Integral projection models
Continuous variable determines
Survival
 Growth
 Reproduction

Easterling, Ellner and Dixon, 2000. Size-specific
elasticity: applying a new structured population
model. Ecology 81:694-708.
The state of the population
0.0 2.5
frequency
Stable distributions for n=50 and n=100
0
1
2
3
4
5
6
alue
size
Relative reproductive value for n=50 and n=1
Integral Projection Model
n( y, t  1)   [ p( x, y )  f(x,y)] n( x, t )dx
Integrate over all possible sizes
Number of
size y individuals
at time t+1
Number of
size x individuals
at time t
=
Babies of size y
made by size x individuals
Probability size x individuals
Will survive and become size y individuals
Integral Projection Model
n( y, t  1)   [k ( x, y )] n( x, t )dx
Integrate over all possible sizes
Number of
size y individuals
at time t+1
=
Number of
size x individuals
at time t
The kernel
(a non-negative surface representing
All possible transitions from size x to size y)
survival and growth functions
p( x, y)  s( x) g ( x, y)
s(x) is the probability that size x individual
survives
g(x,y) is the probability
that size x individuals who survive
grow to size y
survival
s(x) is the probability that size x individual
survives
logistic regression
check for nonlinearity
log(s( x) / 1  s( x))  a  bx
growth function
g(x,y) is the probability
that size x individuals who survive
grow to size y
mean
regression
check for nonlinearity
variance
growth function
0.35
probability density
0.3
0.25
0.2
0.15
0.1
0.05
0
-4
-2
0
2
4
6
8
size y, at time t+1
g ( x, y) 
1
2  ( x)
e
( y   ( x )) 2 / 2 ( x ) 2
Comparison to
Matrix Projection Model
Matrix Projection Model





Populations are structured
Discrete time model
Population divided into
discrete stages
Parameters are estimated
for each cell of the matrix:
many parameters needed
Parameters estimated by
counts of transitions
Integral Projection Model
• Populations are structured
• Discrete time model
• Population characterized by a
continuous distribution
• Parameters are estimated
statistically for relationships:
few parameters are needed
• Parameters estimated by
regression analysis
Comparison to
Matrix Projection Model
Matrix Projection Model



Integral Projection Model
Recruitment usually to a • Recruitment usually to
more than one stage
single stage
• Construction from
Construction from
combining
observed counts
• survival, growth and
fertility functions
into one integral kernel
Asymptotic growth rate
• Asymptotic growth rate
and structure
and structure
Comparison to
Matrix Projection Model
Matrix Projection Model

Analysis by matrix
methods
Integral Projection Model
• Analysis by numerical
integration of the kernel
• In practice: make a big
matrix with small
category ranges
• Analysis then by matrix
methods
Steps






read in the data
statistically fit the model components
combine the components to compute the
kernel
construct the "big matrix“
analyze the matrix
draw the surfaces