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Modeling Population Dynamics with
Cellular Automata
Heiko Balzter, Paul Braun and Wolfgang ~ i i h l e r '
Abstract: When predicting changes in plant populations, spatial interactions must be considered. Cellular automata are models incorporating
both spatial and temporal dynamics. Vegetation data were sampled on a
meadow over three years using the point-quadrat method and population
dynamics of three single species were modeled with a cellular automaton
using stochastic decision rules. Links with the theory of Markov chains
are briefly discussed.
INTRODUCTION
Ecological science is structured in the three major disciplines autecology,
demecology and synecology. While autecology examines the relation between a
single organism and its environment, demecology looks at a whole population of
organisms and synecology investigates several populations of different species
interacting with the environment. In this sense our approach to modeling spatiotemporal population dynamics is an application of statistical methods to a synecological problem. Because there is still a lack of a complete theory of
succession (Huston 1994), the need for building vegetation models to simulate
changes and patterns becomes obvious. Cellular automata provide a class of
spatio-temporal models with a simple basic structure but offer a nearly unlimited
range of possibilities. Their value for modeling purposes is discussed, looking at a
vegetation study performed at the Agricultural School of Giessen University,
Germany, from 1993 till today.
CELLULAR AUTOMATA
A cellular automaton model consists of N elements, or cells, each capable of
several discrete values in a defined state space. The cells can be linked in different
ways. In the simplest case they are connected geometrically according to any
spatial order, such as in a one- or two-dimensional grid, in which the spatially
neighboring cells determine the state of each cell. But the cells may be connected
randomly, too, such as in neural networks (Kauffman 1984). Here, only simple,
Address: Justus-Liebig-University,Dep. of Biometry and Population Genetics, Ludwigstrasse 27, 0-35390 Giessen,
Federal Republic of Germany. Email: gha9Bagrar.uni-giessen.de
two-dimensional automata with spatially symmetrical neighborhood links are
considered. Neighborhood definitions often used are the Moore-neighborhood
consisting of the cell itself and the eight neighboring cells with at least one point
in common, and the von Neumann-neighborhood, that only contains the cell itself
and the four cells with at least one side in common. Each cell can change its state
during discrete time steps t, according to specified decision rules2, which can be
either deterministic or stochastic (Czarin & Bartha 1992).
For a step towards a theory of cellular automata see Wolfram (1984). Cellular
automata have a broad range of applications, for example to model chemical
reactions with spatial diffusion, the development of spiral galaxies, for phase
transitions, crystal growth, but also quite often to model biological and ecological
systems (Wolfram 1983, Molofsky 1994). A literature review concerning ecology
is given by Phipps (1992). Looking exclusively at the modeling approaches to
ecological systems one furthermore must emphasize the following publications.
Flamm & Turner (1994) modeled land cover with a stochastic cellular automaton integrating several data layers and thus increasing precision of the model
output. In vegetation science, Ratz (1994) made forecasts of the development of
spatial structures of boreal forests underlying the influence of fire. The effect of
fire on Banksia populations in Australia was also examined (Marsula & Ratz
1994). On a small scale, Winkler et al. (1994) modeled the dynamics of a dry
grassland community, observing a plot sized one square meter. A very important
study was performed by Silvertown et al. 1992, who modeled population
dynamics of five grass species, paying special attention to competition. They
extended the Markovian property to a cellular automaton model and obtained
results that a non-spatial model would not have been able to explain. A study of
the impacts of a coal-fired power plant to freshwater wetland in Wisconsin was
carried out by Ellison & Bedford (1995) with a quite sophisticated model. Using
rank data the model output agreed well with the observed changes caused by
hydrological impact. A major problem of cellular automata models is to formulate
adequate decision rules determining the state transitions of the cells. A good
solution to this is presented in Wiegand et al. (1994), who modeled five plant
species of a semi-arid shrub ecosystem in South Africa, translating biological and
ecological knowledge into decision rules depending on weather conditions. Their
major problem was the time scale of the model, as approximately every ten years
the seemingly stable vegetation abruptly changed, following a certain weather
event. However, it is possible to successively improve the decision rules by the
"top-down approach" as described by Kummer et al. (1994). Modeling the spread
of rabies they started with a simple, one-dimensional basic model and scaled
down in the investigation of the system by expanding the model spatially to two
dimensions with time steps of one year, and then temporally to finer time steps of
two months. Their experiences were satisfying, for it was possible this way to
gain heuristic insights in the dynamics of the rabies-fox-system.
sometimes called "transition rules" or simply "rules"
704
In a detailed discussion of cellular automata and ecological theory, Phipps
(1992) comes to the conclusion that, as far as application to natural systems is
concerned, probabilistic decision rules usually have a better analogy to the system
than deterministic ones, though sometimes their heuristic value is less.
