Modeling Spatio-Temporal Dynamics of Taiga
Boreal Forest
Andrey Lyushnin and Dmitry Bratsun
Abstract The simple three variable evolutionary model of boreal forest of Perm
region has been proposed. The model is built as a complex system, where each population is represented by individual trees competing for solar light. Other factors
taking into account are growth rate, seed dispersal and mortality. The parameter values used in the model were calibrated from the information available for Perm forests.
This work has a fundamental aspect because a formation of dynamical macroscopic
patterns in ecological systems attracts great interest of researchers. In addition, the
proposed model can have many applications for more effective forest management.
Keywords Forest evolution · Cellular automata · Individual-based models · Spatiotemporal behaviour
1 Introduction
As it is known, the spontaneous formation of spatial and temporal structures is
the general law of functioning of complex ecosystems. Study of such structures is
one of the central problems of the modern ecology [1–3]. In plant communities a
formation of spatially ordered structures which are inhomogeneous in concentration
and qualitative composition of the biomass is also observed [3–5]. It should be noted
that these structures are cooperative in its nature and arise due to the interaction
at the level of individual plants. They should be distinguished from the structures
which originate due to the morphological reason or natural landscape factors [5].
As an example of such self-organization is a “tiger bush”, which is a patterned
A. Lyushnin (B) · D. Bratsun
Theoretical Physics Department, Perm State Pedagogical University, Perm, Russia
e-mail: [email protected]
D. Bratsun
e-mail: [email protected]
A. Sanayei et al. (eds.), ISCS 2013: Interdisciplinary Symposium on Complex Systems,
Emergence, Complexity and Computation 8, DOI: 10.1007/978-3-642-45438-7_24,
© Springer-Verlag Berlin Heidelberg 2014
245
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A. Lyushnin and D. Bratsun
vegetation community consisting of alternating bands of trees or shrubs, separated
by bare ground or low herb cover, that run roughly parallel to contour lines of equal
elevation [6]. This phenomenon can be met in the arid areas of Africa, Australia
and North America. In recent years a number of studies have shown that the regular
pattern formation is quite typical for plant communities. In particular, the similar
structures have been observed in Ural Mountains [7].
Historically, there are two approaches to the consideration of spatially distributed
systems of plants. The first approach is a phenomenological modeling. For example,
in the paper [8] it is proposed reaction-diffusion model of the dynamics of vegetative
plants, which describes the pattern formation even in homogeneous and isotropic
environments. It includes an equation for the concentration of biomass with diffusion
and specially chosen “reactive” term responsible for the plant–plant interaction. Thus,
the cause of pattern formation here is the collective interaction of plants.
Another direction in modeling of the forest evolution is an approach based on
cellular automata [9–13]. This approach has gone through several stages of development. One of early models was the JABOWA forest model [9] developed for northern
forests of New Hampshire. The purpose of JABOWA was to model succession in
mixed-species forests and thereby to predict species composition. It was based on the
notion that the interactions that drive forest dynamics are local. This model was later
modified and extended, for example in [10], to forests ranging from boreal regions
to the tropics (FORET model). The next advance was made in [11, 12]. The authors
suggested that the range of dynamic behavior that JABOWA-FORET models are
capable of exhibiting is too large, since some critical features of the models were
simply unknown at that time. They have proposed their own model SORTIE based on
that their submodels were designed simultaneously with maximum likelihood estimators necessary to estimate them from simple field measurements [11–13]. SORTIE
gave rise to a full pedigree of related models and relevant publications. Probably it
is one of the most successful forest simulation models ever developed.
Currently such models include, typically, the individual-oriented behavior. An
individual-based model is a class of computational models for simulating the actions
and interactions of autonomous individuals (agents) with a view to assessing their
effects on the system as a whole. This approach gives a number of advantages such
as transparency in relation to the objective biological mechanisms, the ability to
describe the system with a high degree of detail to extract more useful information
from the simulation results.
It should be noted that the developers of SORTIE, as biologists were intended
to describe the evolution of the forest community as accurately as possible. For
example, one of the recent variations of the model examines the dynamics of the 12
species [13]. The study of forest pattern formation as the fundamental feature of the
spatially distributed system was not in their plans. Therefore from this standpoint we
prefer the approach developed in [8]. Unfortunately the model of the forest pattern
formation proposed there is phenomenological by its nature.
The main goal of this work is to develop a simple, but still more or less realistic
individual-base model in order to study possible pattern formation. We propose a
three-component model of the evolution of the taiga boreal forest consisting of the
Modeling Spatio-Temporal Dynamics of Taiga Boreal Forest
247
main tree species of the Perm region and explore the features of its evolution and
pattern formation. Our model is based on the submodels of different versions of
SORTIE [11–14] but in a highly simplified form.
2 Three-Component Evolutionary Model of Boreal Forest
The vast majority of Russian forests are in the boreal zone. According to the forest
inventory data for the Perm region, the forest society consists of a fir (55 % of the
total biomass), a birch (26 %) and a pine (12 %). Influence of other trees is not so
significant. That is why we proposed a basic model, which is formulated only for
these edificators of the ecological system.
In deriving the model, we take into account the following factors:
• Sizes of plants. The tallest tree is a pine—up to 40 meters in height. Birch grows
to 20–25 m. Fir may grow up to 15–20 m.
• The fight for the light. Higher trees such as pine, have a competitive advantage.
