## Stability in mathematics thermodynamics and ecology

Das Sein das Sein des Seinden sei. Alexander von Humboldt "Flora Freibergiensis"

6.1. Introduction. Stability concepts in ecology and mathematics

At the end of the XVIIIth century, young Alexander von Humboldt in his "Flora Freibergiensis" proclaimed the methodological principle applicable to natural systems: "Das Sein das Sein des Seinden sei" (let the being be the being of the being). From our viewpoint, this is another formulation of the general stability principle: "Only stable systems can exist".

It is intuitively clear that both an ecosystem and a biological community that exist sufficiently long in a more or less invariant state (such a property is often called persistence) should possess an intrinsic ability to resist perturbations coming from the environment. This ability is usually termed "stability". It is some general, emergent property (the so-called "scalar invariant") of a system. Apparently, we can observe only stable ecosystems, since all unstable ecosystems had disappeared in the process of evolution. The environment destroyed them, since they could not be adapted to it. (You can see that we introduced the new term: "adaptation". Really, these terms are very close, and we can say that only a stable system is able to be adapted to the environment, i.e. to survive sufficiently long under given environmental conditions.)

In spite of being intuitively clear, "an ability to persist in the course of a sufficiently long time in spite of perturbations" can scarcely be defined in a unique and unambiguous way. The reason is that both the "persistence" and the "perturbations" or "fluctuations" in thermodynamics (as well as the "sufficiently long time") are elements of the idea that need further clarification, to say nothing of the scale of a system under consideration. What is understood by "an ability to persist" and what kind of "perturbations" are relevant? How long can this "sufficiently long time" continue? Different answers to these basic questions and a variety of stability concepts have been proposed and discussed in the literature on mathematical ecology (May, 1973; Svirezhev, 1976, 1987, 2000; Svirezhev and Logofet, 1978, 1995; Jeffries, 1988; Logofet, 1993) and in purely ecological literature (see, for instance, Lewontin, 1969; Usher and Williamson, 1974), and yet few of them have attracted a proper mathematicians' attention (Svirezhev, 1983). The main concepts of

stability and their application to the problem of complexity are very well shown in the book of Nicolis and Prigogine (1989).

Though the notion of stability seems obvious, it is quite a problem to provide it with a precise and unambiguous definition. In fact, stability can be defined in quite a lot of ways, both in verbal and formal terms, either in ecology or mathematics. While none of the "ecological" meanings of stability can now be recognised as the most fundamental one, mathematics is "luckier", giving rise to the notion of Lyapunov stability. It appears to be inherent in, or substantial for, any further notion of stability—at least within the theory of dynamical systems. And even then, this heavily overloaded term found no established ("stable") definition. For instance, the theory of stability, which can be considered as a branch of applied mathematics and mechanics, uses about 30 different definitions of stability. So, the definition of stability is some "fuzzy" definition. Paraphrasing von Neumann's sentence, we can say that "... nobody knows what stability means in reality, that is why in the debate you will always have an advantage".

6.2. Stability concept in thermodynamics and thermodynamic measures of stability

Among the different definitions, we can select two large classes differing with respect to the requirements coming under the heading of "stability". The first group of requirements concerns preservation of the number of species in a community. A community is stable if the number of member-species remains constant over a sufficiently long time. This definition is the closest to various mathematical definitions of stability, such as those of Lagrange and Poincare-Lyapunov.

The second group refers rather to populations than to community, which is considered to be stable when numbers of component populations do not undergo sharp fluctuations. This definition is closer to the thermodynamic (or rather, statistical physics) notion of system stability. In thermodynamics (statistical physics) a system is believed to be stable when large fluctuations, which can leave the system far from equilibrium or even destroy it, are unlikely to happen (see, for instance, Landau and Lifshitz, 1995). Evidently, general thermodynamic concepts (for instance, the stability principle associated in the case of closed systems with the Second Law and, in the case of open systems, with Prigogine's theorem) should be applicable to biological (and, in particular, ecological) systems.

There is a very deep connection between thermodynamics as a physical theory and the mathematical theory of stability. One of the most important concepts in the theory of stability is the Lyapunov functions concept. Positive functions, defined in a phase space of a dynamical system, possess the following property: either monotonous increase or monotonous decrease along trajectories. They can be considered as some special class of goal functions. On the other hand, the main thermodynamic laws (the Second Law, Prigogine's theorem) state the similar properties of monotonicity for special functions called potentials, entropy, etc. These functions are Lyapunov functions and thermo-dynamic laws can be considered as applications of the direct Lyapunov method to special dynamical systems.

It was proved in thermodynamics that in any thermally isolated system for any initial distribution, a single equilibrium distribution is established. Moreover, in accordance with the Second Law the entropy monotonously increases and reaches, at this equilibrium, the maximum Smax . Let us introduce the positive function L = Smax — S and name it Lyapunov's function. It is obvious that it is equal to zero at the equilibrium and its derivative dL/dt = —dS/dt < 0 . If we use the thesaurus of the stability theory, we can say that thermodynamic equilibrium is globally asymptotically stable, and the difference between maximal and current values of entropy is Lyapunov's function of the system, which describes such a general property as Lyapunov stability (see below).

