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On superposed systems (I) —an iterative procedure for ecosystem modelling
ON SUPERPOSED SYSTEMS (I) ---AN ITERATIVE PROCEDURE FOR ECOSYSTEM MODELLING He Shan-Yu*
Abstract---In this paper, a straight approach, the Superposition Procedure, is suggested for modelling multicomponent ecosystems. The advantages of this procedure are: (1) the information provided by subsystem models can be utilized to a greatest extent, (2) the models thus obtained are suitable for numerical prediction, and (3) the superposition procedure is also applicable in the identification problem of other complex systems. Keywords--- ecosystem modelling, identification,
complex
systems
I. Motivation Because of the existence of intricate mutual interactions, it is usually impossible
to model a multicomponent
ecosystem by intuition. Fur-
thermore, most simple ecosystem models are not suitable for practical numerical prediction.
Even the famous Lotka-Volterra
equation 11.3]
can be
used only as a start point of rough qualitative analysis. To obtain accurateqantitative
model of complex ecosystems, the au-
thor introduced the concept of Superposed System and suggested a Superposition Procedure. The fundamentals of this are: (1) A whole ecosystem is thought to be formed by superposing its subsystems one after another, (2) we have already satisfactory separate models for all subsystem8 when they were in isolated state.
II. Outline of Superposition
Procedure
Assume that the ecosystem ,$ is composed of N tions). we denote this fact by isolated subsystem
s;,
iz
s=
subsystems(populaFor each
s,vs,v+&
we have already satis-
i,t,.,.,pJ
factory model ML: where
f ca
the control vector used to represent actions from outside, in evolution case tion with the same dimension as xi,
ai30 ;.e, n;,
, and
* Institute of Automation, Academia Sinica, Beijing, People”s China
vector
func-
Republic
of
193
We start the Superposition
Procedure with two subsystems, say, S,
and Sr. Usually we may take the union vector describe the superposed system lity, we can write the model
&=a M,% of
( rc,'i G
S,G,. S_as
to )' No loss of genera-
following
We may naturally assume that the first state variable & (population) of
Sr
, so that if at some instant
the subsystem
Si will disappear for all
corresponding
yi
eat,
is the "size"
+o , %,(te)=o , (extinction!) and the
in (2) should be reduced to $
i.e., l; I 0
for
. This is the Submodel-Fitting Requirement for 9; . t,te When possiblo, we may, under certain initial conditions, apply a (u:i
test control
Uz)'
(e.g. impulse'sequence,
pseudo-random
S,, . With some identification technique (e.g., Volterra series approach) L23, we can obtain some experimental data which may be
code, etc.) to
fitted by approzmate cesed form & =g,(t,u,,u,), #Z=zz[t,&f,,Ufa). Based on & and X3 ' we can obtain a set of differential equations by elimination _
(3)
and F,
are, generally not submodel fitting, although they can fit
the observed data quite well, i.e., data fitting. Let fi be a set of vectors (q,,'i GZ')' , where y, and e, are submodel fitting and have rational comionents. If we choose a suitable norm where
u* 1 9,
of all vectors (~,7i~zT)' (e.g., Euclidean) in the set n are piecewise continuous functions.
and y?
By computation, we can finally find out a vector such that
Then
will be a submodel fitting and best data fitting model of S,*P S,\/S, Continue this procedure by iteration, we can finally arrive at a good model tion.
M,r...rJ
of the whole
s
III. Remarks
194
suitable for numerical predic-
.
lo
In this paper only the procedure itself is given. Rigorous mathema-
tical discussions are left for later papers. 2O
Computers should be used for superposition procedure.
3O
The theory of Superposed System could be developed in different
(including algebraic and geometrical) ways.
Literature B., Fenchel, M.: Theories of Populations in Biologi-
[I]
Christiansen,
[21
Eykhoff, P.: System Identification,
[3]
Volterra, V.: Variations and Fluctuations
cal Communities,
Sec. 2.2, Springer-Verlag
Berlin, 1977.
