Succession Model
• We talked about the facilitation, inhibition,
and tolerance models, but all of these were
verbal descriptions
• Today we are going to consider a
mathematical model that describes those
processes more precisely and generates
predictions that can be tested in the real
world
Succession Model
• To do this, we will develop a matrix model of
ecological succession that describes in general
terms, the ways in which communities change
from one “state” to another through time
• We can also set of the parameters of the
model to describe the processes of facilitation,
inhibition, and tolerance
Succession Model
• We will a Markov model to predict changes in
populations and communities
• Initially we need to define a number of
mutually exclusive stages that represent
different, discrete communities
• These stages may represent entire sets of
species (e.g. algal mats, encrusting sponges),
or they could even represent individuals or
stands of a single species (e.g. red oak)
Succession Model
• When organizing and
classifying communities
into discrete stages, it
may be necessary to
make some decisions
about stages
Succession Model
• When choosing the stages for the model
implicitly sets the spatial scale of the patch
• For example, if the stages represent individual
species, then a single patch must not be so
large as to hold more than one species at a
time
• In addition, there may need to be a open
space stage (following a disturbance)
Succession Model
• Remember, all stages must be independent
and in your model, all possible stages must be
accounted for
Succession Model
• Specifying the Time Step
• Once stages are established, the investigator
must specify the time step of the model
• We will use a discrete time step (hence will
not need to use differential equations)
• This may be years, decades, or days/weeks
Succession Model
• Constructing the Stage Vector
• Suppose we have n possible stages for a patch
and the landscape consists of a large number
of such patches
• We can then create a stage vector that tells us
the number of patches in each of the stages
Succession Model
• Consider a landscape with 4 patch types: open
space, grassland, shrub, and forest
• If we censused 500 patches we might have the
following: s(t) = [250,100,80,70]
• (t) indicates the stage vector at time t
• 0’s are possible as some stages may not be
represented
• We can now predict changes through time
Succession Model
• Constructing the Transition Matrix
• The transition matrix is slightly different than
some other matrices (e.g. Leslie) in both the
possible values entered and the interpretation
• If there are n stages in our model, the
transition matrix A will be square, with n rows
and n columns
Succession Model
• Each column of matrix represents the patch
state at the current time (t), and each row
represents the patch state at the next time
step (t+1). The entries are the transition
probabilities for change from the current state
(column) to the next stage (row)
Succession Model
• Example of a typical matrix. The probability of
moving from grassland (column) to the open
state (row) is 0.23 and the probability of
moving from forest to grassland is 0.10
Succession Model
• The stage transitions are not necessarily
symmetric. For example, although the
probability of moving from grassland to the
open state is 0.23, the probability of moving
from the open state to grassland is 0.15.
• The diagonals of the matrix indicate the
probability that a patch remains in its current
state and does not change
Succession Model
• Probabilities of remaining in the same state
are represented on the diagonals
Succession Model
• This example points out some of the general
properties of transition matrices for
succession models.
• First, notice that all of the entries are positive
numbers between 0-1.0 (thus a probability).
• Second, all values sum to 1.0 (we have
included all possible states)
Succession Model
• Loop Diagrams
• The transition matrix can be represented by a
loop diagram (a circle representing each stage)
• Use a one-headed arrow to connect two
stages
Succession Model
• Loop diagram for the transition matrix
Succession Model
• Projecting Community Change
• The transition matrix summarizes all of the
information on how patches change from one
state to another
• The matrix is a set of probabilistic ‘rules’ that
determine the patterns of succession that will
occur from any possible starting point
Succession Model
• Using vector s at time t, we can use the
transition matrix (A) to determine the number
of patches in each state at the next time step
(t+1)
s(t+1) = As(t)
• In this example, we had 100 patches in
grassland at time t. How many will occur at
time t+1?
