Models of ecosystems thermodynamic basis and methods I Trophic chains

So, naturalists observe, a flea Has smaller fleas that on him prey; And these have smaller still to bite 'em;

And so proceed ad infinitum. Jonathan Swift. Poetry, a Rhapsody.

"Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum. And the great fleas themselves, in turn, have greater fleas to go on; While these again have greater still, and greater still, and so on."

De Morgan: A Budget of Paradoxes.

"Chaos had arisen in the Universe, before all things."

Hesiod

7.1. Introduction

From a thermodynamic point of view, any ecosystem is a typical open system, which exists only due to the permanent exchange of energy and matter between the ecosystem and its environment. As a rule it is at dynamic equilibrium, and this equilibrium is far from thermodynamic equilibrium. The latter can be interpreted as the state of death when there is not any living matter in the ecosystem. In fact, if we interrupt all energy and matter flows both into the system and out of it then the ecosystem will perish.

If also keeping in mind the classic "ecological" tradition going back to Lindeman (1942), in which an ecosystem is described by flows of energy and matter both within the ecosystem and between the ecosystem and its environment (which type of system is a field of interest for phenomenological thermodynamics), then the first idea which comes to mind is to directly apply classic thermodynamic methods and models for description of the ecosystem. But here we collide immediately with the problem of reductionism.

In fact, thermodynamics is a physical science, while in ecology there are certainly some conformities which cannot be reduced to physical laws. Strictly speaking, unless we consider ecological systems as physical-chemical ones there are no prohibitions in

Towards a Thermodynamic Theory for Ecological Systems, pp. 153-188 © 2004 Elsevier Ltd. All Rights Reserved.

principle to applying thermodynamic concepts in ecology. However, as soon as we take into account any purely ecological conformities then we come up against a very serious problem: there is not a direct homomorphism between models (in a broad sense) in thermodynamics and models in ecology. At the same time, since we do not know even one example when physical laws were "cancelled" in ecology then all ecological laws have to be compatible with physical principles. Therefore, physical methods can be used in studying some properties of ecosystems, and the conclusions by analogy with physical systems (if used with a proper degree of caution) may be helpful for the ecologist.

7.2. General thermodynamic model of ecosystem

An ecosystem takes up all matter (nutrients) necessary for it from its environment; it gets free energy from solar energy and organic food, and returns matter producing entropy (in general in the form of heat). In the process, organisms (both individuals and combined into populations and biological communities) permanently dissipate the energy in order to maintain their structures (which might be called dissipative), to develop and to evolve. A measure of this dissipated energy is entropy, which has to be exported out of the ecosystem, since a system which accumulates entropy cannot survive in principle. Hence, the access to free energy is a central problem of existence for any living organism. The struggle for life is first of all a struggle for free energy !

Let the considered ecosystem consist of chemical components with numbers of moles c = {cj,..., cm} and living organisms belonging to n different species with numbers N = {Nj,..., Nn}. The ecosystem occupies some volume V.

The ecosystem considered as an entire system has to satisfy the conditions of energy and entropy balance, i.e. for this the First and Second Laws have to be fulfilled. In addition, the temporal derivatives for internal energy U and entropy S are connected with each other by the Gibbs fundamental equation (see Chapter 2):

Here, T is the temperature, p is the pressure, and V is the ecosystem volume; m is the chemical potential of kth component and Mb is the analogue of chemical potential for ¿th species. In the last case, living individuals are considered as some virtual particles.

The free energy available to the ecosystem is equal to F = U — TS; hence, if T = const then

In order to write the First and Second Laws for the ecosystem, we represent the change of thermodynamic potentials, as usual, as the sum of two items: the item which corresponds to internal processes (i), and the item which corresponds to processes of exchange between the system and its environment (e):

dck dt dck dt

dU _ deU djU dS dt dt dt ' dt deS djS dF dt dt ' dt de F djF

Then the First and Second Laws are written as dU = *f > <2.4, and from Eq. (2.2) for T = const we have

dt dt

It is clear that for a "normal" ecosystem the entropy, on average, should not increase:

dt dt dt

Therefore, ( — deS/dt) $ (diS/dt) > 0, i.e. the export of entropy should not be less than the entropy production within the ecosystem. If the ecosystem is functioning at a constant temperature and either constant volume or pressure, then this relation is represented as de F diS de G di S

dt dt dt dt i.e. if isothermal processes take place in the ecosystem they require the input of either free energy or enthalpy.

