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öl IrkjNkNj

then W is represented as

and its full differential will be dw = x t=1

dNk.

We assume that the functions fk(Nk) and hk(Nk) do not have any singularities at the point N = {N0,...,N0}. Then W(N0) = 0, and this initial state can be taken as some reference level for the thermodynamic potential W. It is natural to assume that W = 0 at the thermodynamic equilibrium. As follows from common sense, life does not exist at the thermodynamic equilibrium, so that all N° have to be equal to zero, i.e. N0 = 0. Note that the origin of coordinates N = 0 is the trivial solution to Eq. (3.1), i.e. N* = 0 is also a dynamic equilibrium.

If we look at formula (4.2) we see that W! —1 when N n and fk and hk are bounded. Therefore, there is the open domain V c Pn, within which the function W is positive, i.e. W(N) > 0 for any N [ V, except the origin of coordinates where W(0) = 0. The thermodynamic equilibrium is unstable iffk(0) > hk(0) even for single k [ [0,1]. If fk(0) < hk(0) for all k, then the equilibrium is stable. The latter has a very simple interpretation: if the metabolism and mortality of individual species are not compensated by external energy flows then all populations are eliminated. In this case V = 0, and W(N) < 0 for any positive N. It is natural to assume that the domain of positiveness for W defines a domain in which the concepts of linear thermodynamics are applicable. It is easy to prove that the non-trivial equilibrium N* also belongs to the domain V. Since the proof is cumbersome, we illustrate it by a simple example for n = 1 .

Let fi and h1 be constants. Then W = (fi — h1)N1 — 2 TnNj2; f1 > h1. One can see that W(0) = 0 and W(N1 ) > 0 if N1 [ (0,Nfr), where Nf = 2(f — h1 )/y1 1. On the n other hand, at the non-trivial equilibrium (9W/dN1)* = (/ — h1) — j\\N\ and Np = (fi — hi)/Gii = Nf/2, i.e. Np [ V : (0, Nf).

Finally, we can say that when such an open system as a community of competing populations tends to its dynamic non-trivial equilibrium then the thermodynamic potential, defined by the function W, monotonously increases, attaining its maximum at the equilibrium. At the thermodynamic trivial equilibrium the thermodynamic potential is minimal.

Since the change of W is caused by biological processes, it is natural to name the potential function W a biological potential o/community. We formally assume that there is such a non-negative function G = G(N) (G(0) = 0) that can play the role of Gibbs' thermodynamic potential. By expanding G into the Taylor series in the vicinity of zero and restricting two first terms we obtain:

Since the function G has to be a full differential,

dNi dNj dNj dNi '

and the differential is represented as

Comparing this expression with expression (4.3) we see that they coincide if / >G \ ( >2G \

The first condition implies that if we would like to remain within classic thermodynamics frameworks not far from thermodynamic equilibrium then the flow functions qk, q1ut, Mk and Dk must be linear functions of Nk, i.e. the functions fk and hk must be constant. In this case Eq. (2.13) is transformed into the classic Lotka-Volterra equations:

The first condition of dynamic (but not thermodynamic!) equilibrium, diS/dt = —deS/dt, which has been already formulated in Section 8.2 (see Eq. (2.8)) in a sufficiently general form, is now written as (Nfq = N*)

The internal production of entropy, diS/dt, is equal to d S 1 n n d7 = r! X (4.8)

and the entropy export out of the system into its environment, deS=dt, is d S 1 n

(Note that we consider isothermal processes, i.e. when the temperature is constant.) Since the outflow of entropy, deS/dt, is negative, we can associate this flow with the flow of Schrodinger's negentropy from environment into the system.

