David Claessen CERES-‐ERTI & Labo « Ecologie & Evolution » UMR 7625 CNRS-‐UPMC-‐ENS
Evolu&onary problems � How do life history traits, behaviour, and other ecological traits evolve?
� How can we understand observed characteristics of organisms?
� How can we predict these traits? � Evolutionary traits, e.g.:
� Age or size at maturation � Number of eggs per clutch � Size of eggs � Semelparous vs iteroparous reproduction � Energy allocation (growth – reproduction – survival) � Dispersal rate, consumption rate, death rate , …
Evolu&on of individual traits � Traits:
� Life history, behaviour, exploitation strategies
� Two contrasting approaches 1. « Optimization principle »
� Life history theory � Optimal foraging theory
2. « Game theory » � Adaptive dynamics
� …that differ in important respects: � How to take into account the (impact of adaptation on) the environment
� How to define fitness
Simplifica&on � If we ignore feedback, we can simplify the problem of « predicting » evolution
� Use the method of « Optimization » � Optimization principle
� Find the « optimal » strategy, which maximizes « fitness » � Classic refs:
� Krebs and David (1993) � Stearns (1992) � Roff (1992)
� But: � How to define fitness? � How valid is this assumption? (see later)
Example: life history traits
From: Mayhem (2006)
From: Mayhem (2006)
Ques&on Timing and extent of reproduction spread throughout an organism’s lifetime
� Example: # offspring produced � Mosquito, perch: >10,000 to >1,000,000 (per season) � Elephants, humans: 1, 2, 3 per lifetime
� Example: timing of reproduction � Salmon: once (then die) � Perch: each year
� Are these differences adaptations to different environmental conditions?
Example: guppies (Poecilia re*culata)
� Live bearing fish � Coastal regions � Sexual maturity in < 3 months � Litters at 3-‐4 week interval � Sexual dimorphism
� Males smaller than females
Guppy distribution
Two habitats “high predation” “low predation”
Predator = killifish Predator = pike cichlid High mortality Low mortality
Are these differences adaptations to different
environmental conditions?
� Life cycle � Population structure � Matrix population models � Population growth rate / fitness
� Life history � Evolution of life histories
Life cycle � Life cycle
� “A series of stages through which an organism passes between recurrences of a primary stage”
� “The course of developmental changes in an organism from fertilized zygote to maturity when another zygote can be produced”
� Life cycle graph (Caswell 2001) 1. Set of stages 2. Projection interval 3. Create a node for each stage, number 1 to s 4. Arcs between nodes (contributions, transitions) 5. Label each arc by a coefficient
Structured popula&ons � Variation between individuals
� Age � Body size � Sex � Location in space � Genotype � etc…
� Population is structured by one or more of these i-‐state variables
� Population structure, for example: � Size distribution � Spatial distribution
“ i-‐state variables ”
Dynamics of structured popula&ons � Matrix-‐vector multiplication � Population growth rate � Eigenvalue � Sensitivity � Elasticity � Euler-‐Lotka equation
Matrix model
Michael Bulmer (1994) Theoretical evolutionary ecology
Year-‐to-‐year dynamics
For x>1 (thus excluding age-‐1)
For age-‐1 only
The same equations, but written in matrix form (a vector-‐matrix product)
L is the transition matrix
(Leslie matrix)
From life table to transi&on matrix Life table:
=Age-‐classified model
(stage classified model; similar analysis, see guppy model)
Asympto&c behaviour Now look at population dynamics when t→∞
(Long term dynamics)
The vector-‐matrix product Analogous to simple exponential growth
The dynamics converge to exponential growth, with growth rate λ λ = dominant eigenvalue of matrix L e1= corresponding eigenvector
Euler-‐Lotka equa&on Assume pop is in stable age distribution
Survival to age x
# newborn offspring in year t
Females of age x have survived since t-‐x
Each age class increases at rate λ
Stable age distribution!
“Euler-‐Lotka equation”
Life &me reproduc&ve success
Discrete time
Continuous time
Euler-Lotka equation Equilibrium
NB. In continuous time: -‐ stable age distribution -‐ mx is the birth rate
R=1 → r=0
R=1 → λ=1
“LTR”
Stable age distribu&on (again)
� λ=1 � λ>1 � λ>1
Distribution biased toward younger age classes
Distribution biased toward older age classes
Distribution proportional to survival curve
Evolu&on of age at matura&on � Why postpone reproduction?
� (Why are there juveniles?)
§ Cost of reproduction § Survival § Growth
§ Delaying reproduction can be advantageous if fecundity increases with age § Bigger → higher fitness
Evolu&on of age at matura&on � Model:
� Juveniles invest all energy in growth and maintenance � Von Bertalanffy growth curve
� Adults do not grow but invest all surplus energy in reproduction
� Fecundity increases with body size � Proportional to body mass L3
t= Optimal age at maturation
k = “growth rate” M = mortality rate
Evolu&on of age at matura&on Optimal age at maturation
Estimates for 30 fish species
k, M, t
The model predicts
higher mortality (M)
→ earlier maturation (t)
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