Ecological factors shaping the genetic quality of seeds and

seedlings in forest trees.

A simulation study coupled with sensitivity analyses

Project BRG-Regeneration 2003-2005

Reproduction cycle in trees

ADULT TREES

SEEDLINGSdispersal then germination

SAPLINGS

SEEDS

dispersal

growth / mortality

Pollen Ovules

fecundation

Sexual allocation

Pseudo -cycle :

Evolution in space

And in demographic and genetic composition

growth / mortality

Experimental « calibration » of input factors: project BRG-Reneration, 6 species

Demographic et genetic evolutions in natural regenerationFrom seed… …..to sapling

Impact of : sur :

A) Stand structure (seed trees density) -> mating system, seed genetic quality (in situ) [1]

B) Temporal variation in fertility, phenology -> mating system, seed genetic quality (in situ + simulation)[5]

C) Seed G.Q. in controlled conditions -> phenotypic value of des saplings (ex situ : germination test in lab, nursery) [2]

D) Seed G.Q. in natural conditions -> demography (survival, growth) : installing sapling plots in forest (in situ) [3]E) Q. G. of natural regeneration -> demography (survival, growth) : monitoring natural regeneration in forest . (in situ) [4]

[1]

[5]

[2]ex situ

[3] in situ [4]

Simulation model (TranspopRege, under Capsis4)

Input and output variables

ADULT TREES

SAPLINGS

Growth / mortality

SEEDS

Pollen dispersal

fecundation

Pollen Ovules

Male versus female fertility

Density, spatial distributionPhenotypic diversityGenetic diversity and structure

Seed dispersal then germinationSEEDLINGS

Genetic quality:― Level of diversity (drift)― Spatial structure

OBJECTIVES

•How these different processes (adult stand characteristics), mating system, survival rate) respectively affect saplings genetic quality (factor screening)•How the way each process is modeled affects the output variable

• “The study of how the variation in the output of a model (numerical or otherwise) can be apportioned (qualitatively or quantitatively) to different sources of uncertainty in the model input” Andrea Saltelli, Sensitivity Analysis

• Originally, SA focuses on uncertainty in model inputs, then by extension to the very structure of the model (hypothesis, specification)

What is sensitivity analyses ?

Sensitivity analyses : Morris method

Screening the factors that mostly affect the variance of output variable (Y)

Economic method in terms of computation/simulation (# evaluations = a# parameters)

Identifying factor(s) that can be fixed without significant reduction in Y variance

Method presentation

• k input factors X

• Each factor Xi takes p values

• Variation space = grid kp

• Elementary effect of factor Xi :

)(,...,,,,...,

)( 111 xyxxxxxyxd kiii

i

=incremented ratio defined in a point x of the variation space

Property : the transformed point x+eiΔ also belongs to the variation space

Distribution of elementary effects associated to factor Xi = Fi

• # of elementary effects = )1(1 pppk

• Gi = distribution of absolute values of elementary effects (Campogolo et al. 2003)

k = 2p = 5Δ = 1/4

X1

X2

How to measure the sensitivity of Y to factor Xi (Fi, Gi)

• μ = mean of distribution Fi • μ* = mean of distribution Gi

• σ = standard deviation of distribution Fi

• High μ* value & low μ value large effect of factor Xi + effects of different signs according to the point in space where it is computed

• High σ value the values of elementary effect are greatly affected by the point in space where they are computed (strong interaction with other factors)

An exemple of graphical representation of Morris sensitivity measures

σ

μ*

Estimation of the distribution statistics (μ*, μ and σ )

• Problem = sampling r elementary effects associated to factor Xi

• # runs needed to obtain r values of each Fi, 1≤i ≤k : n=2rk ↔ économy = rk/2rk=1/2

Morris sampling method• B* = matrix k k+1, each row = input parameter set

so that k+1 runs allow estimating k elementary effects ↔ economy = k/k+1– Choice of p and Δ:

• p uniforme entre 0 et 1• Δ = p/[2(p-1)]

Morris sampling method

• Randomly select an input parameter set x*; each xi drawn randomly in {0,1/p-1, 2/p-1,…, 1}

• 1rst sampling point x(1) : obtained by incrementing one or more elements in x* by Δ

• 2d sampling point x(2) : obtained < x(1) so that x(2) ≠ x(1) only at its ith component (+/- Δ), i Є {1,2,..,k}

• 3rd sampling point x(3): so that x(3) ≠ x(2) only at its jth component (+/- Δ), j Є {1,2,..,k}

