPUS , NSF 1982National Geographic NERC Royal Society Biotechnology and Biological Research Council Field assistants, students and colleagues

Extras for questions…

Sample years in context of historical record Monthly rainfall during the dry season only

Difference from the long-term mean, Years = Environments

Sample years in context of historical record Monthly rainfall during the dry season only Annual Deviations from Mean NAO Year, starting with 1864

Standardized annual deviations (observed/SD) of NAO Historical data

Flowers with trigger

Tongue length Floral tube length

zNext slides exemplify differences due to sequence once frequency is accounted for…

Elasticity of λ s to stage-specific reproduction for each environmental state, normalized for frequency Stasis Dry season Tree-falls Years => Environments El niño

Elasticity of λ s to the first 3 age-specific survivals for each environmental state, normalized for frequency 26 yr Cycle Quality correlated Iid and equal Years => Environments NAO

× = Example: “dry season” driver, envt (elementwise multiplication)

Years => Environments