of equilibrium. If, on the contrary, this distance increases along some trajectory, then (in accordance with Chetaev instability theorem) the equilibrium is unstable and this trajectory recedes from it.

All these results can be easily interpreted from the thermodynamic point of view. Indeed, in thermodynamics the value of L is proportional to the mean square of fluctuations around equilibrium or the power of fluctuations. Therefore, we can state that if the power of fluctuations decreases with time, then the system goes to stable equilibrium. Note that if the movement in the direction of this equilibrium can be called an evolution of the system, then the foregoing statement can be reformulated in the following way: the power of fluctuations decreases during the process of the system's evolution. Since the state N* is a goal of evolution then, if we understand the evolution in this sense, the Lyapunov function L will be a goal function.

In the opposite case, when the equilibrium is unstable, the power of fluctuations increases with time. We can say that the growth of fluctuations in the system, when it leaves thermodynamic equilibrium, is a sufficient condition for its destruction, since upon their action, sooner or later the system will come to the thermodynamic equilibrium again.

6.7. Lyapunov function and exergy

Let the function p be p(J) = J — ln J — 1, then p(1) = 0; (p = 1 — J—1)j=1 = 0; p' = 1/ J2 > 0, and the corresponding Lyapunov function will be n

When we introduced such a kind of Lyapunov function then, generally speaking, we assumed that the equilibrium N* was non-trivial. It is necessary that the function L and its Lee derivative do not have any singularities at the equilibrium point. However, it is easy to see that both the first and the second derivatives of p tend to infinity as J! 0. On the other hand, apparently, there are many cases when such a sort of Lyapunov function can be used even for N* = 0. Namely, there is a similar case to what we have here when lim

] L = £ [(Ni - N*) - N* ln(Ni/N*)] = X N = N. (. i=1 J i=1

Comparing this result and the results of Section 6.4, we can see that in the limiting case, the Lyapunov function belonging to this type coincides with the same one for trivial equilibrium.P

If we re-write Eq. (7.1) as L = £?=1 [N* ln(N*/N) - (N* - N)] and compare this expression and the expression for exergy, suggested in Chapter 5, we can see that these expressions are equivalent (within units, in which the species numbers are measured), but only when assuming that the reference state is equivalent to the current state of the system (5.1) for which the Lyapunov function (7.1) is defined (see definition of exergy). Using frequencies pt = NjN and p* = N*/N* we re-write the expression for L in the form:

where K(p ! p*) = Kpp* = £n=1 P* ln(p*/pi) $ 0 is Kullback's measure, which is equal to the increment of information when the distribution p is transformed to p*:

Let us assume that the general dynamics of the system could be presented as a sum of the fast dynamics of the total biomass N and the slow dynamics of community composition p. Then the total biomass is quickly established at the equilibrium level, so that N = N* and L = N*Kpp* = NKpp*. The last result may be interpreted in the following way: Kpp* is a specific value of information store per one unit of biomass in the community, which is spent when the current structure of the community evolves to the stable equilibrium p*. The other interpretation is: Kpp* is an information distance from a current state to equilibrium.

Let the trajectory be started at some initial point N0: {N°,...,N0} (or (p0,N0): p0: {p?,...,pn}; N0 = £N0). Then the Lyapunov function L(N0, N*) can be considered as a distance from the initial point N0 to the non-trivial equilibrium N*: If the equilibrium is stable then the trajectory has to move to it, and the distance has to be reduced. However, another interpretation is also possible here. If tPhe initial point is situated somewhere near the thermodynamic equilibrium at the simplex £n=1 N0 = N0, the point N* is far from it so that N0/N* = e << 1 and, keeping in mind that the exergy is almost equal to zero in a vicinity of thermodynamic equilibrium, then the value L(N0, N*) can be considered as the exergy which was accumulated by the system in the course of evolution from thermodynamic to dynamic equilibrium. The evolution is a forced movement, which occurs on account of the exchange between the system and its environment. Spontaneous processes, which are accompanied by the exergy dissipation, shift the dynamic equilibrium to the thermodynamic one. Note that when we talk about the evolution we imply the following process.

