## Eqeq

The latter relation means that when the trophic chain is in dynamic equilibrium with its environment, then its total biomass has to be constant (although the biomasses of each trophic level may fluctuate).

We can see that the conditions of stationary state do not depend on the choice of hypothesis about the fate of dead organic matter. It is possible to show that other results would not fundamentally differ from each other. Therefore, in order to avoid overloading our account with superfluous details, we shall consider only the case when the dead organic matter remains within the system, when d S n

Until now we have implied a dynamic equilibrium of the entire chain when the latter was considered as a single thermodynamic system. However, stronger conditions of equilibrium could be formulated if every level is considered as some separate system. In this case, the equilibrium conditions are: (dN1/dt) = ••• = (dNn/dt) = 0, from which it follows that (Nfq = N* = const):

qi(Np) = qi2(Ni, N|) + Di(ND, qi2(Ni, N2) = q23(N2, Np) + D2(N*2)

qk-i,k(Nk-i, Np) = qk,k+i(Np, Np+i) + Dk(Np), (4.8)

qn-1,n(Nn-1, N) = Dn(Nk). These equations give us the co-ordinates of equilibrium.

7.5. Prigogine-like theorems and the length of trophic chain

If we keep in mind now the Prigogine theorem (see Chapter 3) then in the application to our case its main statement says that at a stable dynamic equilibrium the value Yn=1 Dk(Nk) is minimal, and also the minimum is positive. It is obvious that Dk(Nk) have to be monotonously increasing functions and Dk (0) = 0. Then the minimum is attained with certainty at the origin of co-ordinates {N™n = 0, k = 1, •■•, n} if there are no constraints except their non-negativeness. However, the minimum is zero, which contradicts Prigogine's theorem. Therefore, the minimum has to be found under conditions of stationary state. Therefore, at the dynamic equilibrium Neq = {N|q, k = 1, •.., n} where N [ Pn, Pn : {Nk \$ 0, k = 1, •.., n} is a positive orthant, n n

under constraints (4.4).

In this formulation the problem is too general; in order to obtain some understandable results we simplify it. For this we are restricted to the case n = 3 and q1 = q = const. The latter implies that the inflow of energy or a resource does not depend on the state of the autotrophic link. The functions Dk(Nk) are represented as Dk(Nk) = (ri + mi)Ni = diNi, where ri and mi are the coefficients of respiration (metabolism) and mortality. The inflows between trophic levels are represented in the bilinear form: q12 = a1N1N2; q23 = a2N2N3. Then Eqs. (4.4) and (4.5) are written as dN1

where d = £n=1 dkpk is the mean value of coefficients dk, k = 1,2, 3.

Elementary calculations show that the system has three equilibrium points:

(b) Np = d2/«j, Np = (q/d2) - (dj/aj), N3* = 0 and