Ecological Modelling 143 (2001) 33 – 41
www.elsevier.com/locate/ecolmodel
A simple space competition model using stochastic and
episodic disturbance
Michael Liddel *
Uni6ersity of Maryland, College Park, 5100 Wickett Tr., Bethesda, MD 20814, USA
Abstract
Communities of organisms competing for homogeneous and limited resources such as space or light can appear to
coexist indefinitely. There have been several models that have explored the mechanisms by which this can occur.
(Armstrong, Sebens, Levins and Culver, Dayton and others). This paper extends the model developed by Armstrong
(Ecology 57 (1976) 953)) and Sebens (Theor. Pop. Biol. 32 (1987) 430)) to explore the effects of random variability
in disturbance rates and episodic disturbances on community structure. Results of this model show that the combined
effects of variable disturbances and episodic disturbances can lead to major changes in expected community structure.
These changes can include the persistence of one or more species that would normally be out-competed and major
competitive reversals in which the competitive dominant is excluded from the resource. © 2001 Elsevier Science B.V.
All rights reserved.
Keywords: Lotka– Volterra; STELLA; Space competition; Community structure; Stochastic disturbance
1. Introduction
Competition for limited resources has been
shown to be one of the primary structuring factors in many natural communities. In marine
hard-bottomed communities, the competition for
space to live is of great importance. Forest communities compete for better access to light. Other
communities may compete for specific types of
food or another resource. The structure of these
communities is often affected greatly by disturbances that remove individuals and free resources.
Species diversity is thought to be maximized at
intermediate rates of disturbance (Levin and
* Tel.: +1-410-326-7263.
E-mail address: [email protected] (M. Liddel).
Paine, 1974; Levin, 1976). At low disturbance
rates, the superior competitor (i.e. the species that
is best able to take resources away from other
species) will have the advantage. At high disturbance rates, there will be large amounts of resource freed, and the best colonizing or fugitive
species will have the advantage. Both types of
species are thought to be able to coexist at intermediate levels of disturbance (Armstrong, 1976;
Hastings, 1980). There have been several mathematical models that consider this type of interaction. Armstrong presents a two-species
Lotka –Volterra model modified for space competition. This model showed how a good colonizing
species could coexist with a superior competitor at
intermediate disturbance rates. Sebens (1987) extended Armstrong’s model to consider the effects
0304-3800/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 3 8 0 0 ( 0 1 ) 0 0 3 5 3 - 2
M. Liddel / Ecological Modelling 143 (2001) 33–41
34
of situations where there is not absolute competitive dominance and of situations where there are
more than two species interacting.
The analysis of these two models is limited to
solving the systems of equations to find combinations of parameter values (birth rates, disturbance
rates, etc.) where the system is in equilibrium.1
Furthermore, although competitive success was
allowed to be indeterminate in the Sebens model,
the competitive hierarchy was clear. This paper
extends the Sebens model to include the competitive interactions between nine theoretical species,
and examines the effects of stochastic and
episodic disturbances on the competitive interactions between these species. As a result of these
random elements, the system may not always be
in equilibrium. The goal is to examine how the
combination of major episodic disturbances and
variable general disturbance rates affect what
types of species will persist in a hypothetical
community.
2. The model
The model uses an equation for the population
of each species at time (t) that includes: a recruitment term based on the colonization ability of
each species, in terms of the population of the
other species and competitive ability; a density-independent mortality term representing disturbances and episodic events; and a competitive
mortality term based on competitive ability and
the population of other species. The equation for
each species is of the form:
dNi Bi Ni
=
{[K −Ni −(1 −ƒji )Nj
dt
K
− (1− ƒj + 1,i )Nj + 1 −…]} − DNi
1
Mitra et al. (1992) use a different model structure, and the
analysis is focused on finding equilibrium solutions, in a static
model. Donaldson and Nisbet (1999) use a stochastic Lotkka –
Volterra model, and a far more complex computer model.
They focused on predator –prey interactions and finding equilibrium solutions. Also see Abundo (1991), Neuhauser (1998),
Waniewski and Jedruch (1999) for additional modeling approaches.
−
n
Bjƒij
Bj+1ƒi, j+1
N i Nj −
Ni Nj+1 −…
K
K
for species i = 1“9 competing with species j=
1“ 9, where N is the population, B the birth rate,
K the carrying capacity, and ƒij the probability of
species j taking resources from species i. The
values used for the competition coefficients, ƒ, are
derived from a field study conducted by Turner
and Todd (1994) in which competitive interactions
among 16 species of encrusting bryozoans were
observed. Data form the nine species for which
there was the most complete interaction data is
used here. This competition data was mapped to
the theoretical competitors in this study in order
to have a representation of a possible and realistic
complex interaction structure. This paper is directed at studying the effects of stochastic disturbances on competition and community structure,
and is not intended to be a specific or accurate
model of the bryozoan community discussed in
Turner and Todd (1994) or any specific community for that matter. The competition data from
Turner and Todd was used simply to provide a
realistic idea of what a complex indeterminate
competitive structure could look like. Turner and
Todd (1994) was chosen as a source for this
information because it had the most complete and
best organized competition data available.2 Table
1 presents the competition coefficients used in this
study. The values indicate the probability of the
species by rows taking resources from the species
in columns.