Obscuring his crown young trees, they hamper their growth. However, the need
for sunlight in different species of trees is different. It levels the playing field
somewhat. A birch is the most sensitive to light, and a fir on the contrary likes to
grow in the shade. Thus, a fir easy grows in the shade of the birch, but when it
grows, it blocks the growth opportunities for the former. Pine is difficult to rise in
the shade, but if it has risen, because of its height it apart from the competition.
• The lifetime. Birch is the most short-lived—60–80 years. Fir and pine lives up to
250 and 400 years respectively.
• Distribution of seeds. Pine has the largest radius of the distribution, but the survival
rate of seeds is lowest.
The model consists of three populations of trees and landscape. For simplicity, we
consider the area of a square measuring 500 by 500 m. To compute the luminous flux
falling on a certain point the landscape, we introduce a uniform grid N × N , where
N in the most of the numerical experiments was 100. Populations consist of the sets
of individual plants, each of which lives its own life. Wood in the model consists of
two cylinders, one of which simulates the crown, and the other – the barrel. Each tree
has a number of important current values of its states, first of all this is the height of
the whole plant H and the thickness of its trunk D. These variables are tightly related
as follows:
H2
D
,
(1)
H = H1 1 − exp −
H1
where H1,2 are parameters depending on the type of population. The width and height
of the crown is also related to each other
R = C1 d a ,
(2)
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A. Lyushnin and D. Bratsun
where C1 and a are the parameters of the populations. Thus, each tree is characterized
by a single independent variable. Further, it is assumed that the plant evolves through
three stages: (i) seed, (ii) young tree and (iii) mature tree. Seed does not react on light
and cannot grow. However, it can take root at a certain place of landscape and become
a young tree. The difference between young and mature plants consists in the values
of the parameters that determine the rate of growth and the ability to insemination.
Each plant involved in evolution at each time step grows in accordance with its status,
makes insemination terrain, and may die suddenly.
The model assumes that the main resource for which competing trees, is sunlight.
That light is the reason for the growth of plants. The light beam irradiating on each
element of the landscape can meet the crown of the tree. The penetration rate λ is
the characteristic parameter for each population. If we now integrate over the whole
set of plants, one can get the amount of light per unit area of the landscape. After
finding the distribution of area illumination for each tree one can calculate its rate
of growth for this year. The relative increase in the size of the tree growing on the
element of landscape and receiving the light is determined by the Michaelis-Menton
law [12]:
g1 g2 Fi j
d,
(3)
G=
2(g1 + g2 Fi j )
where g1,2 are parameters of populations. Then we can calculate the size of the tree
in the next year:
(4)
Dt+1 = Dt + G,
The next step is the calculation of the death probability of the tree. The probability
depends on how fast this tree grows: for good growing trees it is less:
M = m 1 exp(−m 2 G),
(5)
where m 1 is the mortality rate at zero growth, m 2 is the mortality due to light. If
the tree dies, it is permanently deleted from the population [14]. If the tree dies, it is
permanently deleted from the population.
The second important process in the model is the production of seeds by mature
trees and seeds distribution across the landscape. The insemination is a long-range
interaction. Namely this process distributed in space forms a non-linear relationship
between the members of the population and contributes to the emergence of nonlocal structures. For the probabilistic description of the distribution of seeds, we have
used the Weibull distribution [11]:
Vi = γ
g Dk 2
k=1
30
3
.
exp −r1 Mik
(6)
Here Dk is the diameter of the trunk of parent trees, r1 is the population parameter,
and Mik is the distance from i to the k-th parent tree. In our model, we neglect
Modeling Spatio-Temporal Dynamics of Taiga Boreal Forest
249
Fig. 1 Evolution of three-component forest from non-uniform initial conditions
the anisotropy of the distribution of seeds (for example, due to the wind) and other
possible complicating factors.
The values of parameters calibrated on the basis of available forest inventory data
are given in the Table 1. The Java version of on-line forest simulator based on threecomponent model described above can be found on our website (http://urales.trajan.
ru/).
As an example, let us consider the spatio-temporal dynamics of forest simulated
within the model let us consider the evolution of the system from an initial state
in which the center of empty landscape there are several trees of different species.
Figure 1 shows the consistent frames of forest evolution after every 70 iterations
(70 years). Complete evolution of the system is equal to 1120 years. Each circle
represents a single tree, the width of the circle is proportional to the width of the
crown.
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A. Lyushnin and D. Bratsun
Table 1 Parameter values for the three-component model
Type of tree
λ
g1
g2
m1
m2
r1
Birch
Fir
Pine
0.4
0.064
0.4
0.4
0.15
0.18
0.05
0.15
0.019
0.5
0.077
0.268
2.0
6.0
4.0
3 × 10−4
6.9 × 10−4
1.03 × 10−5
It is clearly seen that the overgrown of the wasteland occurs due to the birch trees
(indicated by red) and pines (green). Firs spread slowly and is only after other species
of trees. However, finally the firs displace other trees. Analysis of the distribution of
the landscape illumination shows that in the central region occupied by the firs, the
light to the earth’s surface does not get. This leads to the fact that the seeds cannot
take root in this area and fir is slowly but surely expanding their habitat areolas. The
very young fir loves twilight, and her living conditions are comfortable.
3 Conclusions
The simple portable individual-based model representing a community of the three
populations of forest (fir-birch-pine) has been proposed. Modeling of forest growth,
based on the dynamics of individual trees that come into competition with each other
for solar energy has shown that the interaction at the micro level lead to the emergence
of spatially distributed macroscopic structures.
The work was supported by the Department of Science and Education of Perm
region (project C26/244) and Perm State Pedagogical University (project 031-F).
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