## Models of ecosystems thermodynamic basis and methods II Competition and trophic level

Struggle for Life is a permanent reality in the Nature and the Society.

T. Malthus

### 8.1. Introduction

While in Chapter 7 vertically structured ecosystems, with interactions between trophic levels, were considered, this chapter is devoted to the interactions between species on a trophic level, which, in turn, is considered as a horizontally structured system. From all the five theoretically possible types of interaction between species on one level (symbiosis, commensalisms, amensalism, neutralism and competition, see also Chapter 9) we shall only consider competition, since in nature it is the most widespread type. The latter is a non-linear process. For instance, if we keep in mind the well-known competitive exclusion principle by G.F. Gause, which forbids co-existence of two species with similar ecological requirements in one habitat, we immediately get a non-linear operator that takes us outside the boundaries of linear thermodynamics. In fact, the third Caratheodory Axiom of Thermodynamics states that intensive thermodynamic variables of a joined system are equal to mean values of partial variables of constituent subsystems. It is easy to see that Gause's principle contradicts this axiom. Nevertheless, some thermodynamics statements and theorems, and in particular Prigogine-like theorems, can be valid and proved for this case.

### 8.2. Thermodynamics of a competing community

So, a trophic level is considered as a horizontally structured biological community of competing populations ("competing community"). Their number (number of species) varies in a wide interval, from one to a hundred, but they are all connected by a relation of competition. From the thermodynamic point of view the competition is a typical dissi-pative process, producing entropy. Thus, the competing community is a typical dissipative structure, the stability of which is maintained by permanent export of entropy from the system into its environment, so that the competing community is also an open system.

Let us consider (as in Chapter 7) an open system, within which an equilibrium in relation to temperature and pressure, T, p = constant. Thus, such irreversible processes as heat transport, viscous flows, gravitational sedimentation, etc. are not included in the system. But equilibrium in relation to the distribution of matter or particles, which are capable of a chemical (and biological) interaction, was not established. It is evident that within such a system the entropy cannot increase by means of heat transfer between the compartments with different temperatures. The entropy can increase only by means of chemical reactions, biological interactions (like competition), transfers of mass (particles, individuals) between different phases and compartments of the system and, generally speaking, by means of any process which can be characterised by the change of chemical and biological potentials. As far as the processes of heat exchange between the system and its environment are concerned, we assume that they are in equilibrium.

In this case the function of free energy F and Gibbs' potential G play the most important role. The value dG < dF = d^ — T dS is a fraction of the internal energy d^, which can be transferred either into work or spent on the change of the number of particles Ni. The term T dS, where S is the entropy, corresponds to a fraction of d^, which is transferred into heat. Since the characteristic functions F = F(T, v,N1;...,Nn) and G = G(T,p,N1;..., Nn), where the vector N = {N1,...,Nn} is the vector of numbers (or moles) of chemical molecules or biological individuals, save the potential properties even for the systems with changed composition N1 ; . ; Nn; their exact differentials are written as

where ¡xk = (>F/>Nk)vT = (>G/>Nk)p T. By integrating Eq. (2.1) we get n fNt

We consider a competing community consisting of n species populations; the population size of each kth population is equal to Nk. If the specific energy of kth specimen is ek then the energy contents of kth population will be equal to ekNk. The population is maintained by either the accumulation of solar energy (if the autotrophic level is considered) or the consumption of biomass of the previous trophic level. The corresponding inflow will be denoted by qf. Note that (as we talked about in Chapter 7) even in the case of the autotrophic level this inflow is not necessarily the flux of solar energy: this may be the inflow of some limiting nutrient. It is very important that these inflows depend on the state of the population, i.e. qkn(Nk,R) and >qkn/>Nk \$ 0 , where R is the concentration of some "resource" in the "donor", which can be associated with the environment. The energy balance of kth population can be represented as q^(Nk, R) = Mk(Nk) + Dk(Nk) + ek(dNk/dt + Fk(N, ., Nn) + q?ut(Nk , St) , k = 1,..., n.

Here we repeat the arguments which were used in Chapter 7. In the course of one unit of time the system receives qf(Nk, R) units of energy. One part of energy Mk(Nk) is spent within the population for different metabolic decay processes. As a result of dying-off and mortality processes another part of energy, Dk(Nk), is transferred into dead organic matter. If the energy spent by metabolism immediately transforms into heat then the energy contained in the dead organic matter either transforms into heat (as a result of its decomposition) within the system or leaves the system. In turn, the part ek(dNk/dt) is spent to maintain reproduction and the population growth. It is obvious that both competition between specimens belonging to different populations (interspecific competition) and competition between individuals of the same population (intraspecific competition) are accompanied by a dissipation of some part of the energy, Fk(N1,..., Nn). The dissipation increases (or, at least, does not decrease) with increase in the size of any population, so that dFk/>Nj \$ 0; k, j = 1,..., n. Finally, a residual part qfut(Nk, St) is transferred to another system, an "acceptor" (in particular, to the next trophic level). The last value depends on both the size of kth population and the state of the acceptor, St.

Without loss of generality we can measure the number of species or their biomasses in energy (enthalpy) units (for instance, evaluating biomasses in carbon units we can use the equivalent 1 gC = 41.8 kJ). Formally it means that we can set ei = 1 in Eq. (2.3) and in the further equations.

By summing these equations from 1 to n we get the energy balance for the entire level:

£ Wl(Nk, R) - qout(Nk, St)] = £ k[Mk (Nk) + Dk (Nk)]

If the total biomass of trophic level, N = £n= 1 Nk, is associated with internal or free energy of the system then its change under constant temperature has to be proportional to the change of total entropy taken with inverse sign, — (dS/dt), so that dN dS

One of the main problems in the thermodynamics of open systems is how to divide the total entropy production, (dS/dt), by the two parts, (deS/dt) and (diS/dt), the first of which is the change of entropy caused by different exchange processes between the system and its environment, and the second is the internal entropy production. Note that a successful solution of the general problem is often dependent on a successful choice of such a type of division.

The sum 23=1 Fk(N1,...,Nn) is the total expenditure (in energy units) for competition; the sum JjL 1 [Mk(Nk) + Dk(Nk)] includes both the metabolism of all organisms of the trophic level and the total outflow of dead organic matter. At first sight, it seems that both the sum £n= 1 Fk, which is equal to the rate of energy dissipation caused by competition, and the sum £JL1 (Mk + Dk), which also describes the energy dissipation processes, have to be included in a general expression for the internal entropy production d^/dt, so that d S n n

Then the intensity of the "entropy pump", sucking the entropy out of the system, is equal to the difference between the total inflow and outflow of energy, taken with the opposite sign, i.e.

However, everything is not so simple, and the problem is to determine the entropy of ensemble, community, etc.

It is obvious that the flows qf, q™', Mk and Dk are determining the states of the single isolated kth populations, and their balance is the current values of net free energy accumulated by each population. In other words, the specific balance bk = (qk — q|ut — Mk — Dk)/Nk is the capability of kth specimen to assimilate and to keep the external energy (or resource), which is later spent on both the population growth and competition. Then the value B = £n=1 bkNk = Y.i=1 (qf — q™' — Mk — Dk) can be considered as some integral characteristic of exchange processes. However, the isolated populations, by their nature, do not create such a structure as the competing community. The latter is formed by the competition, which is maintained by the energy dissipation with the rate F = Yn= 1 Fk. Taking into account all these arguments we can write: