i.e. these expressions have to be symmetrical.

There is a deep analogy between Onsager's chemical system and the community of competing populations. So, the competition coefficients are analogous to the partial derivatives of chemical potentials. We think that these and other analogies will be helpful if we intend to construct some form of phenomenological thermodynamics of biological communities.

Since L (and also correspondingly, S2S) increases when going further from the stable equilibrium, this may be regarded as a peculiar form of the Le Chatelier principle: any displacement from a stable equilibrium increases the competition expenses within the community. It becomes clearer if we re-write the expression for the Lyapunov function as

L = — 1X X Gij(ci— cp)(cj— cj) = — 2 X Dci Dcj; (2 . 8)

where ci are the equilibrium values.

6.3. Model approach to definitions of stability: formal definitions and interpretations

In contrast with an intuitive understanding of stability typical of the "stability versus diversity" speculations (see Chapter 4), the model approach can provide for quite formal, mathematically rigorous definitions.

Let us assume that we have a "good" enough (from the viewpoint of adequacy and descriptive completeness) mathematical model of a biological community or ecosystem, then stability properties of a real system can be deduced from investigating its model by the mathematical technique of stability theory.

But, as was mentioned above, there are a lot of formal definitions of stability. The task is to decide just what kind of model behaviour should correspond to a stable functioning of the real system and to select those of the mathematical stability definitions which are adequate both to a meaningful, say, "ecological", perception of stability and to the mathematics of the model.

Stability investigations are thus dependent on a particular mathematical model, assuming they are adequate enough for the studied system. As far as the assumption being true, the model approach has an obvious advantage in its prognostic ability, as well as in its capability of relating stability to other systems properties such as the structure and particular mechanisms of functioning. Expressed in formal terms, stability conditions of the model promote a formulation of hypotheses concerning the functioning of the real system.

Besides everything else, the adequacy assumption itself may always be questioned, and sometimes answered at least, in qualitative terms, from the outcome of stability analysis too.

Let the system dynamics and its evolution be described by the system of ordinary differential equations:

"df = Fi(x1; •■•> xn'; A1; ..., Am); i = 1;•■•>n (3-1a)

or in a vector form: dx

dt where x = {x1;...,xn} and F = {F1;..., Fn}. The L = {A1,..., Am} is a vector of parameters, which describes an influence of the environment on the system. It is natural that only the environment can change them. The system has an equilibrium x* (L), so that F(x*) = 0. Let SR be a spherical domain llx — x*ll < R and HR be a sphere, which is bounding this domain.

In accordance with Lyapunov's theory (see, for instance, Rouche et al., 1977) the equilibrium x*(L) of system (3.1) is

(a) stable, if for any given e (0 < e < A) such 17(e) (0 < H(e) < e) is found, then the solution x(t, L) of system (3.1) with initial state x0 = x(0) [ Sh will never leave the domain Se;

(b) asymptotically stable, if it is stable in the sense (a), and moreover, if for the certain e x(t, L)! x*(L) by t! i. Later on, if we talk about stability, then, as a rule, we take into account the asymptotic stability;

(c) unstable, if for any arbitrary given e (0 < e < A) and arbitrary 17(e) (0 < 17(e) < e) the solution x(t, L) is always found, which starts from some point x0 [ Sh and attains He at some finite moment of time t > 0.

The theorems which are formulated below are based on the following concept: the scalar function L(x) is called positive (negative) definite in the domain SA if, in this domain, L(x*) = 0 and L(x) > 0 (< 0) at any other points of SA. The total derivative of L(x) with respect to time:

dt along the trajectory x(t, L) of system (3.1) in the domain SA is taken into consideration (the so-called Lee derivative), so that

Such a type of functions is called the Lyapunov function for a system (3.1).

1. Stability theorem. If for some SA such a positive definite function L(x) is found, the derivative L along the trajectories of Eq. (3.1) is a non-positive definite function, then the equilibrium x*(L) of system (3.1) is stable.

2. Asymptotic stability theorem. If the derivative L is a negative definite function in SA, then the equilibrium x*(L) is asymptotically stable.

3. Instability theorem. If some positive definite in SA function L(x) exists, the derivative L along the trajectories of Eq. (3.1) is also a positive definite (except, maybe, the point x*(L)), then the equilibrium x*(L) is unstable. Moreover, we can hit any surface Hs (s < A) moving along a trajectory, which starts at any arbitrary point of Ss (except the equilibrium point).

All these theorems have been formulated and proved by A. Lyapunov. We will also use the following additions.