By representing the expression for L as

L(No, N*) = Ex = N^Kpop, + ln(1/e) - 1 + e J = ex N*

we see that the value ex = Kp0p> + ln(1/e) — 1 + e < Kpap> + ln(1/e) can be considered as a specific exergy per individual or one unit of biomass, which consists of two items: the first is determined by the increment of information (Kpop> ) caused by the evolution of composition (implicitly assuming that the total biomass is not changed), and the second is determined by the increment of information, caused by the growth of the total biomass by (1/e) times under the permanent composition.

6.8. One more Lyapunov function

Let us consider the following function p(J) = J ln J — J + 1. It is obvious that p(1) = 0; (p J1 = (ln JJ1 = 0; p" = 1/J > 0, and the corresponding Lyapunov function is n

If we again compare Eq. (8.1) and the expression for exergy, we can see that the latter coincides with the formally defined Lyapunov function. However, there is a difference between this exergy and the exergy defined in Section 6.7: if in the previous case, a reference state, which corresponds to thermodynamic equilibrium, was a current (initial) state and not at a dynamic equilibrium, then a reference state is a dynamic equilibrium, close to a thermodynamic one. If in the first case, a goal of evolution is the dynamic equilibrium, which is far from the thermodynamic one, then in the second case, a goal of evolution is not defined. The system is simply going away from thermodynamic equilibrium, which is an unstable dynamic one in this case. The value

L(N*, N) = NKp*p + N ln(N/N*) — (N — N*) = N^p + ln(1/e) — 1 + e J, (8.2)

where Kp*p = Yj= 1 pi lnp/p*) and e = N* /N, can also be interpreted as a measure of how the system has receded from thermodynamic equilibrium. The Kullback measure Kp*p is equal to the increment of information in the course of evolution of the structure when it is transformed from some prime pattern to the current one. If we assume e « 1 then the value ex = Kp*p + ln(1/e) also may be considered as a specific current exergy.

Logically, we may say that the origin of life can be considered as the loss of stability for thermodynamic equilibrium and the movement of the system away from it along one trajectory. In this case (in accordance with Chetaev instability theorem), if the exergy increases along this trajectory, i.e. the inequality dL/dt = d(Ex)/dt > 0 takes place, then the thermodynamic equilibrium is unstable. It is easy to see that this is the other formulation for J0rgensen's maximal principle. (Note that here we implicitly assume that the reference state, i.e. the thermodynamic equilibrium, is also one of the possible equilibriums of the considered dynamical system.)

In spite of an outer similarity between these two definitions of exergy, there is a deep difference between them. The point is that in the first case, we deal with a teleological system, which has a goal function; the system "knows" about the final target of evolution (non-trivial equilibrium) and tends to it. We assume implicitly that the system could tear away from a sphere of attraction of the thermodynamic equilibrium, and then it can move to the known target. In the second case, the target is to tear from the sphere, the furthest is not important. Later on, we shall consider the problem in detail.

Finally, it is necessary to note that the "second" exergy has a logarithmic singularity when one or several Nt! 0 when Ex n. The problem of how to regularise the singularity will also be considered later on. Note that one method of regularisation was already suggested in Section 5.5 of Chapter 5.

6.9. What kind of Lyapunov function we could construct if one or several equilibrium coordinates tend to zero

In Section 6.8 the exergy was defined as

where the equilibrium Np is considered as a reference state and interpreted as a thermodynamic equilibrium corresponding to some "pre-biological" state. The chain of these arguments is logically faultless, except for one fact: when no life exists, at the thermodynamic equilibrium, all the concentrations concerning the living matter must be equal to zero. Then the corresponding items in expression (9.1) for exergy tend to infinity and by the same token are "blocking up" an influence of other components. In order to bypass the difficulty, J0rgensen (1992c) suggested the so-called concept of "inorganic soup", when these concentrations are very close to zero but, nevertheless, differ from zero. Another by-pass method using a "genetic" paradigm was described in Section 6.5. Certainly, in this case the corresponding items are finite; however, their influence remains prevalent. Perhaps there is a deep biological sense in this, but we would like to suggest here another by-pass based on a formal application of Lyapunov's theory.