In this set of species, species 1 is the competitive
dominant; however, its superiority is by no means
absolute. In general, species 1, species 2, and
species 3 are superior competitors; species 4 and
species 5 are intermediate competitors; and species
6, species 7, species 8, and species 9 are the
inferior competitors.
2
For completeness, the mapping of theoretical species used
here and the actual species used in Turner and Todd (1994) is
as follows: species 1, Escharoides coccinea; species 2, Membraniporella nitida; species 3, Schizoporella unicornis; species 4,
Alyconidium spp.; species 5, Cribrilina cryptooecium; species 6,
Callopora lineata; species 7, Callopora craticula; species 8,
Celleporella hyalina; species 9, Electra pilosa.
ƒ
P(species
P(species
P(species
P(species
P(species
P(species
P(species
P(species
P(species
1
2
2
3
4
5
6
7
8
takes
takes
takes
takes
takes
takes
takes
takes
takes
space
space
space
space
space
space
space
space
space
from
from
from
from
from
from
from
from
from
…)
…)
…)
…)
…)
…)
…)
…)
…)
Species 1
Species 2
Species 3
Species 4
Species 5
Species 6
Species 7
Species 8
X
0.29
0.75
0.14
0.06
0.12
0
0.05
0
0.55
X
0.33
0.43
0.36
0.28
0.33
0.1
0
0
0.5
X
0.75
0.14
0.26
0.18
0.5
0
0.67
0.36
0.25
X
0.43
0.15
0
0
0
0.72
0.45
0.79
0.36
X
0.14
0
0
0.14
0.8
0.7
0.61
0.67
0.7
X
0.33
0.28
0.14
0.87
0.17
0.53
0.65
0
0.33
X
0
0.33
0.85
0.76
0.5
1
1
0.66
0.8
X
0.32
Species 9
1
1
1
1
0.79
0.74
0.66
0.42
X
M. Liddel / Ecological Modelling 143 (2001) 33–41
Table 1
Competition coefficient values ƒ
35
36
M. Liddel / Ecological Modelling 143 (2001) 33–41
Since clear birth rate information for an appropriate set of species was not available, birth rates
were determined arbitrarily. In keeping with previous models, lower birth rates were given to
inferior competitors. As this study focuses on
disturbance, birth rates were held constant. However, the impact of changing birth rates is discussed briefly in the results.
This data was used to construct a STELLA™
(STELLA, 1996) model to examine the dynamics
of the interactions between these species. Fig. 1
presents the STELLA™ diagram of the general
model used.
Several treatments of this model were run, and
will be described in the following. The first two,
using no disturbance and a constant disturbance
rate, are primarily included for comparison purposes and to provide a baseline example of the
community structure. These first two show results
similar to what previous models have shown.
Other treatments will show that variability that
can result from
disturbances.
stochastic
and
episodic
2.1. Model with no disturbance
This treatment of the model shows the community structure that results from the indeterminate
competition between nine species. Fig. 2 shows
the results of this treatment. In this case, three
species coexist; two of the superior competitors,
species 1 and species 3, and one of the better
competitors of the fugitive species, species 7.
2.2. Constant disturbance rate
When a constant disturbance rate of 0.3 is used,
only two species are able to coexist. The competitive dominant, species 1 and species 7, a good
recruit but overall an inferior competitor. Altering
the constant disturbance rate will change the spe-
Fig. 1. Diagram of the STELLA model.
M. Liddel / Ecological Modelling 143 (2001) 33–41
37
Fig. 2. No disturbance model.
Fig. 3. Random disturbances model.
cies that are able to coexist. As was shown by
previous models, if the disturbance rate is low
then the species that are better competitors will
generally be those who are able to coexist and the
poorer competitors are excluded. At higher disturbance rates, the better recruiting species tend to
coexist, and the better competitors are excluded.
2.3. Model with 6ariable disturbance rates
In this treatment of the model, the disturbance
rate was determined randomly each time step. The
variation was normally distributed with a mean of
0.3 and a standard deviation of 0.15. This allowed
for some fairly substantial variation of the disturbance rate but kept it in the low to intermediate
range. Typically, these random effects allowed
several of the better recruiting species (usually
species 9 and species 6) to simply persist longer
than they would with out the variability. Fig. 3
shows the typical run of this treatment.
It should be noted that species 1 and species 7
are the only species that can be said to truly
coexist in this treatment, the same as in the treat-
M. Liddel / Ecological Modelling 143 (2001) 33–41
38
Table 2
Persistence of multiple species in a model with random disturbance rates
2.4. Random disturbance rates and major episodic
disturbances
Number of species
surviving
150 time steps
200 time steps
At least 2
At least 3
At least 4
100
17
7
100
3
2
This treatment of the model, as earlier, uses
normally distributed disturbance rates with a
mean of 0.3 and a standard deviation of 0.15.