4. Chetaev theorem about instability (Chetaev, 1955). Note that the previous instability theorem has one principal defect: the derivative L has to be positive definite in the entire domain SA: Meanwhile, in order to detect the instability (if it takes place), it is sufficient to detect even one unstable trajectory in an arbitrary small vicinity of equilibrium. For this we have to know the behaviour of trajectories in some part of SA, not in the entire domain. If in an arbitrary small vicinity of equilibrium a domain exists in which L(x) > 0 (L(x) = 0 on its boundary) and L(x, L) > 0 at all its points, then the equilibrium is unstable.

5. Barbashin—Krasovsky supplement about asymptotic stability (Barbashin, 1967). In the stability theorem: (a) SA = Sx and (b) L n together with llxll, then all trajectories converge to the equilibrium x*(L).

The last theorem is a simple conclusion of a theorem about the instability of equilibrium at infinity.

Note that the stability conditions depend on parameters L; therefore, the whole domain of parameters definition could be divided into different sub-domains, in each one of which the system is either stable or unstable.

6.4. Thermodynamics and dynamical systems

Returning to Section 6.3 we again consider a dynamical system (3.1): dxj/dt = F(X1; ... , Xn; A1, ..., Xm), i = 1, ..., n. It wiH be shown in Chapters 7 and 8 that the dynamic equations in ecology can be interpreted from the thermodynamic point of view. If we want to develop a phenomenological thermodynamics for the system describing these equations, then the Lee derivative of the corresponding Lyapunov function can be considered as the entropy production, which is also called a function of dissipation, Diss. In phenomenological thermodynamics Diss = Yj=1 XJi (Chapter 2) where X, are generalised thermodynamic forces and Jt are generalised thermodynamic fluxes. Formally, we can assume Xi = >L/>xt and Jt = dx,/dt, i.e. generalised fluxes are equal to components of the vector (phase) field x. Then Diss = £n=1 (>L/>xi)(dxi/dt) = L, i.e. the Lee derivative of the Lyapunov function can be considered as a dissipative function for the systems, described by Eq. (3.1). Thus, there is a certain connection between the dynamical theory and the irreversible thermodynamics. One can say that the dynamical systems with global non-negative production of entropy have sufficiently simple topological construction of phase space. In other words, the systems, which have Lyapunov functions, have also a sufficiently simple topology of phase space. However, in the general case, there is no constructive algorithm which allows obtaining Lyapunov functions for any dynamical systems. Keeping in mind our thermodynamic analogies, we can assume that a complex behaviour of dynamical systems is determined by a "patchy" structure of phase space with respect to the non-negative production of entropy. The "patchiness" means that, in some domains of phase space, the production of entropy is positive and in others it is negative (i.e. the production of negentropy takes place). Finally, there are domains in which the entropy production is equal to zero. In other words, in this case, the general Lyapunov function does not exist.

Continuing our analogy, we can say that if L = dL/dt is the entropy production, then the value L may be considered as the entropy (to within some arbitrary item). The analogy will be complete if the expression dL = £n=1 (>L/dxi)dxi is a full differential that does not always occur. However, if d2L/dxi >Xj = d2L/dXj >xi then dL will be a full differential, and L (as the entropy) is a function of state. It is easy to see that all the so-called separable Lyapunov functions, which are represented as L(x1,...,xn) = Y.n=1 Li(xi), have full differentials, since d2L/dxi >xj = d(>Li(xi)/dxi)/dxj = 0, >2L/>xj >xi = >(>Lj(xj) / >xj) / >xi = 0.

If we remember the definition of exergy, Ex = T(Sfq — S), where S is the entropy and Sieq is its value at some ith (thermodynamic) equilibrium. Then under isothermal condition T = const, we can interpret the Lyapunov function as exergy, L = Ex. The exergy is positive, and if its derivative is also positive, then it means that the thermodynamic equilibrium will be repulsing. We can say that by receding from it the system is accumulating the exergy. In the opposite case, when the exergy derivative is negative, the system approaching the thermodynamic equilibrium spends its storage of exergy. Since d(Ex)/dt--dS/dt then the curious linguistic pair arises:

the system produces the entropy = the system spends the exergy, the system produces the negentropy = the system accumulates the exergy:

Although at first glance this is not more than a purely philological game, there is, however, a deep sense in this game. In fact, when we deal with closed systems, which are under the action of the Second Law, then the entropy is always increasing, i.e. it is always produced. But as soon as we have encountered an open system, we have seen that the entropy might not only increase but also even decrease, and its derivative could be negative. This contradiction could be resolved in two different ways. The first is to assume that the entropy may be negative (negentropy), then the decrease of entropy corresponds to the increase of negentropy. In this case the system produces the negentropy. The second way is to define the antinomy at the level of derivatives, for instance, "some values are accumulated" versus "some values are spent or dissipated". As follows from the above considerations, this value may be the exergy.

Note that in the classic dynamical theory all the limiting sets (attracting or repulsing) are manifolds, i.e. they have a simple topological structure (point, line, curve, cycle, torus). Our thermodynamic speculations allow us to formulate the following statement: dynamical system (3.1) has not any attracting set, which is not a manifold, if and only if the Lyapunov function L(x) ^ 0 [dL(x)/dt $ 0] exists for system (3.1).

It became clear (Lorenz, 1963) that even for the system of three ordinary differential equations with quadratic non-linearity in the corresponding phase space, there is an attracting invariant set, which is bounded and connected. It differs from both a point and a limiting cycle, and its value is equal to zero. Such a type of set is called a "strange attractor" (Ruelle and Tackens, 1971). Locally, a strange attractor is the direct product of a two-dimensional manifold and some Cantor set. The fate of the representing point is explicitly unpredictable within the attractor, i.e. its trajectories will be irregular and chaotic. The system behaviour will be very sensitive in relation to initial values. Another simple example of ecological system with chaotic behaviour will be considered in Chapter 7.

It is possible that the dynamical systems, which demonstrate such a type of "strange" behaviour, do not have the corresponding Lyapunov functions. In other words, if the system in certain domains of phase space produces entropy, and negentropy in others, then the behaviour of the system may become chaotic and irregular. Certainly, instead of entropy we can use exergy. Then the last sentence may be reformulated as: if in some domains of phase space the system accumulates exergy, in others the system dissipates it, then its behaviour may be chaotic and irregular.

It is known that the Lyapunov function can be constructed for any linear system. On the other hand, all the limiting sets of linear systems are manifolds.

And finally, let the considered dynamical system be a system of gradient type, i.e. dx/dt = grad F(x) . The Lyapunov function exists for these systems, and is: L = (grad F, grad F) $ 0 . Therefore, gradient systems do not have strange attractors. We shall deal with such a type of equations in Chapter 8, when we consider the dynamics of competing populations in a certain way transformed space. Since the Svirezhev-Shahshahani transformation (Burger, 2000; see also Section 6.5) used for this does not change types of singularities, there are no any strange attractors in the real space if they were absent in the transformed one. From this, it follows that the dynamics of the community of competing populations are always regular; the community cannot principally have irregular and chaotic trajectories.

6.5. On stability of zero equilibrium and its thermodynamic interpretation

Let us consider the following dynamical system: dN

dt i n where vector N = {N1,...,Nn} is non-negative, i.e. belongs to the positive orthant Pn of Euclidian space En . We also assume that all Ft are analytical with respect to their variables, i.e. they are expanded to Taylor's series in the vicinity of any point of phase space. This is a typical form of equations describing the dynamics (kinetic) of a system; each state, in turn, is described either by the number of individuals (populations, communities) or by the number of molecules (chemical concentrations). It is natural that all these values are non-negative. In particular, in mathematical ecology the values N, are interpreted as either biomasses or densities of corresponding species, age cohorts and other ecological groups constituting a biological community. If we assume that the system occupies some fixed spatial volume, then the values Ni can be considered as either biomass densities per volume unit or a volume concentration of some substances. Later on, we choose the first interpretation, namely, the value Ni is the biomass of ith species in biological community.

The trivial equilibrium N* = 0, which corresponds to the full absence of biological "particles" in the system, is naturally interpreted as thermodynamic equilibrium. Since N* = 0 is an equilibrium then F,(0, ..., 0) = 0, i = 1,..., n. It means that a Taylor-series expansions of Fi in a vicinity of N* = 0 has to begin with linear terms, so that n

A natural question arises: under what kind of conditions will the trivial equilibrium be unstable? In other words, what kind of conditions has to be fulfilled in order for the system to be able to leave the thermodynamic equilibrium and begin to recede far from it, i.e. for the life to arise within the system?

It seems that the problem's solution is very simple when using the Lyapunov instability theorem. However, it is not the case, since the equilibrium lies on a boundary of admissible domain (positive orthant), and any negative variations which lead out of it are senseless. In other words, keeping in mind the definition of Section 6.3, we cannot surround the equilibrium point by spheres which are fully situated in the admissible domain. But the thing which is impossible in the usual phase space of population numbers becomes possible in the space of new variables zi = ± V2N,, i = 1,..., n. This is the so-called Svirezhev-Shahshahani transformation, which transforms the positive orthant into the entire phase space. In this connection, a boundary singularity is transformed into an internal one, and the origin of coordinates (the boundary point of positive orthant) becomes an internal point.