Let us consider one more concrete model, the so-called model of "dilute solution" (Feistel and Ebeling, 1981; Mauersberger, 1981). In particular, we apply the Planck theory of a dilute solution when living organisms are considered as molecules of different chemical substances submerged into an environment, which is considered as a solvent. About the latter it is assumed that it is at thermodynamic equilibrium and characterised by the numbers of substrate moles c1,..., cm, the given chemical potentials of substrate M1,..., Mm, the given temperature T and the pressure p (Fig. 7.1).

The chemical components are considered as the solvent with a surface which coincides with the surface of the ecosystem, the volume of which is equal to Vs: The pressure within the ecosystem is equal to p + ps. Continuing our analogy we can say that the "ecological" pressure ps corresponds to the osmotic pressure of a solution, and living organisms are "dilute solution" in the sense of Planck's theory.

Fig. 7.1. The "dilute solution" as a model of ecosystem.

We also assume that the numbers of individuals of ith species (i = 1,..., n) within the volume Vs (areal) are subjected to the following phenomenological equations:

It is clear that an arbitrary extensive thermodynamic potential F of our system has to depend on the number of moles of substrate c1;..., cm, the number of individuals of each species N1;...,Nn, the pressure p and the temperature T, so that

The main assumption for the theory of the dilute solution is that the potential F can be represented as the sum of two items:

F = Fo + Fs, where F0 = F(cu..., cm, 0,..., 0;p, T).

In this case, the fundamental Gibbs equation for the living subsystem is written as dUs dSs dVs , n sdNi a" = Ta" 2ps"dT + 2 . (2.9)

Here the index "s" indicates belonging to the living subsystem and Ms is the chemical potential of ith species.

Since Us = 0 for N1 = ••• = Nn = 0 then the linearised form of Us is written as n

where the energy of individual ut is represented as the sum of kinetic (ek) and potential (ep) energy, and also the energy of interaction between the individual and substrate (&iT) : U = ek + ep + sT. Assume that the individuals are in thermal equilibrium with the environment, and their heat capacity is st. For entropy, the Planck approximation is written as

n Ni

where k is Boltzmann's constant and st are the specific entropies. If pi = NjN and Nis the total number of individuals then the entropy per individual is S m m N

— = > pisi — k> p ln p — k ln —. (2.12)

If all specific entropies are approximately equal, s, < s, then Ss m N

— = s — k> p, ln p, — k ln—. (2.13) N = i i Vs V J

It is easy to see that the value of Ss=N reaches a maximum for uniform distribution p, = 1/n : max(Ss/N) = s — k ln(N/Vs) + k ln n; and this maximum grows with an increase of the number of species, but decreases with growth of the total number. If only one species is dominant in the ecosystem, for instance p1 < 1, p2 = ••• = P < 0, then the entropy will be minimal: min(Ss/N) < s — k ln(N/Vs). However, if the specific entropies strictly differ from each other, then the distribution where the entropy attains maximum will also significantly differ from the uniform one. For free energy we obtain:

Fs = Us — TSs = £ Nj ek + ep + ctT + kT ln N — Tst . (2.14)

Knowing free energy we can calculate an "ecological" pressure

9Fs kT n kTN

Since in ecosystems the number of "particles" is much less than 1023, then this pressure is vanishingly small.

For the chemical potential of ith species we have

n Mi Vs where M (0) = ek + ep + criT + kT — Tsi. Using this expression for chemical potential, we can represent free energy and Gibbs potential in a standard form:

It is obvious that if the input of free energy is interrupted then, according to the Second Law, the system tends to thermodynamic equilibrium, in which both F and G have a minimum. The minimum has to be reached under additional constraints, which are given by kinetic equations for "particles". These equations are standard equations of mathematical ecology dN

For instance, these could be the Lotka-Volterra equations.

