F ll

where Sx* and Sq are variations of equilibrium and inflows. By assuming that the inflows do not depend on state variables and taking into account that ll(9f/Sx)*II — II(9y/9x)*II is nothing more or less than the Jacobi matrix A, we can rewrite Eq. (7.5) as Sq + ASx* = 0, whence (if the reciprocal matrix A—1 exists)

On the other hand, since x* = x*(q) then Sx* = IISx*/SqIISq- In accordance with the standard definition the matrix S = IISx*/SqII is a sensitivity matrix. Comparing it to Eq. (7.6) we see that S = —A—1 and Sx* = SSq-

Generally speaking, we can define arbitrary norms for the vectors Sx* and Sq, and also for the matrix S: ISx* l, ISql and iSl, then lSx*l = iSSql # iSl-lSql- If we assume iSl < 1, x i.e. all eigenvalues of iSl, Mi, must be situated inside a unit circle Ml = 1, then l8x*l < lSql. (7.7)

The norms lSql and lSx*l can be considered as some measures of external perturbations and the system response in relation to these perturbations; and inequality (7.7) as the formulation of Le Chatelier's principle: the systems, organised by the principle, attempt and, very importantly, are capable, to relieve external perturbations. In our case, if lSl < 1 then input perturbations are always relieved.

Let a, be the eigenvalues of Jacobi matrix, then Mi = — (1/L) and the inequality lSl < 1 holds if la, l > 1. On the other hand, the equilibrium must be stable, i.e. ReA,- < 0; then an intersection of these conditions gives the new general condition for fulfilment of the Le Chatelier principle: ReAi < — 1, which is stricter than the previous ones.

It is obvious that the norm of some vector is identified only with its magnitude, but not with its direction. We cannot say the vector is either positive or negative, it is "good" or "bad" until we shall define ourselves what is "good" and what is "bad", but we may connect the definition of either "good" or "bad" vector z with the sign of scalar product (z, e) where e is a unit vector. For instance, if (z, e) > 0 then the vector z is "good", and vice versa, if (z, e) < 0 then the vector z is "bad". This is a definition in average. In fact, the (Sq, e) = £f=1 Sqi = Sq and (8x*, e) = £f=1 Sx* = Sx* are the variation of the total inflow and the total shift in equilibrium, caused by the variation. These are the following possible combinations:

1. Increase of the total inflow Sq > 0 causes an increase of the total equilibrium storage Sx* > 0.

2. Decrease of the total inflow Sq < 0 causes a decrease of the total equilibrium storage Sx* < 0.

3. Increase of the total inflow Sq > 0 causes a decrease of the total equilibrium storage Sx* < 0.

4. Decrease of the total inflow Sq < 0 causes an increase of the total equilibrium storage Sx* > 0.

It is intuitively clear that if the first two reactions could be called normal, then reactions 3 and 4 are paradoxical.

Since Sx* = 5j=1 sySqp where sij are elements od the matrix S then the condition dx* > 0 is equivalent to £n=1 siSqi > 0 where si = 5j=1 sji. It is clear that the planes I?=1 Sqi = 0 and 5j=1 siSqi = 0, passing through the origin of the co-ordinates, divide the space of the co-ordinates {Sq1,..., Sqn} on two domains (see Fig. 9.8): in the first domain the system has a normal reaction (in this domain the product SqSx* is positive), in the second domain the system has a paradoxical reaction (the product SqSx* is negative).

9.8. Stability and reactions of a bog in the temperate zone

Let us return to the diagram shown in Fig. 9.3, which is a compartmental scheme of the carbon local cycle of a bog in the temperate zone.

Fig. 9.8. Domains of normal and paradoxical reactions.

The choice of flows type is based on an expert opinion: the rate of the uptake of plant's carbon by animals depends on both storages, so thatf12 = y12x1 x2; analogously, the flows f24 and f34 have the donor-type, the flowsf42 and f43 have the Volterra-type. An application of the above-described algorithm results in the non-linear system of differential equations describing the dynamics of the ecosystem (Zavalishin and Logofet, 2001):


dx2 dt dx3



= ql — ml Xl — au Xl — yl2 Xl X2, = — m2X2 — A24X2 + ^42X4 + Gl2Xl )X2,

= q4 — m4 X4 + Al4 Xl + A34X3 + A24X2 — X4 (y43 X3 + G42X2 ).

The Jacobi matrix of Eq. (8.1) in equilibrium will be

A =

ql xl

f2 X2

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