Succession Model
• There are four ways in which grassland
patches can appear at time t+1: there can be
transitions from open space, shrub, or forest
and some grassland patches will remain
grassland patches
GP (t+1) = (0.15)(250) + (0.70)(100) +
(0.25)(80) + (0.10)(70) = 134.5
Succession Model
e.g. Grass Patches (t+1) = (0.15)(250) +
(0.70)(100) + (0.25)(80) + (0.10)(70) = 134.5
Succession Model
Remaining patches: OP(t+1)=(0.65)(250) + (0.23)(100) +
(0.25)(80) + (0.40)(70) = 233.5
SP(t+1)=0(250)+(0.07)(100)+(0.25)(80)+(0.15)(0.40)(70) = 37.5
FP(t+1)=(0.2)(250)+0(100)+(0.25)(80)+(0.35)(70) = 94.5
Succession Model
• So we started with the vector:
s(0) = [250, 100, 80, 70]
and after one time step, ended up with:
s(1) = [233.5, 134.5, 37.5, 94.5]
• Although the patch states changed, the total
number of patches remains the same (500)
• You can repeat and calculate t+2…
Succession Model
• Determining the Equilibrium
• Even though the content of the stage vector
keeps changing, remember that the transition
matrix A remains the same throughout the
process. Consequently, in relatively short time
the stage vector reaches an equilibrium state
s(t) = [223.03, 164.7, 31.52, 80.75]
Succession Model
• Once this equilibrium has been reached, there
will be no further change in these numbers
• Interestingly, the same equilibrium vector is
reached irrespective of initial patch
distribution
Succession Model
• Successional trajectories for two initial patch
vectors: a) [250 100 80 70] & [500 0 0 0]
Succession Model
• Pg 193
Succession Model
• Model Variations: Facilitation
• We can utilize strategically placed 0’s in the
matrix, which ensures that patches move
through the sequence in orderly fashion
Succession Model
• Idealized loop diagram for the facilitation
model
Succession Model
• Model Variations: Inhibition
• In this model, we can assume that each of the
3 community states (grass, shrub, forest) can
be replaced by another community state only
through an intervening disturbance that frees
up space
Succession Model
• Consequently, stages can only remain as they
are or go to the open stage (and then
transition into any one of the others)
Succession Model
• Idealized loop diagram for the inhibition
model
Succession Model
• Model Variations: Tolerance
• For the tolerance model, assume that all
states, including the open state, are equally
likely. This generates a transition matrix with
identical transition elements.
• Each community neither inhibits nor facilitates
replacement by other communities (therefore
equally likely)
Succession Model
• Notice the equal likelihood in the tolerance
model
Succession Model
• Idealized loop diagram for the tolerance
model
Model Comparison
• Patch trajectories for
simple successional model.
In each model the initial
vector is s(0) = [1000 0 0 0]
• The inhibition and
tolerance matrices settle
into their equilbria after a
single time step (facilitation
takes ~20 steps)
Succession Model
• Empirical Examples: desert vegetation
• Can get a good measure of the dynamic
nature of desert vegetation (the slow decay
allows for relatively good estimate of mortality
and patch transition)
Succession Model
• This matrix is not clearly any of the idealized
matrices we have discussed thus far (e.g.
Larrea rarely colonizes open space and the
seedlings of Larrea are almost always found
beneath Ambrosia.
Succession Model
• This is a type of facilitation, although it does
not lead to orderly species replacement as in
the classic facilitation model (plus it takes a
very long time between steps, a gradual
replacement)
• The diagonals are close to 1.0. So what?
Succession Model
• It turns out the observed matrix matches the
equilibrium stage vector relatively well
• There is an underrepresentation of Larrea
(2.8%) compared to the predicted (9.9%),
perhaps due to density-dependent mortality
Succession Model
• Observed and expected frequencies of patch
states
Succession Model
• This model can be used to forecast how this
desert community may respond to a human
disturbance
Succession Model
• Following severe human disturbance, it takes
a very long time (~2000 yrs) to settle into
equilibrium