Since we associate thermodynamic equilibrium with the death of living matter, then the tendency to this is described by the limit transition Ni ! 0. Therefore, in the vicinity of thermodynamic equilibrium we can neglect all non-linear terms in Eq. (2.18) and vary the values of Ni independently. As at thermodynamic equilibrium n

As soon as we try to do the limit transition N,! 0 we obtain:

We see that the first limit vanishes only if 5j=i SN, = 0, i.e. the total number of "particles" of the ecosystem is constant. The second limit is equal to (negative) infinity: we again meet the logarithmic singularity as in Chapter 6. In chemical thermodynamics all these paradoxes are resolved rather simply (see, for instance, Landau and Lifshitz, 1995), but how can we do this in our model? For this we have to keep in mind that the population sizes N, have to be varied in accordance with linearised "ecological" equation (2.18): SN, = a—fit. The latter means that in the vicinity of thermodynamic equilibrium all the species grow exponentially, but each with its own exponent a,. If this condition is fulfilled then both free energy and Gibbs potential (free enthalpy) have a minimum at zero. Note that the constraint is very strong. In particular, from this follows that the function F, in Eq. (2.18) has to be represented as F, = Nf-Nj,..., Nn) where f(0,..., 0) = a, > 0 (compare with Section 6.5).

There is also another method to solve the problem. Assume that:

(a) The matter conservation law is valid: £n=i SN,- = 0.

(b) All organisms have a similar chemical composition: Mi(0) < ••• < Mn(0) < Ms(0).

(c) In the vicinity of zero the distribution of "particles" is close to uniform:

This means that we have to assume that in the vicinity of thermodynamic equilibrium the living "particles" already exist, at least in small quantities. In other words, we come again to the concept of "inorganic soup".

In this state let N{ = Nfq be extremely small and (8Fs/8N,-)TV = (dGjdN)T!Ps = mT be equilibrium values of chemical potential, which are determined by the chemical composition of organisms. Then

It is interesting that if M;q = 0; then we immediately get nN

As a rule, ecosystems are always far from thermodynamic equilibrium, so that Ni >> Nfq, whence M >> kT; Fs >> NkT; Gs >> NkT. These inequalities are not fulfilled for gases and solutions studied by classic thermodynamics; therefore, they express a fundamental difference between physical-chemical and ecological multiparticles systems.

Note that if we know the solution of "ecological" equation (2.18) then in the framework of the "dilute solution" model we can calculate all the thermodynamic functions using formulas (2.14)-(2.17).

7.3. Ecosystem's organisation: trophic chains

The ecosystem structure, presenting the transfer of energy from one species to another trapped in food, linked together by prey-predator (or resource-consumer) type relations, is known in ecology as the trophic chain. In every successive transfer a significant part of energy (70-80%) is lost, being spent on respiration and heat. This restricts the number of "links" in the chain usually to four or five, though in all ecosystems there either already exist (if only in negligible numbers), or are introduced from outside, the specimens which could form the next trophic level (panspermia hypothesis). When the amount of energy coming into the ecosystem abruptly increases, the possibility of forming a new level using the generative material available is realised, and a new trophic level appears and establishes itself in the chain.

Trophic chains are not isolated from each other, but interweave to form a trophic web. An example of such a trophic web is displayed in Fig. 7.2.

Sunlight

Sunlight

matter

Producers

Primary consumers

Intermediate level

Fig. 7.2. Part of a trophic web in a stream ecosystem in South Wales. (From Jones, J.R.E., 1949. J. Anim. Ecol. 18(2), 142-159.)

I Input of organic matter

Producers

Primary consumers

Intermediate level

Secondary consumers

Fig. 7.2. Part of a trophic web in a stream ecosystem in South Wales. (From Jones, J.R.E., 1949. J. Anim. Ecol. 18(2), 142-159.)

In order to judge about the direction of the energy transfer in ecosystems which is realised along trophic chains and the stability of these processes (and hence about their possible evolution), it is desirable to have some energetic criteria for these inferences. It is natural that the first idea is to apply the methods of the thermodynamics of irreversible processes.

Processes in ecological systems occur, as usual, under the conditions of constant temperature, volume and pressure. In this case, the function of free energy F and Gibbs potential G play the most important role. Since the physical volume of living organisms does not practically change in the process of energy exchange (slowly increasing as a result of energy accumulation and growth) then (pdV < Vdp < 0) dU < dH and dG < dF.

Consider a simplified scheme of energy exchange between different trophic levels in the chain with length n (Patten, 1968). We assume that each level is an open thermodynamic system, so that dFk = deFk + diFk = dUk — TdSk = dUk — TdeSk — T^; (3.1)

Because each trophic level exists, it has to export entropy, deSk < 0; ldeSk l $ diSk > 0, and we have deFk $ dUk + TdiSk. (3.2)

Since in our case (isothermal system at constant pressure) TdeSk = dHk — deGk then deGk $ dHk + TdiSk. (3.3)

Thus, in order to provide the export of entropy out of the kth trophic level, the inflow of free energy (enthalpy) from the environment has to overlap both the change of enthalpy of the trophic level and its expenditure for the internal entropy production, when the energy is dissipated in different decay processes, such as metabolism and dying-off processes. The equality in Eq. (3.3) holds in the case of dynamic equilibrium.

At the first autotrophic level, the inflow of free enthalpy, deG1, corresponds to accumulated solar energy. Note that deG1 does not necessarily have to be solar energy. It may be the chemical energy of detritus for a so-called "detritus trophic chain". If we assume that the Liebig "Limiting Factors Principle" is valid, then even for the first autotrophic level the value of deG1 may also be the chemical energy of some limiting nutrients. At the other levels, the free enthalpy input is determined by the part of enthalpy transferred from the previous level.

Let us represent the change of total enthalpy, dHk, at the kth level, as dHk = dHk + LkdH'[ + (1 — Xk)dH'l, where dHk is the enthalpy of the increment of biomass at this level and dH'l is the enthalpy transferred to the next (k + 1)th level. The (1 — Lk)th part of this value is lost in the process of transfer, so that only Lkth part reaches the next level: deGk+1 = LkdHf. Assume that the lost parts of energy, TdiSkk+1 = (1 — Lk)dH'[ are dissipated; therefore, they have to be included into the total entropy production when all these levels are joined into a single ecosystem.

Thus, conditions (3.3) for the entire trophic chain are written as deG1 $ dH1 + X1dH'i + (1 - A1 )dHi + TdS, deGi = A1dHjf $ dH2 + A2dH"2 + (1 - X2)dH'2 + Td^, deGk = A—dHk-1 $ dH'k + AkdHl + (1 - Ak)dH'[ + TdiSk, (3.4)

Since the chain ends with the nth link then dHn = 0. By summing these inequalities, we obtain n n n deG1 $ £ dHk + ^ (1 - Ak)dH + T X di^k k=1 k=1 k=1

From this expression we can see that the part of the free enthalpy of solar energy accumulated by autotrophic organisms is spent on increasing both its biomass and the biomass of other levels forming part of the chain (it is equal to £n=1 dHk), and the other part, ^Sn=1 (di Sk + di5kk+1), is irreversibly lost in the form of metabolic heat and dead organic matter.

Certainly, we can complicate system (3.4) by taking into consideration the energy inflows from adjacent chains and outflows to them, i.e. by considering so-called branching trophic chains. However, we shall not follow this path, since this is not fundamental to our analysis.