Let the considered system include not only biotic components, Nk, k [ v, but also abiotic ones, Ns, s [ V, for instance some chemical elements or substances. We assume that v and V are some subset of the set of indices so that v U V = [ 1, 2,..., n]. It is obvious that at an exact thermodynamic equilibrium N* = 0, k [ v and N* — 0, s [ V. Instead of the exergy defined by Eq. (9.1) we shall consider the function:

Since only^the positive variations, SN*, k [ v, are admissible then the second differential S2Ex will be positive definite, Ex(N*), and the function Ex(N) is the Lyapunov function. If the exergy, defined in such a way, increases along even one trajectory, then the reference state is unstable.

We illustrate this by a simple example. Suppose there is a system consisting of a living biomass, N, and an inorganic resource, C. Then the exergy will be Ex = C ln(C/C*)(CC*)N.

Being given the dynamic equations for N and C in the form describing a simplest biological cycle, dC/dt = -aCN + mN, dN/dt = aCN - mN, so that N + C = C*, we get: dEx/dt = ln(C/C*)(dC/dt) + dN/dt = ((dN/dt)(1 - ln(C/C*)) > 0.

Since N = C* — C < C* for any N > 0 then the second multiplier is always positive.

In order to increase the exergy, it is necessary that dN/dt = N(aC - m)= N (aC* - m) - aN > 0.

In the vicinity of N = 0 this is fulfilled if C* > m/a. This means the following: in order for a biological cycle to start "turning", an initial value of "turned" matter will be more than the ratio (m/a). With a small amount of matter when exergy does not increase, the cycle would not turn!

Of course, the example is trivial, and the same result could be achieved in a different way, without the use of the exergy concept. But in our opinion, the general idea is more important than a concrete result in this case.

6.10. One more ecological example

Note that we have forgotten about another "trivial" idea, i.e. how to by-pass the singularity of exergy in the "trivial" equilibrium of ecological models: we can simply "shift" the coordinates' system of phase space (J0rgensen et al., 1995a,b). As a result, the trivial equilibrium becomes non-trivial. We shall illustrate the method with the help of a simple example.

Consider the simple point model of a pond used by Mejer and J0rgensen (1978) for illustration of the exergy concept. The model has the following equations (this is a typical chemostat model):

dPs Q

where the state variables Ps and Pa are concentrations of soluble and algal-bound phosphorus. Pin is the soluble phosphorus concentration in the inflow, Q is the rate of outflow (or dilution rate), V is the volume; m = MmaXPs/(K + Ps) is the P-uptake rate and m is the rate of remineralisation. An obvious constraint for such a model is the positiveness of state variables. By summing these equations, we obtain the equation for the total phosphorus Ptot = Ps + Pa:

dPtot Q

We assume that the total amount of phosphorus is conserved in the system. It is possible if V « Q, and the equilibrium with respect to Ptot is established very quickly. Then the system of equations (10.1) is reduced to one equation:

where the reaction coordinate J is defined by any one of the conditions:

Pa = Paq + j, ps = Peq - j, while Paq and Psq designate concentrations at the thermodynamic equilibrium. If the inorganic soup represents the state of thermodynamic equilibrium for living nature, we have Peq < 10-50 gP/ m3 from estimates by Morowitz (1968). Thus, even at the beginning of organic evolution, which corresponds to the thermodynamic equilibrium state of J = 0 in Eq. (10.3), the concentration of organic matter is not zero but a negligibly small quantity. Therefore, if we are precise, Pa $ Paq. This minor adjustment of the constraints must not affect the dynamic behaviour of a model in other regions of the phase space. Therefore, we have to transfer the origin of our coordinate system to the thermodynamic equilibrium point while retaining the same phase portrait as before, within the shifted positive semi-axes.

In the general case of n living components with the thermodynamic equilibrium state Neq, the positive orthant, Pn, of the n-dimensional space must be transformed to Pn =

Fig. 6.4. Phase portrait of a prey-predator type system consisting of equations for two populations: (a) formal change of variables Ni = Ni — Nfq retains the invariance of the positive orthant Pn: Ni > 0, i = 1,n, (b) "old" equations for "new" variables result in the invariance of the "shifted" orthant Pn: Ni > Nfq.

Fig. 6.4. Phase portrait of a prey-predator type system consisting of equations for two populations: (a) formal change of variables Ni = Ni — Nfq retains the invariance of the positive orthant Pn: Ni > 0, i = 1,n, (b) "old" equations for "new" variables result in the invariance of the "shifted" orthant Pn: Ni > Nfq.

Pn — Neq. Practically, this means replacing the state variables of the model by their shifted values.

