Case Study 4:
Data requirements, limitations,
and challenges:
Inverse modeling of seed and
seedling dispersal
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
The Probability Model


Observations consist of counts
Assume the counts are either Poisson or Negative
Binomial distributed
0.3
m = 2.5
Poisson PDF
Poisson PDF:
prob( x ) 
x e 
x!
Prob(x)

m=5
0.2
m = 10
0.1
0.0
0
5
10
15
20
25
x
Where x = observed density (integer), and  = predicted
density (continuous)
30
Negative Binomial PDF

“shape” of the PDF controlled by both the
expected mean (m) and a “shape” parameter (k)
( k  n )  m 
p( X  n ) 
1  
( k )n! 
k
k
 m 


mk
n
This is the notation for
the gamma function…

As k varies, the distribution can vary from over- to
under-dispersed (i.e. variance > or < mean)
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
N
Seed rain (# /m )   f sizei g ( disti )
2
i 1
How does total seed production vary
with tree size?

Common assumption: seed production is a function of
DBH2 (following Ribbens et al. 1994)

DBH

i
seed production of tree i  STR 
 30 
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):
g (dist )  e
-
Lognormal:
 disti

 disti 
ln(
)

X0
1
 

2 X b 


g (disti )  e
*See Greene et al. (2004), J. Ecol. for a discussion…
2
One more trick…

Normalizing the dispersal function [g(dist)] so that STR
is in meaningful units…
g ( dist ) 
1

e
 disti

Where  is the “arcwise” (i.e. 360o) integration of
the dispersal function
So, the basic scientific models…
Lognormal form:

 dbhi  1
e


 30.0  
n
seed rain (# /m 2 )  STR 
i 1
 disti 
ln(
)

X0
1
 

2 X b 


Exponential form:
n
seed rain (#/m 2 )  STR 
i 1
1
dbh
 disti 

i 
e


 30.0  
2
The Scientific Models
2.5
Lognormal
2.0
Exponential (apha = 2)
1.5
1.0
0.5
0.0
0
10
20
Distance
30
40
Anisotropy: does direction matter?
•
 disti 
ln(
)
For the lognormal dispersal function:

X0
1
 

2 X b 
1
n


dbh


2
i
seed rain (# /m )  STR  
e

i 1  30.0  
2
• Incorporate effect of direction from source tree on modal dispersal
distance1:
X 0  X 0  X p cos( anglei   

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.
Shape of the wind direction effect
18
16

14
Xp
Xo
12
10
8
6
4
2
0
0
45
90
135
180
225
270
315
360
Angle
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
Beware of simplifying assumptions
in your model...
n
seed rain (# /m 2 )  STR 
i 1

 dbhi 
 ...

 30.0 
Total Seed Production
Effect of alpha on fecundity
1000
alpha
alpha
alpha
alpha
800
600
=0
=1
=2
=3
400
200
0
0
25
50
DBH (cm)
75
100
Parameter Estimation – Varying 
r2
STR
alpha
X0
Xb
Estimated
0.96
1056.04
3.08
10.71
0.62
=3
Theoretical Estimated
0.84
1000
3
10
0.6
487.26
2.17
11.60
0.59
= 2
Theoretical Estimated
0.83
500
2
10
0.6
952.89
1.22
11.65
0.59
=1
Theoretical
1000
1
10
0.6
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.
How could we determine the effective minimum
reproductive size?

One approach – estimate the minimum (don’t assume it)
n
seed rain (# /m 2 )  s 
i 1

 dbhi 

 ...
 30.0 
 STR if dbhi  dbhmin
s
0 if dbhi  dbhmin
Parent size and seedling production in a
Puerto Rican rainforest
Log (Seedlings produced)
20
15
Casearia arborea
Dacryodes excelsa
Guarea guidonia
Inga laurina
Manilkara bidentata
Prestoea acuminata
Schefflera morotoni
Sloanea berteriana
Tabebuia heterophylla
10
5
0
0
20
40
60
80
100
Parent tree DBH
Source: Uriarte et al. (2005) J. Ecology
120
140
Scaling reproductive output to tree size:
Maximum likelihood parameter estimates
Species
Casearia arborea
Dacryodes excelsa
Guarea guidonia
Inga laurina
Manilkara bidentata
Prestoea acuminata
Schefflera morototoni
Sloanea berteriana
Tabebuia heterophylla

0.14
0.51
2.06
2.38
0.01
0.15
3.22
1.70
0.01
min. size (cm)
13.7
NA
48.13
16.39
44.04
13.89
9.61
11.06
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:
n
seed rain (#/m 2 )  b  STR 
i 1

dbh

i 

 ...
 30.0 
Where b is an average input of seeds even when there
are no parents in the neighborhood…
For seedlings: does light influence
germination?
n
seed rain (#/m 2 )  m(GLI ) * STR 
i 1

 dbhi 

 ...
 30.0 




if GLI  Lopt , m(GLI )  1  Llo ( Lopt  GLI )

m( GLI )  if GLI  Lopt , m(GLI )  1

if GLI  Lopt , m(GLI )  1  Lhi ( GLI  Lopt )
1.0
0 < M(GLI) < 1
0  m(GLI)  1
Lopt
Llo = slope to Lopt
Light multiplier
0.8
0.6
0.4
0.2
Lhi = slope away from Lopt
0.0
0
10
20
30
Light level (GLI)
40
50
Light Availability
1.0
Light multiplier
0.8
Casearia arborea
Guarea guidonia
Manilkara bidentata
Prestoea acuminata
Schefflera morototoni
Inga laurina
Tabebuia heterophylla
0.6
0.4
0.2
0.0
0
10
20
30
Light level (GLI)
40
50
Is there evidence of density dependence
in seedling establishment?
Add yet another multiplier...
Actual Re cruits  exp
 C Pr ed .Seedlings
1.2
DD Effect (0-1)

1
0.8
0.6
0.4
0.2
0
Conspecific seedling density
δ
Negative Conspecific Density Dependence
1.0
Casearia arborea
Dacryodes excelsa
Guarea guidonia
Manilkara bidentata
Prestoea acuminata
Schefflera morotoni
Tabebuia heterophylla
Survival
0.8
0.6
0.4
0.2
0.0
0
20
40
60
Seedling density (#/sq. meter?)
80
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)
How do you determine whether this is true?

Moran’s I and other indices of spatial autocorrelation
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

A “best-case” species…
1
Weinmannia racemosa
Moran’s I
0.8
Total Predicted
Total Observed
Residual
0.6
0.4
0.2
0
0
10
20
30
40
50
-0.2
Distance class (m)
Examples from a study of seedling recruitment in a New Zealand rainforest (data from Elaine Wright)
Another species…
A worse case…
1
Nothofagus menziesii
0.8
Moran's I I
Moran’s

Total Predicted
Total Observed
residual
0.6
0.4
0.2
0
0
10
20
30
-0.2
Distance Class (m)
Distance
class (m)
40
50
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?