By representing Eqs. (3.4) and (3.5) in the infinitesimal form we get deG1 n dHk n di(Sk + Skk+1)

Using the standard representation of the entropy production: (dS/dt) = (deS/dt) + (diS/dt) and taking into account that (deS/dt) = -(1/T)(dH1/dt) and (diS/dt) = (di(Sk + Sk,k+1)/dt) (deS/dt) = -(1/T)(deG1/dt) we obtain:

If we measure the total biomass of the trophic chain, N = Y1=1 Nk, in enthalpy units then ^n=1 (dHk/dt) = (dN/dt), and Eq. (3.7) is represented as

To increase the total biomass, the entropy has to decrease; moreover, the rate of entropy decrease has to be higher than the rate of biomass growth: ldS/dtl $ (dN / dt). In the steady state (dynamic equilibrium), when the processes of energy inflow and utilisation balance each other, (dS/dt) = 0, then (dN/dt) = 0, i.e. the total enthalpy of the system contained in the biomass of the ecosystem does not change, although the enthalpy (biomass) of each level separately may change. The decrease of entropy in the course of the ecosystem evolution implies the accumulation of free enthalpy (energy); conversely, the increase of entropy can be interpreted as the free enthalpy dissipation.

One can see that these considerations give us a simplified scheme of energy balance in ecosystems. At the same time, in order to conclude that the transfer of energy from kth to (k + 1)th level is accompanied by some losses, and the increase of free energy at the acceptor level has not to exceed its decrease at the donor level, it is sufficient to use the matter and energy conservation laws; it is not necessary to apply the methods of non-equilibrium thermodynamics. This would be justified if, prior to estimating the rate of energy dissipation in the system, we could predict the direction of its transfer. For this we would need to know in turn the values of velocity and moving forces of the trophic transfer of energy. Of course, if the increment of biomass at each trophic level is known, there is not a problem to estimate these velocities, but that is concerned with moving forces, which is a problem.

It seems that if we know the energy contents of a biomass unit at the level of donor, Hd, and acceptor, Ha, then the moving force X = Hd - Ha. There are well-known methods of thermochemistry which allow the enthalpy of 1 g of dry biomass to be determined by the thermal effect in the reaction of a burnt sample. Then, determining, for instance, the values of enthalpy differences X1 = DH1 = Hd - Ha1 and X1 = DH1 = Hd - Ha1, which correspond to the energy trophical transfers from donor d to two different acceptors a1 and a2, we obtain the estimation for moving forces. If the velocities of energy transfers /1(d ! a1) and J2(d ! a2) are also known, the functions of dissipation can be found: fi1 = X1J1 and B2 = X2J2. They describe the process of energy transfer from the trophic donor to each of the two acceptors.

Nevertheless, if we want to forecast the direction of energy in irreversible trophic transfers, this method could be applied only in an ideal case where the chemical compositions of 1 g of the biomasses of the donor and acceptors are identical. This would be equivalent to the statement that the character and direction of the trophic transfer are only determined by the difference in the energy contents of the biomass unit between different components of the ecosystem. A lot of ecological data bear witness against such a suggestion and, on the contrary, the change of energy contents in biomasses is said to be a result of the combined action of many factors, just as the growth of organisms is the final result in the course of the whole complex of cell metabolic processes (see, for instance Tables 2.2-2.4 in Chapter 2).

This means that equal values of energy contained in the biomasses of interacting species are not actually equivalent in a biological sense, so that the values of the energy content of biomass units cannot be used to define the moving forces in trophic transfer. It is appropriate to draw an analogy with the thermodynamic questions of the growth and development of organisms discussed above (Chapter 3). Obviously, the final state of a system in both cases depends on actions within a whole complex of processes for which we now have no adequate description in terms of chemical kinetics. Even if such a description is performed, then the forecast of a system's behaviour over time based on it would depend, in the first place, on the character of kinetic factors of regulations. Calculation of the functions of dissipation using methods of non-equilibrium thermodynamics for systems far from equilibrium could only be important for estimation of their energetic effectiveness and plays no "forecasting role".

