Qualitative Analysis of Forest Dynamics Models Using Elliptic Functions
Professor Mikhail J. Antonovsky
“…Mathematics gives the deep description of the Nature, but any attempt to express the Nature,
based upon philosophical Principe or an intuitive mechanical analogy never brings to the
seriously scientific results.”
Paul Dirac,
Nobel Price on Physics, 1933
The mathematical approach to description and investigation of the biological and ecological
phenomena has attracted the attention of a large number of researchers in the XX century. E.
Wittecker, one of the outstanding mathematician of the Great Britain of the XX century, has
written on the survey of the works of Vitto Volterra, one of the most famous scientist of the XX
century, the President of the Accad Lincei: “Biologists have criticized Volterra for his extremely
careful workout abstract mathematical models, that are far off complexity of the Nature. Now
this principle is a main-on which is based on the triumph of physical sciences”. Lately several
influential physical phenomena were discovered on the result of investigation of corresponding
mathematical models.
In some sense the continuation of the mathematical approach to biological and ecological
problems is the research of Robert May, President of the Royal Society (2000-2005), see, for
example, his book “Theoretical Ecology: Principles and Applications”.
Another example is the article of M.Ya. Antonovsky et al “Forest – Pest Interaction Dynamics:
The simplest Mathematical Models”, Theoretical Population biology, vol. 37, no. 2 (1990), 343367 pp. I have got almost 600 letters from all continents and from almost all Universities of the
USA.
In this Report I would like to introduce (apply) another mathematical device to the analogue
problems of forest dynamics. From one side it establishes the connections with the models of
classical and quantum mechanics through the equations of the type of Korteveg de Vries. From
the other side we are convinced that some of the models of forest dynamics are of Hamiltonian
one that permits us to apply the methods of classical mechanics.
In an assumption that the solution of model has the shape of a running wave we have shown that
explorated model is equivalent to the ordinary non-linear differential equation that has exact
analytical solution. This solution is written in terms of the elliptic Weirstrass function with the
parameters of initial models. It in their turns permits us to find effects of periodicity and quasiperiodicity (see Appendix 3). Let us stress that far from every model is a Hamiltonian one, it is a
special property (the same rare enough that the equation has exact analytical solution).
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Picturesquely saying the Hamiltonian is a way. But how the “car” moving on road – it is
uniformization. Mathematics tells about the possibility, the selection of necessary is turn of
forester. The result of Weirstrass is that elliptic curves have such uniformization. When we found
such one to one presentation we can recognize the hidden periodicity and quasi-periodicity,
which could not see on phase plane.
These mathematical models of Forest Dynamics could be considered as the tools for monitoring
of the latent phenomena in Forest Dynamics. First we found the effects mathematically and then
– in reality, creating appropriate notions.
Let us consider the Frame of our Report
Qualitative Description of the Forest Dynamics
Models of the Forest Dynamics ≡ the system of differential
equations
Analytical
methods
A
Geometrical methods
Ordinary non-linear
differential equation
Phase portraits, parametric
space and space of
dynamical variable
Hamiltonian Nature of
the system
Synthesis of two methods
and their amplification
Construction of a general
analytical solution in the
terms of elliptic
functions
The computer simulations
Increase of knowledge. New questions, hypothesis to ecologists,
forest scientists. Program for investigation. Application to Forest
Service.
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In the beginning we shortly describe each block of this scheme and then we present the results on
the base of this scheme.
First of all, we cite some of our works which we use in this Report:
1) M.Ya. Antonovsky et al.: “Forest-Pest Interaction Dynamics: The Simplest Mathematical
Models”, Theoretical Population biology, Vol. 37, No. 2 (1990), 343-367 pp.
2) M.Ya. Antonovsky: “Impact of the factors of the environment on the dynamics of populations
(mathematical models). In: “Comprehensive Analysis of the Environment”. Proc. SovietAmerican Symp. Tbilisi, USSR, 1974, 218-230 pp., Leningrad, Hydromet, 1975.
3) M.Ya. Antonovsky, E.A. Aponina, Yu.A. Kuznetsov: “On the stability analysis of the
standing forest boundary”. WP-91-010, Laxenburg, Austria, International Institute for Applied
Systems Analysis, 1991.