Note that the above procedure does not correspond merely to the formal change of variables, Ni = Ni — Nfq, i = 1,n, since the latter keeps the phase portrait unchanged (see Fig. 6.4a), whereas the purpose is to have it shifted and invariant for the transformed equations of the model (Fig. 6.4b). Since the values of Nfq are negligibly small, the dynamics of the model far from the equilibrium are not affected by this transformation.

Applying this general idea to the phosphorus model, described by Eq. (10.3), we replace the state variable Pa by Pa — P^q and obtain j = [m — m — Q j. (10.4)

Furthermore, (L(J) < m(J) as Ps » Peq. It has to be determined now whether the exergy function is the Lyapunov function by using the equilibrium J = 0, corresponding to the thermodynamic equilibrium, as the reference state. We have

when Ex(0) = 0 and Ex( j) > 0 when J > 0. The derivative dEx/dJ = RT0 ln[(Peq + j)Peq/(Peq — J)Ptq] (10.6)

is always positive (since J is always non-negative), except at the point J = 0 where it vanishes. Note that d2Ex/d j2 = RT0[(P^q + J)—1 + (P^q + j)—1 ] > 0, whenever J # Peq, so that function Ex( J) has its local minimum at point J = 0: Ex(0) = 0 (which is also a global minimum in accordance with thermodynamic theory).

It follows from Eqs. (10.5) and (10.6) that the derivative of Ex(J) by virtue of Eq. (10.4), i.e. along the trajectories, is defined as d(Ex) SExdJ „,,, , (Peq + J)Ptq[ Q1 - =--= RT0 ln—^—eq- MJ ~ m — —

everywhere in some finite domain (excluding the point J = 0, in which the model equation gives a positive increase in the concentration of algal phosphorus, i.e. whenever m(J) > m + Q/V.

Thus, as seen, exergy Ex( J) possesses all the basic properties of the Lyapunov function; the derivative has the same sign as the function itself. According to Chetaev instability theorem, Ex(J) verifies local instability at the equilibrium state J(t) ; 0. This means that any initial deviation, however small, from the thermodynamic equilibrium state t = 0 will cause an increase in Ex. Such behaviour of model trajectories is consistent with the concept that life began as small fluctuations in the vicinity of thermodynamic equilibrium and a gradual movement away from it (Schrodinger, 1944).

Notice that we have only used (1) the invariance of the phase orthant for the model equations and (2) the mass conservation principle of the system to test the "Lyapunov" properties of exergy. We have not used dynamic equations (10.1) beyond property m(J) > m + Q/V.

Hence, we may expect these properties to occur in a sufficiently wide class of ecological models possessing the above-mentioned characteristics. Such models could be referred to as exergical at zero, with the idea that model trajectories go away from thermodynamic equilibrium and exergy increases along these trajectories.

6.11. Problems of thermodynamic interpretation for ecological models

We are here dealing with a principal distinction between ecological models and those of theoretical mechanisms or chemical kinetics, where the minimum of a potential function (or the minimum of a thermodynamic potential) corresponds to a stable steady state. The dynamics of such systems are determined by the use of potential functions, while for ecological models (especially models of mathematical ecology), such a simple form of dynamic equations becomes practically unacceptable so that the equations have to be chosen by other considerations. These represent, in essence, a phenomenological description of the ecosystem. Nevertheless, in Chapter 7 we try to give a thermodynamic justification for the basic equations of mathematical ecology. We think that, after this procedure, we would be easily able to interpret the obtained results.

From the standpoint of thermodynamics, any ecological system should be exergical at zero but, on the other hand, the exergy function of the classical Lotka-Volterra "prey-predator" model:

dN1 dN2

dt dt is not the Lyapunov function. To write the expression for exergy, we should first expand system (11.1) with non-living environmental variable N0 (N1 and N2 are the biomasses of prey and predator) such that the mass conservation law holds: N0 + Nj + N2 = N = const. Then, bearing in mind the modification to which any model should be subject in order to be considered an exergical system at zero, we have the following expression for exergy according to definition:

Differentiating Eq. (11.2) along trajectories of Eq. (11.1) results in d(Ex) f / Nj \ dNj , / N2\ dN2

It is possible to prove (J0rgensen et al., 1995a,b) that this derivative is neither positive nor negative and, furthermore, in any small vicinity of (Neq, N2q) such points (Nj, N2) > (Neq,N2q) exist, so that d(Ex)(N1;N2)/dt < 0.