However, apart from the energy content there is one more value that characterises biomass: exergy (see Chapter 5). How could we apply the exergy concept to determine the direction of energy transfer between the kth and (k + 1)th levels? For instance, if we now introduce the specific exergies exk and exk+1 of the corresponding biomasses then we can define the active energy transfer in the form of biomass flow from kth to (k + 1)th level as qk k+1 ~ ln(exk+1/exk)NkNk+1. It is obvious that in this case the flow of matter is always directed to such a level where the biomass has a higher specific exergy.

7.4. Dynamic equations of the trophic chain

Let the biomass of the species belonging to the kth trophic level be equal to Nk. A standard estimation of the energy contents of a biomass unit in ecology is the measurement of biomass in terms of carbon units (the quantity in grams of carbon contained in the biomass) and then its multiplication by the enthalpy of 1 g of carbon (~ 10 kcal = 42 kJ). From this point of view the concepts of biomass and energy are equivalent. The existence of the trophic chain is maintained by an inflow of solar energy q1, accumulated at the first autotrophic level as new biomass. Note (as we already mentioned above) that the flux q1 does not necessarily have to be a flux of solar energy; it may be either the flux of chemical energy of detritus or the chemical energy flux of some limiting nutrient (resource). It is very important that the flux is regulated, q1 (N1, R), 9q1 /9N1 $ 0, i.e. a feedback exists in the system. The parameter R is either the external flux of solar radiation or the current value of the resource. One part of the energy is spent within the trophic level, lost in different decay metabolism and dying-off processes (mortality), D1(N1). Another part, in turn, is spent in maintaining reproduction (accumulated in seeds, eggs, support of growth, reaching maturity, etc.) and the growth of population, dN1/dt. The last, residual part q12(N1; N2) is partially transferred to the next level. Then the law of energy (matter) conservation for the first level is presented as q1(N1 )= ^ + q12(N1 , N2) + D1N1). (4.1)

If the transfer is realised without any losses, and the flux of energy accumulated at the second level, q2 = q12, then the energy conservation law for the second level will be q12(N1, N2) = ^ + q23(N2 , N3) + D2N2). (4.2)

Finally, for the chain of n length we have q1(N1) = "N + q12(N1 , N2) + D1(N1) , q12(N1, N2) =

dNT + q23(N2, Ns) + D2(N2), qk-i,k(Nk-1,Nk)= ~Nr + qk,k+i(Nk,Nk+i) + Dk(Nk), k = 3, ..., n - 1; (4-3)

Finally, we get the balance (with respect to matter or energy) dynamic equations of the trophic chain as:

^ = 412(N1, N2) - q23(N2; N3) - D2(N2), d:Nk::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: (4:4)

--I" = qk-1,k(Nk-1, Nk) - qk,k+1(Nk, Nk+1) - Dk (Nk), k = 3, n - 1,

As one can see from these equations, the direction of energy movement given here is not concluded from thermodynamics. This means that, as we assumed beforehand, the input of chemical resource from the environment and the transfer of matter between levels are a result of active transport. In this case, there are conjugating reactions within the system, which have to provide energy for the transfer processes.

dN n

The sum 23=1 Dk (Nk ) includes both the proper metabolisms of organisms of the trophic chain and the outflow of dead organic matter. If the energy spent by the metabolism is immediately transformed into heat, then the energy contained in the dead organic matter is either transformed into heat within the system (as a result of its decay) or exported out of the system. In the first case, the sum Xn=1 Dk(Nk) can be interpreted as the rate of entropy production within the system, T(diS/dt). The intensity of the "entropy pump" sucking the entropy out of the system is equal to the rate of accumulation of solar energy, T(deS/dt) = -q1(N1). Before a consideration of the second case, we present each item of the sum as Dk(Nk) = Df (Nk) + D^(Nk). The term Dkm(Nk) describes the rate of production of metabolic heat, while the term Dd(Nk) describes the mortality at kth level. Then the rate of entropy production within the system T(djS/dt) = YP=i DTk(Nk), but the intensity of the "entropy pump" will be

We say that the chain as a whole is in a dynamic equilibrium when these entropy fluxes are balanced. Then (we shall denote this state by the upper index (eq)):

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