One of the important characteristics of community is its total number of individuals, or biomass N = Y1=\ Nk. Summing Eq. (2.13) in our case we obtain (see also Eq. (2.5))

where S is the total entropy of the system. This equation allows us to give a thermo-dynamic interpretation to different stages through which the system passes in the course of its succession. At the initial state, when Nk are still small, both the free energy inflow and the internal entropy production are also small. Therefore, the absolute values of the rates of biomass and entropy change, IdN/dtl and IdS/dtl, will also be low, but in order for the total biomass to grow it is necessary for the entropy to decrease. It is possible, if the system could possess a powerful energy inflow with low entropy. At this stage the exchange processes, which are described in linear terms, play the main role. And the power of the entropy pump, IdeS/dtI, exceeds significantly the internal entropy production. Now the rate of decrease of the total entropy grows, which promotes the increase of (dN/dt). But with the growth of N the inhibiting role of quadratic terms, which describe the internal entropy production, becomes more and more tangible. The rate of increase of the total biomass decelerates until it becomes equal to zero. At the same time the internal entropy production becomes equal to the power of the entropy pump.

By integrating both sides of Eq. (4.10) with respect to time and assuming that the integration starts from the thermodynamic equilibrium, when N = 0 and S = S0 we get a relation between the total biomass and entropy, or exergy:

We can say that if the entropy of the community is minimal then its total biomass is maximal. Correspondingly, the exergy of community will also be maximal. In other words, any well-organised competing community has a maximal biomass and its exergy is maximal.

Note that since dN = £JLl dNk, the total biomass N is the exact differential as well as the total entropy S:

Let us keep in mind that the biological potential of community, n i n n

monotonously increases and tends to maximum. This result also allows us to do a sensible interpretation. The value Gr(N) = Y1= 1 (fk ~ hk)Nk in essence accounts for the rate of biomass gain in the case that competition and any kind of limitation by resources are absent, and the growth is determined only by the physiological fertility and natural mortality of the organisms. Therefore it is natural to define Gr as the reproductive potential of the community. The value Diss = £¿=1 5j=i JkjNkNj may be used as a measure of the rate of energy dissipation resulting from inter- and intraspecific competition. Therefore we shall refer to Diss as the total expenses of competition. Hence the increase in Diss in the process of evolution may be interpreted as the goal of the community to maximise the difference between its reproductive potential and the total expenses of competition. This goal can be achieved in several ways: either the reproductive potential is maximised at fixed expenses of competition (r-strategy) or the competition expenses are minimised for a limited reproductive potential (K-strategy). There may also be some intermediate cases.

The main statement of Prigogine's theorem is: at the stable dynamic equilibrium the value of diS/dt, i.e. the internal entropy production, has to be minimal, and this minimum has to be positive. However, as is seen from Eq. (4.8), the minimum of diS/dt is attained at the origin of coordinates N* = 0 (certainly, if there are not any constraints except the non-negativeness of Nk). The stationary conditions (4.7) are also satisfied at N* = 0, i.e. the trivial solution N* = 0 is equilibrium. We have mentioned above that this equilibrium is thermodynamic, although it is also a dynamic equilibrium of kinetic equations. This minimum is equal to zero, which contradicts Prigogine's theorem. To resolve the contradiction, at least one equilibrium coordinate has to differ from zero. Then the requirement for positiveness of the minimum becomes understandable: Prigogine's theorem deals with dynamic equilibriums, which differ from thermodynamic ones.

By analogy with the main statement of Prigogine's theorem we postulate:

At the stable non-trivial dynamic equilibrium N* E Pn

min nepn diS n n

under constraint (4.7): ^=1 Nf ~ hk) = XLi X"=i 7kjN*kN*.

We shall assume that the minimum is attained at some internal point of Pn. If it is attained on some orthant faces, so that N* = {N* > 0, s E vk; N* = 0, l E Vk} where vk is the set of k indices because the corresponding equilibrium coordinates are non-zero, vk u Vk = 1,..., n, then the minimum will be internal in relation to the sub-orthant Pk: {Ns \$ 0, s E vk}. Therefore, the problem can always be reduced to the problem of internal minimum, which will be considered below.

It is not evident that the statement is true for our case. Let us test it, but first we introduce the following notations:

Using the method of Lagrange multipliers we find the minimum of function Z — T{(di5/dt) +A[(di5/dt) — q + h]} where X is the Lagrange multiplier. Then the necessary conditions of minimum under constraints will be:

— = 2Gk* + A(2Gk* -fk + hk) = 0, k = 1, ..., n. (4:13)

Multiplying both sides of Eq. (4.13) by N* and summing all these equations we get the formula for the determination of X: 2G* + XG* = 0, whence X = -2: Then the minimum necessary condition is:

Comparing these conditions with the necessary conditions for the maximum W that are written in our case as fk — hk — Tp = 0, k = 1,..., n, we can see that they coincide. The sufficient condition of minimum di5/dt = (1/T) YJi=1 Sj=1 JkjNkNj is a positive definiteness of this quadratic form, but this is also the sufficient condition for the maximum W: Since the entropy production within the system is always positive, diS/dt > 0, the corresponding quadratic form has to be positive definite in the entire positive orthant. This is the condition of uniqueness of the minimum. Thus, this Prigogine-like theorem is applicable to our case.

Note that instead of the problem of minimisation for the internal entropy production we can consider the problem of maximisation for the negentropy flow. Formally the problem is formulated so as to find the maximum of linear form Gr = f — h under the constraint f — h = G The latter is the condition of equilibrium. This is the well known duality principle in the optimisation theory (Handbook of Mathematical Economy, 1991). Since diS/dt has to be positive, the constraint can be fulfilled only if Gr = —deS/dt > 0, i.e. we really deal with the negentropy. Using the method of Lagrange multipliers we find minimum of the function Ze = R + X(R — G) where X is the Lagrange multiplier (we use the notations that were introduced above). Then the necessary conditions of minimum under constraints will be:

—e = fk — hk + X(2Gk* —fk + hk ) = 0, k = 1,..., n: (4:15)

Multiplying both the sides of Eq. (4.15) by N* and summing all these equations we get: q — h + XG* = 0, whence X = (q — h)/G* = — L Then the necessary conditions of maximum will be: 2(fk — hk — 2Gk*) = 0, k = 1,..., n, i.e. they coincide with the necessary conditions for the minimum of diS/dt: Hence, we proved that at the equilibrium the flow of negentropy into the system is maximal. It is possible (and it is the principal difference from the situation of classic Prigogine's theorem) because this flow depends on the system's state and, by the same token, it can be controlled by the system.

If we assume that all the populations compete for the single resource R, then the resource compartment can be considered as the system environment, and the resource flow into the system can be identified with the negentropy flow from the environment into the system. We also assume that if the rate of resource uptake is equal to 5j!=i akRNk, then (deS/dt) = n=i AkRNk. Note that in all our considerations we assume a constancy of the environment that, in turn, implies a constancy of resource. It is possible if an external reservoir containing the resource is very large in comparison with the ecosystem (the latter is a standard assumption for different thermodynamic considerations) and an inverse influence of the system on its environment can be neglected. Then R < R* = constant, and fk(R) — hk(R) = akR < akR* are also constant. In this case all the previous considerations and results are conserved. The coefficients sk in the Lotka-Volterra system (4.6) have to be sk = akR*. However, it is interesting to consider a situation when we cannot neglect the inverse influence; then we have to supplement the system (4.6) with one more equation

If we assume that the resource is restored by a constant inflow qR then the equation will be:

A constancy of resource concentration is maintained if the constraint £n=i akNk = a = constant is fulfilled. As was proved earlier, the form has to be maximum at the dynamic equilibrium (under constraint £n=i 5j=i Jk]NkNj = constant). Since R* = qr/a, the maximisation of a is equivalent to the minimisation of R*. These results could be interpreted in the following, somewhat speculative manner.

