Seminar 3

Data requirements, limitations, and challenges:

Inverse modeling of seed and seedling dispersal

Likelihood Methods in Forest Ecology

October 9th – 20th , 2006

Approaches to Estimation of Seed and Seedling Dispersal

Functions

Direct sampling around isolated trees(David Greene)

Develop mechanistic models with directly measurable parameters (Ran Nathan)

Inverse modeling using likelihood methods and neighborhood models(Eric Ribbens, Jim Clark, and a rapidly growing community of practitioners…)

The questions…

What are the shapes of the dispersal functions? How does fecundity vary as a function of tree size? What other factors determine the spatial

distribution of seeds and seedlings around parent trees?- Wind direction (anisotropy)

- Secondary dispersal

- Density and distance - dependent seed predation and pathogens

- Substrate conditions

- Light levels

The basic approach: field methods

Map the distribution of potential parent trees within a stand

Sample the density of seeds or seedlings at mapped locations within the stand

Measure any additional features at the location of the seed traps or seedling quadrats

Poisson PDF

0.0

0.1

0.2

0.3

0 5 10 15 20 25 30

x

Pro

b(x

)

m = 2.5

m = 5

m = 10

The Probability Model

Observations consist of counts Assume the counts are either Poisson

or Negative Binomial distributed

Poisson PDF:

!)(

x

exprob

x

Where x = observed density (integer), and = predicted density (continuous)

Negative Binomial PDF

“shape” of the PDF controlled by both the expected mean (m) and a “shape” parameter (k)

As k varies, the distribution can vary from over- to under-dispersed (i.e. variance > or < mean)

nk

km

m

k

m

!n)k(

)nk()nX(p

1

This is the notation for the gamma function…

The basic “scientific” model

Seed rain at a given location is the sum of the input of N parent trees, with the input from any given tree a function of the:- Size (typically DBH) and

- Distance to the parent

)()1

i

N

ii distgsizef

2/m(# rain Seed

How does total seed production vary with tree size?

Common assumption: seed production is a function of DBH2 (following Ribbens et al. 1994)

30iDBH

STRi tree of production seed

where = 2, and STR = total standardized seed production of a 30 cm DBH tree

Is this a reasonable assumption?

Is it supported by either independent data or theory?

How does seed rain vary with distance from a parent tree?

Two basic classes of functions are commonly used*:

- Monotonically declining (negative exponential):

- Lognormal:

2)ln(

21

)(

0

b

i

i

X

Xdist

edistg

*See Greene et al. (2004), J. Ecol. for a discussion…

idistedistg

)(

One more trick…

Normalizing the dispersal function [g(dist)] so that STR is in meaningful units…

idistedistg

1)(

Where is the “arcwise” (i.e. 360o) integration of the dispersal function

2)ln(

21

1

0.30)

0

1

b

i

n

i

i

X

Xdist

edbh

STR /m(# rain seed 2

Lognormal form:

Exponential form:

i

n

i

i diste

dbhSTR

/m(# rain seed 2

1

0.30)

1

So, the basic scientific models…

The Scientific Models

0.0

0.5

1.0

1.5

2.0

2.5

0 10 20 30 40

Distance

Lognormal

Exponential (apha = 2)

Anisotropy: does direction matter?

ip angleXXX cos(00

• For the lognormal dispersal function:

• Incorporate effect of direction from source tree on modal dispersal distance1:

1Staelens, J., L. Nachtergale, S. Luyssaert, and N. Lust. 2003. A model of wind-influenced leaf litterfall in a mixed hardwood forest. Canadian Journal of Forest Research.

2)ln(

21

1

0.30)

0

1

b

i

n

i

i

X

Xdist

edbh

STR /m(# rain seed 2

Shape of the wind direction effect

When would this matter? (just to increase goodness of fit and improve parameter estimation?)

Potential Dataset Limitations

Censored data: not all parents are accounted for

Insufficient variation in predicted values: parents are too uniformly distributed

Two different populations treated as one: not all potential parents actually produce seeds

Lack of independence: spatial autocorrelation among nearby samples

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50

Distance From 50 cm Parent Tree (m)

See

dlin

g D

ensi

ty (

#/m

2) Theoretical

25 m radius

50 m radius

100 m radius

Effects of Search Radius

Theoretical 25 m radius 50 m radius 100 m radius

Likelihood -528.79 -493.59 -492.55r2 0.83 0.84 0.84slope 0.96 0.99 1.00mean dispersal 24.59 28.64 27.95 25.21STR 500 1449.51 487.26 462.77alpha 2 1.63 2.17 2.10X0 10 15.25 11.60 11.45Xb 0.6 0.50 0.59 0.56

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50

Distance from a 50 cm Parent Tree (m)

See

d R

ain

(#/m

2)

Theoretical

25 m radius

50 m radius

100 m radius

50 m edge

100 m edge

Sampling along the edge...

Theoretical 25 m radius 50 m edge

Likelihood -528.79 -560.23r2 0.83 0.71slope 0.96 1.04mean dispersal 24.59 28.64 23.95STR 500.00 1449.51 4384.73alpha 2.00 1.63 0.68X0 10.00 15.25 13.09Xb 0.60 0.50 0.52

What if all of the trees are uniformly spaced?

This produces relatively similar neighborhoods for all observations…

Random vs. strategic sampling…

Theoretical 10/ha 50/ha 100/ha 500/har2 0.14 0.25 0.65 0.45mean dispersal distance 36.83 24.51 30.99 39.93 29.39STR 1000 1551.22 1241.98 1723.11 1402.88alpha 2 1.47 1.81 1.53 1.62X0 15 14.01 16.30 12.41 15.20Xb 0.6 0.54 0.55 0.71 0.53

0.0

0.5

1.0

1.5

2.0

2.5

0 25 50 75

Distance from a 50 cm DBH tree

See

d r

ain

(#/

m2)

Theoretical

density = 10/ha

density = 50/ha

density = 100/ha

density = 500/ha

Goodness of Fit – Sites with Different Densities

Predicted Seed Rain

0 50 100 150 200 250 300

Obs

erve

d S

eed

Rai

n

0

50

100

150

200

250

300

10/ha50/ha100/ha500/ha

What if all of the trees are the same size?

Tradeoffs between STR and

30iDBH

STRi tree of production seed

0.0

1.0

2.0

3.0

4.0

5.0

0 5 10 15 20

X

Y

xy = a(a=2)

What if not all trees produce seed?

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50Distance From 30 cm Parent Tree (m)

See

dlin

g D

ensi

ty (

#/m

2) Theoretical - 50%reproductive

100 m radius

50 m radius

25 m radius

Theoretical 100 m radius 50 m radius 25 m radiusLikelihood -563.80 -564.24 -595.97r2 0.73 0.73 0.73slope 1.00 0.99 0.96MDD 24.59 25.95 31.83 28.36STR 500 444.21 479.13 1445.32alpha 2 2.17 2.27 1.65X0 10 10.70 10.14 15.38Xb 0.6 0.60 0.68 0.49

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 25 50 75

Distance from a 50 cm DBH Parent Tree (m)

See

d R

ain

(#/m

2)

Theoretical

Lognormal - 100 m

Weibull - 100 m

2Dt - 100 m

Theoretical Lognormal Weibull 2DtLikelihood -492.548 -500.289 -499.061r2 0.839 0.819 0.821slope 1.000 0.993 0.998mean dispersal 24.59 25.213 25.278STR 500 462.772 411.536 426.169alpha 2 2.103 2.218 2.183X0 10 11.454Xb 0.6 0.562

Can the data discriminate between different functions?

Beware of simplifying assumptions in your model...

Effect of alpha on fecundity

0

200

400

600

800

1000

0 25 50 75 100DBH (cm)

Tota

l See

d P

rod

uct

ion alpha = 0

alpha = 1

alpha = 2

alpha = 3

...0.30

)1

n

i

i2 dbh STR/m(# rain seed

= 3 = 2 = 1Estimated Theoretical Estimated Theoretical Estimated Theoretical

r2 0.96 0.84 0.83

STR 1056.04 1000 487.26 500 952.89 1000alpha 3.08 3 2.17 2 1.22 1

X0 10.71 10 11.60 10 11.65 10Xb 0.62 0.6 0.59 0.6 0.59 0.6

Parameter Estimation – Varying

What is the minimum size of a reproductive adult?