Therefore, there are no obstacles to use the Lyapunov instability theorem for the transformed system (5.1), which is dZi Fi(z2, .■•, z2>

dt zi where z [ En, i.e. it could be used for the entire phase space. Since the origin of coordinates z = 0 is an equilibrium point then lim Fi(z2, .■■>z2) = 0, i = 1, . n. (5.4)

llzl!0 zi

This condition has to be fulfilled for any path along which the representative point moves to the origin of the coordinates. This is possible if the functions Fi(N1,...,Nn) are represented in the form Fi = Nf (N1,...,Nn), where f are again analytical with respect to its variables. Then a. — 0, a¡j = 0 for all i — j in expansion (5.2). In this case, Eq. (5.2) is written as dz. 1

It is obvious that the function L = Xn=1 z2/2 is equal to zero at the point z* = 0 and positive definite in the entire phase space. If the derivative dL/dt is also positive definite in some domain SA, then the thermodynamic equilibrium is uPnstable.

Returning to our real variables, Nt, we get L = Xn=1 N., and the Lyapunov function is nothing more or less than the total number (biomass) of communities, N. All Lyapunov's spheres Hc : llzll = c in the transformed space correspond to the simplexes Sc : N = Xn=1 N. = c in the real space of numbers. The surfaces of constant biomass are the surfaces of constant "energy" in the transformed "mechanistic" space.

The derivative of the Lyapunov function is the rate of biomass, dL/dt = dN/dt. Then we can say that, in accordance with Chetaev instability theorem, the increase of total biomass, even if along a single trajectory of the system, is a sufficient condition for the instability of thermodynamic equilibrium to arrive. If the total biomass increases in its entire vicinity SA, in accordance with Lyapunov instability theorem, then any trajectory of the system which starts at any arbitrary point of Se (e < A) (except the thermodynamic equilibrium point) attains the sphere He. In the real phase space of species biomasses this geometric result can be interpreted in the following manner.

It is clear that sooner or later the total biomass of community is stabilised at some equilibrium level N*. The level depends on the flow of external energy and resource, which can be assimilated and dissipated by the system. The simplex Yn=1 N* = N* bounds the domain SA, in which the derivative dN/dt is positive. Then any trajectory started within the simplex will, in any case, reach the simplex £?=1 N. = N* — e, which will differ arbitrarily little from the equilibrium simplex.

Finally, we can say that although the results of the chapter seem trivial at first sight, the fact that the total biomass is the Lyapunov function for thermodynamic equilibrium allows us to consider this value as a macroscopic variable of the system.

6.6. Stability of non-trivial equilibrium and one class of Lyapunov functions

Let system (5.1) have a single non-trivial equilibrium N*, which is situated within orthant Pn. We shall consider the following class of functions, which may be candidates for Lyapunov functions (Svirezhev, 1998b):

L = £ NpWNi/N*) or, if J = N./Np, L = £N*w(J), (6.1)

where the function w(J) possesses the following properties:

Since N [ Pn then the vector J = {JJ,..., Jn} also belongs to the positive orthant. It is obvious that the point Ji = J, i = J,..., n corresponds to the equilibrium N* . In other words, the function p(J) has to be convex for positive J. It is obvious that L(N*) = 0; therefore, the first variation of L in the vicinity of N* is equal to

then SL(N*) = 0 for any arbitrary variations SN* . By calculating the second variation we get d2l = - y y-sNi m = - y w (jo^-T-

for any non-zero variations SN . Thus, L is a convex function of N, which has an isolated minimum at the point N* and monotonous increases with increase in the norm llNll in the orthant Pn . In accordance with Lyapunov stability theorem the equilibrium N* is asymptotically stable in some domain SA # PN if the derivative dL/dt taken along the trajectories of Eq. (5.J) in SA will be negative: dL n

where the equality is fulfilled only at N* . On the other hand, the equilibrium N* is unstable in SA, if the derivative dL/dt is positive, except N* where it is equal to zero (Lyapunov instability theorem). Generally speaking, in accordance with Chetaev instability theorem, it is sufficient for instability if dL/dt would be positive even along a single trajectory of Eq. (5.J) in a small vicinity of equilibrium.

Note that, in Eq. (5.J), if Fi(NJ,..., Nn) = Nifi(NJ,..., Nn), i.e. the origin of coordinates is also equilibrium, then only the interior of orthant can be considered as the domain SA . It becomes evident by expression (6.5) for dL/dt: in this case the derivative dL/dt, in addition to the non-trivial equilibrium, is equal to zero also at the point N = 0

Let us consider one partial form of the function p: p(J) = N*(1 — J)2 .It is easily seen that p(J) possesses all the necessary properties in order to be the Lyapunov function. In fact, p(1) = 0, p(J) = 0, p'(J) = 2N* > 0 . Then the Lyapunov function is written as (seeEq. (6.1))

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