4) Yu.A. Kuznetsov, M.Ya. Antonovsky et al: “A Cross-diffusion model of forest boundary
dynamics”, Journal of Mathematical Biology (1994), 32, 219-232.
5) M.Ya. Antonovsky, V.M. Buchstaber, J.Yu. Bunkova: „Wave processes in Forest Ecosystem“
(in publication, ДАН, 2012)
6) M.Ya. Antonovsky, M.D. Korzukhin, V.K. Mackjavichus: “Periodically behavior of agedistributed population of trees”, WP-89-012, Laxenburg, Austria, International Institute for
Applied Systems Analysis, 1989.
7) M.Ya. Antonovsky, M.D. Korzukhin: “Mathematical modeling of economic and ecologicaleconomic processes” in: Integrated global monitoring of Environmental Pollution. Proceedings,
2nd International symposium, Tbilisi, USSR, 1981, pp. 353-358, Hydromet, 1983.
8) R.A. Fleming, M.Ya. Antonovsky, Yu.A. Kuznetsov, “The Response of the Balsam Fir Forest
to a Spruce budworm Invasion…”, WP-87-71, International Institute for Applied Systems
Analysis, Laxenburg, Austria (1987).
Let us remember that the goal of this Report is to show that the study of mathematical solutions
of corresponding formalized forest models which we offer to the professional forest people could
be interpreted on real phenomena in real forest ecosystems. As it is well known, in the forest
ecosystems permanently proceed the physical, chemical and biological processes. For example,
age waves on ecosystems arise due to concurrence for the light, for the water (roots and crowns),
for minimal elements of soil and so on. Moreover the forest fires, ageing, transition in categories
“to be ready” to be burned. For example, at Boreal belt each lot of forest burned down with big
probability once at 120 years. For example, in 2010 and 2012 were burned several millions of ha
of high-quality forest. Beside, arise outburst of vermin, illness and other internal causes of
disappearing of some kind of trees. The waves of heat and waves of cold hit on mechanism of
productivity and so on. Moreover, the year cycle also create some type of wave motions. Thus,
phase space of boreal belt is some tor in which there exist periodically and quasi-periodically
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motions with different frequents and amplitudes, including solutions, for example, age groups. In
fact, as we have stressed, if one short period (1 year) of changing physiological processes in
trees, the other, like successions, and even their abrupt, fire and so on, create large circle of tor.
Mathematical findings could give to forester the possibility to think if it is possible to organize
the forest ecosystem move to such states from which it can go to chaos – destruction from fire,
strong wind, outbreak of parasites, and so on.
The phase picture of the state of given part of forest and doesn’t let to go to another state with the
help of forestry or possibility to transfer into other states of equilibrium.
Let us note that the global and local forests are extremely important for mankind. From one side
forest mainly define Biodiversity on the Globe, from the other side, forest is a time bomb and
even the weapon of mass destruction: the forest fire last 5 years.
As I have told above, this Report is based on our works [1-8]. At the beginning we construct
mathematical models (phenomenological) of special-temporal dynamics of trees population, and
investigate them. The first such a model we have proposed in the work [1] and it has the
following form:
Let u – the number of “young” trees (or density), υ – the number of “old” trees (or density) on
some surface of a region. Here for simplicity we consider only two age classes. In the work [6]
we consider situation with arbitrary number of age classes. Parameters ρ, ƒ and һ are coefficients
of reproduction, ageing and mortality of “old” trees respectively, while γ (υ) is a mortality rate of
“young” trees. Our hypothesis that there exists some optimal value of “old” trees density under
which the recruitment of “young” trees is maximal, it is possible to choose γ(υ)=a(υ -b)2+c with
positive parameters a,b,c. Our model is phenomenological
(1)
It is clear that any spatial distributed generalization of model (1) should have spacehomogeneous solution corresponding to two stable state of model (1). To account for spatial
effects one has to consider in greater detail the process of regeneration which was described in
model (1) in a phenomenological way by a reproduction coefficients. It is well known that the
regeneration usually proceeds through four main stages involving seeds, namely: seed
production, seed transport, seed deposition and seed establishment. The seed transport (in many
cases) is the main spatial effect that should be taken into consideration in the forest modeling.