Most studies have arbitrarily assumed that all adults over a low minimum size (10 – 15 cm DBH) contribute seeds.

One approach – estimate the minimum (don’t assume it)

min

min

1

0

...0.30

)

dbhdbh

dbhhSTR if dbs

dbhs

i

i

n

i

i

if

/m(# rain seed 2

How could we determine the effective minimum reproductive size?

Parent size and seedling production in a Puerto Rican rainforest

Source: Uriarte et al. (2005) J. Ecology

Scaling reproductive output to tree size:

Maximum likelihood parameter estimates

Species min. size (cm) Casearia arborea 0.14 13.7Dacryodes excelsa 0.51 NAGuarea guidonia 2.06 48.13Inga laurina 2.38 16.39Manilkara bidentata 0.01 44.04Prestoea acuminata 0.15 13.89Schefflera morototoni 3.22 9.61Sloanea berteriana 1.70 11.06Tabebuia heterophylla 0.01 20.93

Source: Uriarte, M., C. D. Canham, J. Thompson, J. K. Zimmerman, and N. Brokaw.

2005. Seedling recruitment in a hurricane-driven tropical forest: light limitation, density-dependence and the spatial distribution of parent trees. Journal of Ecology 93:291-304.

Should there be an “intercept” in the model?

Allowing for long-distance dispersal via a “bath” term:

...0.30

)1

n

i

idbhSTRb /m(# rain seed 2

Where b is an average input of seeds even when there are no parents in the

neighborhood…

For seedlings: does light influence germination?

0 < M(GLI) < 1Lopt

Lhi = slope away from Lopt

Llo = slope to Lopt

...0.30

*)()1

n

i

idbhSTRGLIm /m(# rain seed 2

Light Availability

Light level (GLI)

0 10 20 30 40 50

Ligh

t m

ultip

lier

0.0

0.2

0.4

0.6

0.8

1.0

Casearia arboreaGuarea guidoniaManilkara bidentataPrestoea acuminataSchefflera morototoniInga laurinaTabebuia heterophylla

Is there evidence of density dependence in seedling

establishment?

Add yet another multiplier...

Seedlings.edPrexpcruitsReActual C

δ

0

0.2

0.4

0.6

0.8

1

1.2

Conspecific seedling density

DD

Eff

ect (

0-1)

Negative Conspecific Density Dependence

Seedling density (#/sq. meter?)

0 20 40 60 80

Sur

viva

l

0.0

0.2

0.4

0.6

0.8

1.0

Casearia arboreaDacryodes excelsaGuarea guidoniaManilkara bidentataPrestoea acuminataSchefflera morotoniTabebuia heterophylla

Spatial autocorrelation

Dealing with spatial autocorrelation among observations…

Remember - the formula for calculating log-likelihood assumes that observations are independent…

We have been conditioned to assume that two observations taken at locations close together are likely to be not independent (a legacy of Stuart Hurlbert)

Moran’s I and other indices of spatial autocorrelation

How do you determine whether this is true?

A critical distinction…

Remember – the issue is whether the residuals (the error terms in the probability model) are independent. NOT whether the raw observations are…If your scientific model “explains” why two nearby

observations have similar values, then the fact that they are similar is NOT evidence of lack of

independence*…

*despite assertions to the contrary in some papers on the subject

So, examine your residuals for spatial autocorrelation

Distance class (m)

Mor

an’s

I

Weinmannia racemosa

-0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

Total PredictedTotal ObservedResidual

A “best-case” species…

Examples from a study of seedling recruitment in a New Zealand rainforest (data from Elaine Wright)

Another species…

Nothofagus menziesii

-0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

Distance Class (m)

Mo

ran

's I

Total Predicted

Total Observed

residual

A worse case…

Distance class (m)

Mor

an’s

I

Causes and consequences of fine-scale spatial autocorrelation…

The causes are probably legion:- Many trees don’t produce seed in any given mast

year,

- Many factors can cluster input of seeds or survival of seedlings

The consequences are important but not fatal:- Generally very little bias in parameter estimates

themselves,

- But estimates of the variance of the parameters will be biased (low) Do the thought experiment or test this with real data – what would happen

if you duplicated some observations in the dataset and then redid the analysis?