If such a system as the biological community of competing populations in a process of interaction with its environment tends to a stable equilibrium with non-zero values of population sizes then at the equilibrium:

• the system tends to minimise the internal entropy production system (Prigogine's theorem);

• the system tends to arrange its structure and interaction with the environment in such a way that the negentropy flow out of a system into an environment will be maximal (the principle of maximum for the negentropy flow);

• the concentration of resource in the environment tends to be minimum (the principle of maximal utilisation).

In conclusion we show how the methods of phenomenological thermodynamics can be used for the description of a competing community.

It is obvious that the thermodynamic equilibrium is interpreted as "not a life", i.e. all N0 ; 0. Therefore, the values grad Ni 0 N, — N0 = Ni could be considered as the generalised thermodynamic forces Xk. We assume that generalised thermodynamic flows can be presented in a linear form Jk = 5j=i GkjXj = 5j=i JkjNj where ykj = jjk \$ 0.

for R

These are well-known Onsager's reciprocal relations. In this formalism the value of dissipated energy, which is proportional to the entropy production caused by irreversible processes within the system, is equal to d S n n n

In accordance with Prigogine's theorem, this value—named the function of dissipation—is minimal at the stable dynamic equilibrium. It is obvious that this statement is equivalent to the statement of the above-proved Prigogine-like theorem. In this formalism the reciprocal Onsager coefficients correspond to the competition coefficients in Lotka-Volterra equations, and they are equal to specific energy, which is lost in the single act of competitive collision. The latter always takes place, since a competition is a typical irreversible process leading to the dissipation of energy and the loss of biomass.

### 8.5. The system of two competing species

In order to illustrate all these theoretical results we consider one partial case, namely two competing species. The corresponding equations are (f 2 — h1 2 = e1 2):

Let N1 and N2 be microscopic variables, then the reproductive potential Gr = e1N1 + e2N2 and the function of dissipation Diss = y11N^ + y12N1N2 + y21N2N1 + y22N| are also macroscopic ones. However, only in the case n = 2 can system (5.1) be written in the variables, Gr and Diss, since their number is equal to the number of phase variables, i.e., two. We maximally simplify the problem setting e1 = e2 = 1 and y11 = T22 = 1, T12 = T21 = G, so that the system state will depend only on the single parameter y, which can be considered as a bifurcation parameter. It is easily seen that the system has three equilibriums: trivial Np = Np = 0; semi-trivial Np = 1, Np = 0and Np = 0 , Np = 1; nontrivial Np = Np = 1/(1 + y) , where the latter is stable if y < 1, and unstable if y > 1. The value y = 1 is critical. At this time the semi-trivial equilibriums are stable (the trivial equilibrium is always unstable), and the system (depending on the initial state) comes either to one or another equilibrium. In this case the function of dissipation is Diss = D = N2 + 2yN1N2 + Nf and the reproductive potential is equal to the total size of the community, Gr = N = N1 + N2. After simple calculations we obtain:

We shall study this system analytically. It is easy to see that Eq. (5.2) has the trivial equilibrium N* = D* = 0, and two non-trivial ones, which are determined from the equation (1 + y)(N*)2 - (3 + y)N* + 2 = 0: N* = D* = 1 and N* = D* = 2/(1 + y). The trivial equilibrium is always unstable; the equilibrium N* = D* = 1 is stable if y > 1, and unstable if y < 1. The latter case corresponds to the semi-trivial equilibriums N* = 1, N* = 0 and N* = 0, N* = 1 in the phase space {N1; N2}. At last, the equilibrium N* = D* = 2/(1 + y) is stable if y < 1, and unstable if y > 1. The phase portrait of the system (5.2) is shown in Fig. 8.3 for two values of y: y = (1/2) < 1 and y = 2 > 1. Notethatfrom a biological point of view only domains of the phase plane, those corresponding to the nonnegative values of the variables N1 and N2, make sense. They are non-negative if the inequalities