We take a hypothesis that there is no prevailing wind direction and, therefore the Gaussian
(diffusion) model can used for describing seed dispersion for a general treatment of windinduced seed dispersal. Let introduce an additional w-variable (density of air-borne) seeds into
the model:
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(2)
where α,β and δ are seed production, deposition and establishment rates respectively, and D is
the diffusion coefficient along a space variable X. The system (2) is of “reaction-diffusion” type.
It is possible to derive an approximate system of equations for “old” and “young” tree classes
densities by eliminating the seed variable from (2) under the hypothesis of different time scales
for dispersing seeds and trees dynamics.
(3)
Note that model (3) has a “cross-diffusion” term while (2) has a classical diffusion term.
By a linear change of variables, time and space scale we will transform system (3) into the
following dimensionless form
(4)
The analysis of the model (4) begins with the search for spatially non uniform stationary
solutions:
u
The second equation is algebraic and allows one to find U(x) if V(x) is known. Thus the problem
of finding out the stationary solutions of eqs (4) reduces the analysis of the equation:
5
,
which can be rewritten as a system of two first-order differential equation (where ´ indicates
space derivatives); introducing W=
we obtain:
(5)
Solutions of (5) define profiles of stationary solutions of system (4).
This system differs only by space scale from the corresponding standing wave system for model
(4).
Therefore, we can use those results in our present analysis.
From (5) we obtain (6)
(6)
Equation (6), as it is known, (see Appendix 2) has analytical solution:
V=-2
(7),
where a,b,c – are constants.
Here
where
(t)and
– constants (see Appendices 1,2).
It is said that our model is uniformized by function (7).
We go back to the system above
(6)
Let us show that our system is a Hamiltonian system, i.e. there exist such function H=H(V,W),
that
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We look for H(V,W) in the form (which is prompted by system (1)). To this Hamiltonian
corresponds a conservation law
+P(V)=const,
where
corresponding to Kinetic energy of the system, and P(V) – potential energy. The
proving of this assertion consist in finding of P(V):
Hence, according to the system (6) we obtain:
P(V)=h
h
polinom of 4 deg
Obtaining P(V) we can study lines of level of the Hamiltonian
H(V(t),W(t))=const
What is sapid sense of Hamiltonian? The phase plane with the coordinates (V,W) is subdivided
on the line of level of the function H(V,W). The motion at phase plane defined by dynamical
system (6) is the motion along the lines of the level correspondingly.
Thus we obtain, that
H(V,W)=
7
We use the well known fact that the equation H(V,W)=const is the equation of elliptic curve and
the equation H(V,W)=const under the condition V=
gives elliptic curve. For each point of
the trajectory we have H(V(t), W(t)=c.
The standard Weierstrass model of elliptic curves is
Let us stress, that the main interest in both methods – phase portraits and elliptic description how
to combine the potential of both methods to reach the best description of the phenomena is study.
We try to demonstrate both cases, using the development we presented above (see equation (6)).
The parametric and phase portraits of Hamiltonian of the system (6) (see Fig. 1) that describes
forest standing boundary. On the line Q we will obtain it below the system has a heteroclinic
orbit connecting
an
Lines
on Fig. 1 are bifurcation lines equilibrium of the system (6) in region 1 have
and
(see calculations below).
the following coordinates
and
Equilibriums
and
, where
are saddles while equilibrium
.
is a center. Bounded nontrivial
orbits of the system (6) either are closed or connects saddles. The equation of line Q on which a
separatrix connecting the saddles exists and can be found analytically from the condition that
isoline H=0 goes through saddle
:
(*)
For parameter values satisfying (*) the equation ( ) can be explicitly written
Therefore, we have established a possibility for the existence of a standing boundary in the forest
modeled by a system (**), but only for a specific relation between parameters (
.
(**)
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Fig. 1 (the meaning see Article 2)
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Let us now introduce an insect pest into our initial model and consider two extremely simple
situations.
a) the pest feed only on the “young” trees
b) the pest feed only on the “old” (adult) trees
c) the general case was considered by on of my summer student (Samarskaya, E.A.: forest-pest
interaction dynamics in temporal and spatial domains. WP-89-16, Laxenburg, Austria,
International Institute for Applied Systems Analysis, 1989).