These observations can be explained from general Lyapunov stability concepts, since Chetaev instability theorem actually covers only the cases that, in a certain sense (that of time reversion, changing t to - t), are opposite to those of asymptotic stability, i.e. to phase patterns of the generalized topological node type (Fig. 6.1a and b). However, the prey-predator case corresponds to neither of these two extreme types of phase patterns (Fig. 6.1c).

Generally speaking, the "prey-predator" model is a curious example of a dissipative system having (as a conservative system) an integral, (N\)m(N2)a exp[-y(Nj + N2)] = const: Maybe it is more correct to speak of a through-flow (not a dissipative) system, a proper existence of which is maintained by a permanent flow of energy through it. Such systems are well known in mechanics (gyroscopes). Let us calculate the function of dissipation for this system Diss = A1(dN1/dt) + A2(dN2/dt), where Aj = RT0 ln(N1/N1*) and A2 = RT0 ln(N2/N2) are the affinities of prey and predator with respect to their equilibriums Np1 = m/y and N2 = a/y, and the derivatives are taken from Eq. (11.1). Then

Diss = RT0-|aN1 ln- mN2 ln^ - yNiN^ln^ - lnNL^|. (11.4)

It is easily shown that in the domain, lying above the line N2 = (a/m)Nj, the function of dissipation is negative, and it is positive in the domain lying below (Fig. 6.5); it is identically equal to zero along this line.

Since the function of dissipation is the entropy production, we can interpret the "prey-predator" system as some entropy machine, which produces entropy along the part of trajectory abc and releases it along the part cda, disturbing formally the Second Law.

Fig. 6.5. Phase portrait of the "prey-predator" system: (+ )-domain, in which the system produces entropy; (— )-domain, in which the system releases it.

However, if we integrate the function of dissipation along the entire close trajectory abcda, then as a result we get a zero, i.e. if locally the entropy can both increase and decrease along the close trajectory, then globally, on average, the entropy does not change.

If we keep in mind that the derivative of exergy is equal to the function of dissipation taken with the opposite sign, then all these results can be re-formulated in "exergical" terms. For instance, we can say that along the part cda the exergy is accumulated, and it is dissipated along abc. As we have already said this derivative is either positive or negative along the entire cycle. On the whole, when the cycle is completed, then the value of exergy is returned to its initial value.

We do not use the missing "exergical" property in a model as an argument against the model in general. Many theoretical models do not claim to be adequate in all regions of the phase space and, in particular, not in a critical region such as in the vicinity of zero equilibrium. Most models were developed to provide a reasonable description in a region where a biological community or ecosystem was believed to function normally, for example in the vicinity of a non-trivial or feasible steady state, N * > 0 (with plausible values of state variables).

We could see that when we tried to define a correspondence between exergy and Lyapunov function, the principal difficulty is how to define a reference state in the case of exergy and an equilibrium state in the Lyapunov case. Despite all the similarity between expressions for exergy and Lyapunov function, there are principal differences in their meanings. The expression for exergy contains values of state variables at the thermodynamic equilibrium point Neq, whereas the Lyapunov function is referred to the steady state N*. As accepted theoretically, N* is associated with a state to which the ecosystem normally evolves or at which it has its normal functions. In principle, such a state is irreversible to the thermodynamic equilibrium one.

However, to represent the system evolution as a sequence of stages, where at every one of them the system evolves from one unstable to another stable equilibrium, we can consider the unstable equilibrium as some intermediate reference state. The point when the system, at the previous stage, reaches the stable equilibrium can be considered as a new thermodynamic equilibrium for a closed at a given moment system. When the system is open, this equilibrium becomes unstable as a result of interaction between the system and its environment, and the next stage of evolution to a new stable equilibrium is starting, while the systems exergy begins to increase as well.

The conclusions of this section concern the behaviour of a system with the same values for its internal and external parameters (i.e. coefficients in model equations). However, as stated in the introduction, the "exergy principle" considers a slow evolution of ordering or structure of the system in the sense that the parameters may also change in addition to the state variables. Could the exergy principle indicate the direction in which such reorganisation will evolve? Could it predict the change in system parameter in response to perturbations? These questions are discussed in Chapters 7 and 8.

6.12. Complexity versus stability

Among ecologists it is almost taken as an axiom that ecosystems which are more complex in structure are more stable. There is a wide variety of arguments in favour of this thesis, supported by observations of real ecosystems.

Thus, for instance, laboratory systems of only two species, the predator and prey, most usually prove to be unstable; population explosions of pests are more typical for agroecosystems than for natural ecosystems, and their effect is more disastrous when the crops are monocultural. In contrast, the communities of rain forests, rich in specific composition and interspecific connections, demonstrate very stable functioning: there are no population explosions and population oscillations are much less pronounced in forests of the sub-arctic zone with less species diversity and relatively greater populations. It is believed that ecosystems of complex structures are more stable under perturbations of environmental factors and the random oscillations in populations of some species, whereas more simple structures, such as the sub-arctic fauna communities, e.g. when subject to sharp population oscillations, are unable to dampen out the perturbations.

Thus, the greater stability of natural ecosystems versus agroecosystems could be attributed to the longer co-evolution period of species composing natural ecosystems. The greater stability of trophic communities compared to the communities of the sub-arctic zone can be explained by the destabilising effects of sharp oscillations in climatic conditions.

But practically all of these facts and observations could also have another interpretation, not appealing to ecosystem complexity. We may remark that all these speculations are not only based on strong and correct definitions of complexity and its measure, but also rest on the intuitive idea that an increase in such ecosystem characteristics—the number of species and trophic levels—makes the number of interspecific connections and their strength comply with higher structural complexity. And though we do not have a formal definition for ecosystem complexity, which would be similar in universality, say, to the definition of stability via stability of equilibrium in an appropriate model, still the mentioned ecosystem characteristics are explicitly present in systems of model equations, thus enabling one to judge the influence of these characteristics on the model stability. The problem is thereby transmitted to the field of dynamical models of ecosystems, where the corresponding analysis is usually referred to as complexity versus stability.

The stability analysis of models of the various ecosystems has shown (see, for instance, Svirezhev and Logofet, 1978) that higher complexity might:

• reduce the probability of stability,

• increase this probability and

• in no way affect stability.

Using the terminology of the well-known physicist, we shall not "obscure this already well-confused subject further". The only obvious conclusion is that within a framework of mathematical models, there is no use in looking for a unique relation between complexity and stability that in the particular case is determined by peculiarities of structures under consideration and the specific character of mathematical formulations.

6.13. Summary of the ecological important issues

The concept of ecosystem stability has several meanings expressed as resilience, resistance, persistence, Lyapunov stability, thermodynamic stability and buffer capacity. The concept is under all circumstances multi-dimensional because the question is: stable in what context? It is also important to quantify the stability. Resilience, meaning ability to return to normal, can hardly be quantified, while resistance and buffer capacity can be quantified. They express what can be considered the inverse sensitivity, defined as the change of a state variable relative to the change caused by the forcing functions (impacts). The higher buffer capacity and the more resistance, the more change in the forcing functions (impact) to change the system. The change becomes multi-dimensional because the change of the system can be described by several (many) state variables. It can be shown that, by the use of model results, exergy and the sum of several buffer capacities are well correlated.

It has been shown in this chapter that exergy has all the properties of the Lyapunov function and the derivate has the same sign as the function itself. Exergy and biomass are both zero at thermodynamic equilibrium, but the system will move towards any point with higher exergy according to Lyapunov stability provided that the new point is stable. Exergy can be considered accumulated by the system in the course of the evolution from thermodynamic equilibrium to a dynamic equilibrium. According to Chetaev instability theorem, exergy has local instability at the thermodynamic equilibrium. This implies that any initial deviation, even a small one, from the thermodynamic equilibrium will cause increase of exergy. The system will move further away from thermodynamic equilibrium. It is also consistent with Schrodinger (1944) that life began as small fluctuations in the vicinity of thermodynamic equilibrium and a gradual movement away from it took place afterwards.

As previously discussed there is no simple relationship between the complexity of the system and the stability. The spectrum, not necessarily the size, of buffer capacities is, however, wider the higher the biodiversity is because the presence of more species must give increased probability that one of the species is at least able to cope with the focal problem.

Chapter 7

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