Additionally, every 50 time-steps, the disturbance
rate becomes exceptionally high. At these times,
0.75 is added to the disturbance rate determined
by the random factor. In these trials, the structure
of the community was often dramatically
changed. Occurrences of more than two species
persisting were common, and there were also a
significant number of runs in which substantial
reversals in which species were dominating the
system. Table 3 presents the number of species
persisting in 100 runs of this model.
As is shown in Table 3, the structure of the
community is no longer completely deterministic.
The most common occurrence is still the situation
in which only two species survive, occurring 29 of
100 times. However, it is clear that wide variability from this standard is not only common, but
likely. Furthermore, in 19 of the 100 runs, major
reversals in spatial dominance occurred. In these
cases, a species that was neither species 1 nor
species 7 was the major holder of space.
Figs. 4 and 5 illustrate the types of persistence
and reversals that can occur in this model due to
major disturbance episodes.
Table 3
Persistence of multiple species in a model with episodic disturbances
Number of
surviving species
150 time steps
200 time steps
0
1
2
3
4
5
6
4
8
23
26
21
9
1
10
14
29
21
19
4
0
ment without the random variability in the disturbance rate. However, in many runs of the model,
more than the two species coexist for unusually
long times. Table 2 presents the results 100 runs
of the model showing how many species persist
for 150 or 200 time steps.
Fig. 4. Episodic disturbance model — persistence of three species.
M. Liddel / Ecological Modelling 143 (2001) 33–41
39
Fig. 5. Persistence of multiple species and reversal of dominance.
Fig. 6. Episodic disturbances only.
For purposes of comparison, Fig. 6 shows the
community structure if only the major episodic
disturbances are included and the normal disturbance is set at a constant rate of 0.3. In this case,
species 1 and species 7 reach an apparent equilibrium of sorts.
3. Impact of birth rates and carrying capacity
As a result of the arbitrary assignment of birth
rate, the models presented may be sensitive to
changes in this parameter in some respects. Variation in birth rates will affect which species in
particular will persist in the community but will
not affect the characteristics (competitive ability,
birth rate) of the species that are persisting in the
models. For example, in a scenario with nine
species at a constant intermediate disturbance
rate, there will be two to three species that reach
equilibrium. One will be a good competitor but a
poor recruit, and one will be a good recruit
existing as a fugitive species. There may also be an
intermediate species coexisting. A change in birth
rate is a change in the characteristics of the theoretical species. As such, varying birth rate will
affect which particular theoretical species has the
necessary characteristics to fill a particular role in
the community, but will not change the available
roles.
40
M. Liddel / Ecological Modelling 143 (2001) 33–41
Likewise, this model is structured such that
birth rates, mortality, and disturbances are all
proportional over population sizes. As such,
changes in carrying capacity should have little
impact on this model. However, it is important
to note that carrying capacity may play a role
in determining community structure, but this is
not explored in this model.
4. Conclusions
The processes that determine community
structure are many and varied. Competition between species for limited resources is a major
contributing factor. However, this competition
appears to be greatly influenced by the type of
disturbances that the community is subjected to.
In this paper, two main factors were explored
through the use of a simple space competition
model. These factors, random variation in disturbance rates and major episodic disturbances,
appear to have limited influence alone, but in
combination their effects can dramatically
change community structure.
In the nine-species bryozoan community example presented in the present paper, the clear
dominants were species 1 and species 7. In most
treatments, species 1 was able to survive because
of its superior competitive ability, whereas species 7 exists as a fugitive species. This is exactly
the type structure that was predicted by Armstrong (1976), Levin (1976), Sebens (1987), and
other workers. Randomly varying disturbance
rates alone appears to allow species that would
not normally survive to persist for substantially
longer times. In the long run, randomly varying
disturbance rates appears to have little effect. In
the end, the same species (species 1 and species
7 in this model) eventually out-compete the others and are able to exclusively control the resource.
When stochastic disturbance rates are combined with major episodic events, the results can
be quite different. In many cases, episodic disturbances allowed other species to coexist with
species 1 and 7, apparently indefinitely. Further-
more, in 19 of 100 trials, competitive reversals
occurred. In these situations, a species that
would be driven to extinction in the constant
disturbance or the non-episodic random disturbance models would not only be able to persist,
but would become the dominant species in the
community.
In the model presented in this paper, it appears that the community structure is determined primarily by the competitive interactions
between the species competing. This interaction
forms a baseline, and the most probable structure. This structure is then affected by the variability in the regular disturbances. This allows
the less competitive species to persist for a
longer amount of time. Finally, through this
variability, a less competitive species can work
its way into a position where it can exploit a
major disturbance event, and subsequently gain
control of the resource. Episodic disturbances
alone or low level variability alone are insufficient to allow substantial reversals or changes in
community structure. It appears that the combined effects of both these factors are needed.
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