• … Two consecutive points differ only for one

component, and each component iof the base vector x* is selected at least once to be increased by Δ

Visualisation

)1(

)2(

)1(

...kx

x

x

Orientation matrix B*Example of trajectory for

k = 3

Estimation

• For a given trajectory, k+1 evaluation of the model, and each elementary effect associated with each factor ican be computed as : :

)()1(

)(ll

li

xyxyxd

)1()(

)(ll

li

xyxyxd

ou

• With r trajectories, one can estimate :

r

ji rd

1

/

r

ji rd

1

2 /

Implementation

11

01

00

B

Triangular matrix, (k+1)k, with two consecutive rows

differing only for one column But the elementary effects produced would not

be random

13/2

3/13/2

3/10

3/10

1

1

1

' BB

X*

Jk+1,1

1. Which orientation matrix B* ??

Consider a model with 2 input factors taking their values in {0, 1/3, 2/3, 1}; we have a k=2; p=4; Δ=2/3.

**22/3/10

1

1

1

* ,1,1 PJDJBB kkkk

X*

Jk+1,1

11

11

11

22

02

00

02

22

20

11

11

11

11

11

11

1. Which orientation matrix B* ??

11

11

11

10

01

Diaginal D matrix with either 1 or -1 randomly

2. Choice of p = number and value of the levels of the input factors

• If Xi follows a uniform law divide the interval of variation in equalsegments

• For any other distribution, select the levels in the quantiles of the distributions

• # of p-values ?– Linked to r : if r small, p high is of no use

– Simulation study show that p=4 and r=10 not bad

Implementation

Conclusion on Morris method

• Elementary effect are basically local sensitivity measures

• But through μ* & μ, Morris method can be seen as global

• Do not allow to separate the effects of interaction between factors from that of non linearity of the model.

Simulation model (TranspopRege, under Capsis4)

Adult stand

Input parameters in TranspopRege

Density (1P)

Spatial distribution : Neyman Scott (1P)

Mean and sd diameter (2P)

# locus, # allèles (1P)

Spatial genetic structure (1P)

Mating system

Growth, mortality

Input parameters in TranspopRege

1. Density/distribution of adult trees

Poisson, 100 trees, DBH = 50 cm, σ = 7 cm Neyman Scott, 100 arbres, 10 agrégats (~ 50 m) DBH = 50 cm, σ = 7 cm

Poisson, 100 trees, DBH = 50 cm, σ = 7 cm Poisson, 100 trees, DBH = 50 cm, σ = 14 cm

Input parameters in TranspopRege 2. Phenotype/Genotype of adult trees

Adult stand

Mating system

Growth, mortality

Input parameters in TranspopRege

Pollen dispersal type (panmixy/ibd = 1TP)

Mean distance and form of pollen dispersal function (2P)

Mean distance and form of seeds dispersal function(2P)

Male fecundity = f (diameter) (1P)

Female fecundity = f(diameter, year, individual) (3P)

Input parameters in TranspopRege 3. Panmixy/ isolation by distance

Random pollen dispersal

Adult under considerationMaternal progenyPaternal progenySelfed progeny

Dispersal folowing a gaussian law

b

a

yx

ba

byxba

22

exp )/2(Γ π2

),;,(2

b = 2 Normale b = 1 ExponentielleAutres b : Exponentielle puissance

b > 1 « light-tailed »b < 1 «fat-tailed»

Input parameters in TranspopRege 4. Pollen/seed dispersal function

Input parameters in TranspopRege 5. Fecundity = f(diameter)

Input parameters in TranspopRege 5. Fecundity = f(diameter)

Depends on tree growth model

Model with year effect : cones ~ A * (cir - 100)^0.25 - (2.8 * A + 25.7)+ stochastic variability

700

4571977873982210

58478227743

Ne=31

Ne=92Ne=76Ne=36Ne=57

Ne=59Ne=85Ne=83

Ne ~ (4N-2) / (V+2)

(Krouchi et al, 2004)

Input parameters in TranspopRege 6. Stochastic variations in female fecundity

(example : cedrus atlantica)

Input parameters in TranspopRege 7. Male fecundity vs female fecundity

Adult stand

Mating system

Growth, mortality

Input parameters in TranspopRege

Mortality = f(genotype, survival rate on plot) (2P)

Adult stand

Mating system

Growth, mortality

Input parameters in TranspopRege

4P

9P

2P

15 parametersr = 100 > 20 trajectories

1600 runs > 320 runs

Problems…solutions ?

• Script mode OK, but within simulation, out of memory errors

• Necessity to include routine for population genetics computation