Let z is insect density,
is the mortality rate of insects and the terms with xz and yz represent
the insect-forest interaction.
Thus for the case where the pest feeds on “young” trees we obtain the dynamical model
(
)
while for the case where the pest feeds on adult trees we obtain the (equations) dynamical model
=fx-hy-Ayz
(
)
The main tools for our investigation are the bifurcation theory of dynamical systems and the
numerical methods of this theory.
By a linear transformation of variables, parameters, and the time the system (
) will transit
into
(AM)
where the previous notations are preserved for new variables and parameters which have the
same sense as in initional model. The new parameters can be represented in terms of the old one
as
10
;
;
;
;
.
In the first octant (i.e. where the variables take on biological possible value), the system (AM)
can have from one to four equilibrium points:
,
,
,
(see Antonovsky et al [ ]).
We show the parametric portrait of system (AM) and the phase portraits of the model (AM).
This presentation is useful device for exact notions in a forest science. For example, the notion of
probability of outbreak of insect could be formulated only on the base Fig. 2.
The parametric and the phase portraits of system (AM): only 3 of 9 possible states of the system
(AM) could be ready for outbreak. It means that probability of outbreak in the system
described by (AM) is 1/3.
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Fig. 2 (the meaning see Article 1)
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Appendix 1.
Sake for convenient of readers we will refresh some notions that we are using and that can be
found in any mathematical reference book. The elliptic functions are meromorphic double
periodic functions. Two important notions – primitive periods and parallelogram of the periods
(on complex plane). If Ω and
, so called primitive periods of an elliptic function, then
parallelogram with vertexes O, Ω, Ω+
don’t has the points of periods nor inside, neither
,
at the sides, not taking into account vertexes. It arises some net of parallelograms that cover all
complex plane. The Weierstrass function
(u) by definition is
(u)=
It turns out that
.
(u+2
)=
(u) and
(u+2w)=
(u), i.e.
(u) is an elliptic function of the
second order. It is the main function in the Weierstrass theory.
As it is well known in special case of degradation elliptic functions transit into trigonometric
functions. Let us remember that
ctg u=
(u)=
(u;w,
(u)2=4
If w=
3
)=
(u)-g2
and w´=
(u,
),
ln sin u
=-
ctg u
(w,
);
=
(w,
), and
(u)-g3 – differential equation for Weierstrass functions.
, then
problem and instead of
(u)=
, g2=g3=0. In this case we get the general solution of our
and
we substitute these values. Contently it means that the
conditions when periodic solutions disappears. In case when only one periodic equal w´=
and
finite, then
(u)=-
+
13
Discriminant
-
of Weierstrass functions will be considered for different values.
Appendix 2. (see in any math. Reference book)
Let us consider well known equation of mathematical physics
- =
+6
+6
,
when
Korteveg de Vries (KdV); when
=0,
=1,
=0 it became the equation of
we obtain modified equation of Korteveg de Vries
(mKdV). The problem of solutions of this equation in the form of traveling wave
Is reduced to the solution of equation
4c
=
+6
+6
(4)
The general solution of this equation with the initial conditions
(0)=0,
(0)=1
is the function
(W),
where
,
),
=
velocity of travelling boundary, parameters
( ,h,s), i.e. c=
( , h, ),
=
+ ,
and
=-
-
, where c is a
through the parameters of initial model
( ,h,s), = , h,s,f,…).
Thus the traveling wave of the initial system of the equation (B) will be expressed through
Weierstrass function. It is important to see all four transformations of the initial parameters. It is
worth to note that the shape of equation (4) is preserved under the transformation
,
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where
0 and
are free parameters. Under this transformation we have
(c,
(
,
,
,
)
-2
,
);
. It takes place
Lemma. General solution of equation (1) with initial condition
(0)=1,
=x is given by formula:
=-2
where
,
-
) – Weierstrass function
,
=
-
;
=
-
.
In conclusion I would like to stress again that the goal of mathematical modeling is to find and
present the mechanism of observed effects. This is exactly what we did in this Report on
example of age waves on Forest.
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