If now we keep in mind one of the definitions of exergy, Ex = T0(Smax — S), and compare it with the definition of L, we can say that exergy is also Lyapunov's function. A negativeness of the exergy derivative in this case means that exergy is dissipated along a trajectory towards thermodynamic equilibrium. It is clear that the opposite inequality for the exergy derivative is possible only if dS/dt < 0, i.e. the system is open.

In a number of simulation experiments on aquatic ecosystem models, Mejer and J0rgensen (1979) and J0rgensen (1982) observed an increase in exergy in response to certain changes in external conditions, i.e. certain perturbations of the system. The exergy correlated positively with values of "ecological buffer capacities", defined (Mejer and J0rgensen, 1979) as the inverse sensitivity of a state variable to changes in the driving force on a model. Basically, the buffer capacity concept, representing the capability of the ecosystem to absorb perturbations, is a measure of stability with respect to perturbations of a certain kind. Therefore, the idea arises that there may be an internal link between the "increasing exergy" concept and the "Lyapunov stability" concept, the most fundamental one in mathematical stability theory. Heuristically, Lyapunov's function is also natural (as well as consistent) for quantifying how far the current state of a system is from some reference state (see, for example, La Salle and Lefschetz, 1961; Rouche et al., 1977). This idea seems consistent with the concept of exergy, which measures the free energy that a system possesses relative to its environment and shows how far the current state of the system is from the state of thermodynamic equilibrium with the environment.

Lyapunov functions exist for a number of theoretical models of population and community dynamics (e.g. Goel et al., 1971; Svirezhev and Logofet, 1978). These functions look similar to the exergy expressions presented in Chapter 5, which has motivated us to investigate the link between exergy and Lyapunov stability functions. It should be emphasised that Lyapunov functions, depending on their particular forms and properties, may represent either the stability or instability of a system state.

From the "stability" point of view, it means that thermodynamic equilibrium becomes unstable. The illustrations of asymptotic stability and instability are shown in Fig. 6.1a-c. Note that here the equilibrium domain, to which all the trajectories tend (or depart from) at t! consists of a single point. In the general case, the domain, also called the limiting set, can possess a more complex structure. These sets can be either attracting (Fig. 6.1a) or repulsing (Fig. 6.1b). The attracting set is often simply called the attractor. Some intermediate case is possible when trajectories along some directions are attracted, but are repulsed along others (Fig. 6.1c).

In the general case the following situation is possible: if perturbations (fluctuations) are relatively small and do not exceed some threshold, then the system remains in the vicinity of a certain equilibrium; if perturbations exceed it, then the system recedes from it. In this case the equilibrium is locally stable but globally unstable. In other words, if the system is Fig. 6.1. Typical phase patterns in the vicinity of the zero equilibrium state: (a) asymptotic stability (any trajectory tends to zero); (b) instability as covered by Chetaev's instability theorem (any trajectory evolves from zero); and (c) instability in the Lotka-Volterra case (where there is a trajectory that tends to zero).

Fig. 6.1. Typical phase patterns in the vicinity of the zero equilibrium state: (a) asymptotic stability (any trajectory tends to zero); (b) instability as covered by Chetaev's instability theorem (any trajectory evolves from zero); and (c) instability in the Lotka-Volterra case (where there is a trajectory that tends to zero).

stable for any initial perturbations, then we talk about global stability, and the attractor is named a global attractor. For instance, thermodynamic equilibrium in a thermally isolated system is a global attractor; but as soon as the system is opened the attractor ceases to be global. Moreover, since in an open non-equilibrium system the entropy may increase or decrease, and the Second Law does not already determine the sign of entropy change (besides other function of states), the existence of some general, universal Lyapunov function becomes very problematic (for details see Section 6.4). Immediately the problem of stability for states far from thermodynamic equilibrium arises; a loss of stability (under certain conditions) causes transition phenomena, which can lead to order and structure arising within the system, making it more complex. A typical example of similar phenomenon is the exergy growth, when thermodynamic equilibrium loses stability, and the exergy begins to increase.

A complexity of the system can be seen as the appearance of several locally stable and unstable states. Consider one simple example of a population with two equilibriums: trivial N0p = 0 and non-trivial Np = K. The population dynamics is described by the logistic equation:

^ = f<N,= an( 1 - K). «2.1, where N is the population size (or biomass) and a is the intrinsic rate. In Fig. 6.2 the phase portraits of the population are shown for two values of a: a+ > 0 and a~ < 0 (see also Section 5.4).

In the first case (a = a+ > 0) the trivial equilibrium N0 = 0 is unstable, and non-trivial equilibrium NK = K is stable. If a = a~ < 0 then the latter becomes unstable, and N0 is stable. In this case the trivial equilibrium, which can be associated with the thermodynamic one, is locally stable, and the population always decreases. Let us now consider the open system so the environment can work on the system (population), creating conditions for reproduction, i.e. the shift of intrinsic rate from a~ to a+. This work is equivalent to some amount of exergy stored by the system. Then the majority of