Chap. 4, John-Wiley,
1977.
of the Number of Indivi-
duals in Animal Species Living Toget.her. J. Conseil,a
(1928),1-51.
195
Abstract---In this paper, a straight approach, the Superposition Procedure, is suggested for modelling multicomponent ecosystems. The advantages of this procedure are: (1) the information provided by subsystem models can be utilized to a greatest extent, (2) the models thus obtained are suitable for numerical prediction, and (3) the superposition procedure is also applicable in the identification problem of other complex systems. Keywords--- ecosystem modelling, identification,
complex
systems
I. Motivation Because of the existence of intricate mutual interactions, it is usually impossible
to model a multicomponent
ecosystem by intuition. Fur-
thermore, most simple ecosystem models are not suitable for practical numerical prediction.
Even the famous Lotka-Volterra
equation 11.3]
can be
used only as a start point of rough qualitative analysis. To obtain accurateqantitative
model of complex ecosystems, the au-
thor introduced the concept of Superposed System and suggested a Superposition Procedure. The fundamentals of this are: (1) A whole ecosystem is thought to be formed by superposing its subsystems one after another, (2) we have already satisfactory separate models for all subsystem8 when they were in isolated state.
II. Outline of Superposition
Procedure
Assume that the ecosystem ,$ is composed of N tions). we denote this fact by isolated subsystem
s;,
iz
s=
subsystems(populaFor each
s,vs,v+&
we have already satis-
i,t,.,.,pJ
factory model ML: where
f ca
the control vector used to represent actions from outside, in evolution case tion with the same dimension as xi,
ai30 ;.e, n;,
, and
* Institute of Automation, Academia Sinica, Beijing, People”s China
vector
func-
Republic
of
193
We start the Superposition
Procedure with two subsystems, say, S,
and Sr. Usually we may take the union vector describe the superposed system lity, we can write the model
&=a M,% of
( rc,'i G
S,G,. S_as
to )' No loss of genera-
following
We may naturally assume that the first state variable & (population) of
Sr
, so that if at some instant
the subsystem
Si will disappear for all
corresponding
yi
eat,
is the "size"
+o , %,(te)=o , (extinction!) and the
in (2) should be reduced to $
i.e., l; I 0
for
. This is the Submodel-Fitting Requirement for 9; . t,te When possiblo, we may, under certain initial conditions, apply a (u:i
test control
Uz)'
(e.g. impulse'sequence,
pseudo-random
S,, . With some identification technique (e.g., Volterra series approach) L23, we can obtain some experimental data which may be
code, etc.) to
fitted by approzmate cesed form & =g,(t,u,,u,), #Z=zz[t,&f,,Ufa). Based on & and X3 ' we can obtain a set of differential equations by elimination _
(3)
and F,
are, generally not submodel fitting, although they can fit
the observed data quite well, i.e., data fitting. Let fi be a set of vectors (q,,'i GZ')' , where y, and e, are submodel fitting and have rational comionents. If we choose a suitable norm where
u* 1 9,
of all vectors (~,7i~zT)' (e.g., Euclidean) in the set n are piecewise continuous functions.
and y?
By computation, we can finally find out a vector such that
Then
will be a submodel fitting and best data fitting model of S,*P S,\/S, Continue this procedure by iteration, we can finally arrive at a good model tion.
M,r...rJ
of the whole
s
III. Remarks
194
suitable for numerical predic-
.
lo
In this paper only the procedure itself is given. Rigorous mathema-
tical discussions are left for later papers. 2O
Computers should be used for superposition procedure.
3O
The theory of Superposed System could be developed in different
(including algebraic and geometrical) ways.
Literature B., Fenchel, M.: Theories of Populations in Biologi-
[I]
Christiansen,
[21
Eykhoff, P.: System Identification,
[3]
Volterra, V.: Variations and Fluctuations
cal Communities,
Sec. 2.2, Springer-Verlag
Berlin, 1977.
Chap. 4, John-Wiley,
1977.
of the Number of Indivi-
duals in Animal Species Living Toget.her. J. Conseil,a
(1928),1-51.
195