Kareiva & Wennergren (1995) pointed out that a strength of this class of
models is that it is capable of modeling ecosystems with respect to spatial segregation, which often is a condition for coexistence of predator and prey
populations. Possibly cellular automata can contribute ideas to solving the
problem of competition and coexistence of plant species (Grace 1995).
MATERIAL AND METHODS
The meadow is located in Giessen, Germany. It
is regularly mown about ten times a year apart
from an area in its centre. The mown plant
community
was
classified
as
loliocynosure tum following from the phytosociological system of Braun-Blanquet (1964), whereas
the unrnown plant community had a quite
different composition. For reason of simplicity
this difference in use was ignored in our model.
The survey method used was the point-quadrat
Figure 1.- Point-quadrat
method. An extensive statistical study of pointframe as described by Kreeb
quadrat methods was performed by Goodall (1983) and used in this study.
(1952). The frame used in this study holds three
pins above the vegetation, and during sampling
each pin passes through a guide channel down to the ground and the number of
contacts of the pins to each plant species is counted (figure 1). In this manner a
grid of 10 parallel transects, each consisting of 12 frame positions was sampled,
resulting in 120 subplots. Methodological issues are discussed elsewhere (Balzter
et al. 1995). Data sets for Lolium perenne (perenniel ryegrass), Trifolium repens
(white clover) and Glechoma hederacea (ground ivy) from May 1993, June 1994
and June 1995 were used.
The 120 subplots on the meadow are used as the cells of the two-dimensional
cellular automaton. When looking at a certain plant species, it can either be absent
(=O) or present (=I) in each cell, resulting in a dualistic, discrete state space. First,
the transitions from 1993 to 1994 and from 1994 to 1995 were pooled. In order to
set adequate decision rules, two different approaches to stochastic probability
estimation were then carried out:
1) Under the assumption that only the state of the cell itself at the preceding
time step determines its actual state, a matrix of transition probabilities (transition
matrix) as known from the theory of Markov chains was estimated from the data.
'
Thus, the ,,neighborhood" incorporates only the subplot itself. We will refer to
this approach as ,,spatial Markov chaincc.
2) Assuming an influence of the spatially neighboring subplots on the subplot
itself, the Moore-neighborhood (the cell itself and the eight neighboring cells) was
used in the second set of decision rules. Transitions from any possible neighborhood into one of the two states must be considered. Because this would result in
36 different neighborhood states, they were classified, so that the transition matrix
is stochastic3.Obviously this classification has to be changed for frequent and rare
species (table 1). In the following text this approach is called ,,Mooreneighborhood".
Table 1.- Neighborhood classifications. Numbers mean neighboring cells with species
present. Percentage cover was smallest for Glechoma hederacea ,followed by Lolium perenne
and finally Trifolium repens.
neighborhood class
Lolium perenne
Trifolium repens
Glechoma
hederacea
Once the transition matrix was estimated, the distributions of cells with
absence or presence of the species at several time steps were predicted. During
each time step every cell randomly took a new state depending on the
corresponding probability distribution of the transition matrix.
RESULTS
The transition matrices for the spatial Markov chain estimated from the
vegetation data are given in table 2.
Table 2.- Transition matrices for the s ~ a t i aMarkov
l
chain and three ~ l a n s~ecies.
t
Lolium perenne
Trifolium repens
Glechoma hederacea
from
to
0
1
0
1
0
1
0 (absent)
I (present)
(::: :1"3
(:: ::::)
0.92 0.08
(062 0.3
8)
For the Moore-neighborhood the matrices (table 3) must not be confused with
the transition matrices of Markov processes, because the neighborhood state space
is not the same as the cell state space, expressed in the different numbers of rows
A matrix is called ,,stochastic ", I f i) all row sums equal one, ii) no element is less than zero or greater than one and iii)
at least one element in each column differsfiom zero.
and columns of the matrices. Because of this property is it far fiom easy to
interpret the matrices in the way we do with the spatial Markov chain.
Table 3.- Transition matrices for the Moore-neighborhood using five neighborhood
classes for the three plant species.
Lolium perenne
Trifolium repens
Glechoma hederacea
fiom neighborhood
class
to
0
1
0
1
0
1
Figure 2 shows the dynamics of Lolium perenne. The relative frequency of
cells with the species present quickly reaches a limiting distribution and the
following variation is caused by the limitation of the number of cells. If an infinite
number of objects were to pass the Markov chain, the limiting distribution would
be stable. Interested in the value of the limiting distribution we can calculate a
first estimate from the last 20 time steps t = 8 1,82,...,100. For Lolium perenne
this results in xLp = 0.50.
For Trifolium repens the predicted
frequencies show similar results, but
stabilize on a higher level (F, = 0.67,
figure 3) and Glechoma hederacea tends
= 0.12 (figure 4).
against
The theory of Markov chains allows
us to calculate the limiting distributions,
assuming that every subplot is changing
its state according to the Markovian
transition matrix. Because all three
transition matrices
of the
colonized by Lolium perenne over 100 time
Markov chains are aperiodic classes of steps (mean of 10 simulation runs)
positively recurrent states and the state predicted by the spatial Markov chain.
space is finite, the limiting distributions x:time; Y: relative frequencyp ( m ) are ergodic and can be determined
&y eq. 1 (Heller et al. 1978). Note, that the ergodic property means, that the
limiting distribution is independent of the initial distribution. In other words, the
plant population reaches the same equilibrium, wherever it starts. The results
(table 4) agree to the rough estimates from the data.
p(m) = lim P'
-
t+oo
707
9
Figure 3.- Relative frequency of cells
colonized by Trifolium repens over 100 time
steps (mean of 10 simulation runs)
predicted by the spatial Markov chain.
x:time; y: relative frequency.
Figure 4.- Relative frequency of cells
colonized by Glechoma hederacea over 100
time steps (mean of 10 simulation runs)
predicted by the spatial Markov chain.
x: time; y: relative frequency.
Table 4.- Ergodic distributions of the spatial Markov processes
Lolium perenne
Trifolium repens
Glechoma hederacea
0 (absent)
1 (present) 0 (absent)
1 (present) 0 (absent)
1 (present)
0.33
0.67
0.89
0.1 1
0.50
0.50
-
state:
rel. fieq.:
--
-
--
Using the second set of decision rules, under the assumption of the Mooreneighborhood determining the future state of a cell, the model still shows
asymptotical behavior. Figure 5(a) shows the results for Lolium perenne.
Surprisingly they do not differ from those of the spatial Markov chain, as the
estimated limiting distribution xLp = 0.50 is the same. To examine the model
behavior two further approaches were made. First the transition matrix was
applied to various initial distributions. All initial distributions tended to vary
randomly around the same mean after a specific number of time steps. Secondly,
the cell grid of the cellular automaton was expanded to 100 .lo0 = 10000 cells.
(4
(b)
Figure 5.- Relative frequency of cells colonized by Loljumperenne over 100 time steps as
predicted by the cellular automaton using the Moore-neighborhood. x:time; y: relative
frequency. (a) mean of 10 simulation runs; (b) one simulation run, 100x100 cells.
As expected the variation was decreased strongly by this expansion (figure
5(b)), while the mean of the limiting distribution is shown more clearly than using
120 cells.
Looking at Trifolium repens the results are similar (figure 6), but setting the
complete initial distribution to zero no plant can ever establish itself. The reason
for this is the transition matrix (table 3), where a neighborhood of all zero
produces a cell in state zero with probability poo = 1. The limiting distribution of
Trifolium repens is thus dependent on the initial distribution. Neglecting this
special case, because for practical purposes it is very unlikely that no single plant
of Trifolium repens would be found on the meadow, the estimated limiting distribution is ZTr = 0.72 differing slightly from the forecast of the spatial Markov
chain.
Figure 6.- Relative frequency of cells colonized by Trgoliurn repens over 100 time steps
predicted by the cellular automaton using the Moore-neighborhood. x:time; y: relative
frequency. (a) mean of 10 simulation runs; (b) one simulation run, 100x100 cells
A completely different result is obtained for Glechoma hederacea in figure 7.
Although the model behavior is still the same as for the other species, the value of
the limiting distribution IT,, = 0.66 is much greater than that given by the spatial
Markov chain. Which model comes closer to reality can hardly be judged, but will
be subject to further validation in the following years. Because Glechoma
hederacea does not typically cover two thirds of an area the spatial Markov chain
is expected to be more reliable, but this is only supposition. The importance of
selecting appropriate decision rules based on the right neighborhood definition
can be seen in this example. If the neighboring subplots do not have a major
impact on the species in the subplot, the Moore-neighborhood will be the wrong
modeling approach and a spatial Markov chain will be preferred. But, if the
species is able to colonize adjacent subplots by stolons or seed dispersal, the
Moore-neighborhood is likely to be preferable. Using stochastic decision rules
whose probabilities are estimated from the data has the major advantage that the
system behavior can be modeled without knowing exactly quantitative relationships between certain factors (which also would result in a quite complicated
deterministic model). The effects of all major factors influencing the development
of the population add up to a stochastic probability density function that can easily
be determined.
Figure 7.- Relative frequency of cells colonized by Glechoma hederacea over 100 time
steps predicted by the cellular automaton using the Moore-neighborhood. x:time; y: relative
frequency. (a) mean of 10 simulation runs; (b) one simulation run, 100x100 cells
DISCUSSION
The theory of cellular automata has not yet been satisfactorily investigated.
Despite the underlying simplicity of this class of models, cellular automata exhibit
numerous different behaviors. The most important work was done by Wolfram
(1984), but there is still a need for mathematical examinations. In our study the
model behavior can be explained partly using the theory of Markov chains. In
addition, the proved existence of a limiting distribution is important for the
assumptions of vegetation science. A limiting distribution is equivalent to the
concept of a climax of succession as proposed by Clements (1916). Whether there
is a climax of the observed vegetation changes or not shall not be discussed in
detail, because various conditions necessary for the maintenance of the successional path are likely to change before this question arises, e.g. the size of the rabbit
population. These changing conditions erect limits for the predictions of the cellular automaton model. 100 time steps are surely too many to interpret in a serious
way, this number was simply selected to examine the long-term model behavior.
But the model output described above probably gives a rough impression of the
expected development of cover and random variation of the plant species.
The two cellular automata presented here show differences in model output for
two of the three species. Because the only thing that was changed was the
definition of the stochastic decision rules, the differences in model output must be
due to assumed spatial interactions between adjacent subplots. The way in which
cells are connected thus plays an important role. Furthermore, the decision rules
used here are very simple: They cover only one single species and do not contain
information about biological conditions of the species, such as mean length of
stolons or distance of seed dispersal, which were included in the studies of
Wiegand et al. (1994) and Ellison & Bedford (1995). In fact, it is intended to
improve successively the decision rules of our model following the top-down
approach of Kummer et al. (1994) by using vegetation data of four time steps per
year, information about the seed bank in the soil, wind dispersal and interactions
between species (the latter was done for two species by Balzter et al. 1996).
However, a practical advantage of strongly simplified decision rules in the
model is that data sampling is not as laborious as if a lot of factors had to be
measured. This feature is even more important, if the difficulty of sampling spatial
data is considered. This may be one reason, why over a long time only a few
scientists have applied spatial data analyses to their problems, although in the
meantime the importance of both spatial and temporal structures and dynamics are
generally recognized. In the hture more spatial data will be usable, because
Geographic Information Systems (GIs) and large data bases are becoming more
and more established. Linking cellular automata with raster-based GIs seems to
be a promising approach. The great fieedom in setting up decision rules allows the
adaptation of a wide range of knowledge to the model and prediction of the
expected development under the condition that the assumptions made in the
model are true. This variety of possibilities might promote the use of cellular
automata in future ecology.
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