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The choice of such a type of division could be explained from the purely physical point of view. In fact, the metabolic processes are very fast (in comparison with the processes of formation and evolution of the community). Since we consider a partially equilibrium system, which is in the state of heat equilibrium with its environment, the metabolic heat has to dissipate very quickly. This process reduces an effective value of consumed free energy and does not influence the competing processes. In this case we can consider the metabolic processes as exchange processes.

The decomposition of dead organic matter is also a dissipative process. If it happens outside the system then there are no problems, and the mortality can be considered as an exchange process. In the opposite case, when the dead organic matter remains within the system (and decomposes there), it seems that the decomposition has to be included in internal entropy production. However, for this, we have to include in the system a variable, which has to describe the state of dead organic matter. In this case, we are not prepared to do it; we will have to consider the mortality as an exchange process.

By keeping in mind one of the basic statements of the thermodynamics of open systems, that at the dynamic equilibrium (deS/dt) + (diS/dt) = 0, we immediately obtain from Eq. (2.4) that at the dynamic equilibrium (we shall denote this state by the upper index (eq)):

(a) X feW, R) - <ftut(^eq, St)] - X Mk(Neq) + Dk(Nekq)] k=1 k—1

The latter relation means that when the trophic level is in a dynamic equilibrium with its environment and other external systems, its total biomass has to be constant (although the biomasses of each population may fluctuate; compare with Chapter 7). We can see that the conditions for a stationary state do not depend on the choice of hypothesis about the division of total entropy production.

All these results are absolutely correct but they are too general. Their concretisation of all functions, used in the description of energy and entropy balance, can give us the results that will be interpreted in a more interesting manner.

It is obvious that all the functions qik(Nk, R), q°k\Nk, St), Mk (Nk), Dk(Nk) at the point Nk — 0 are also equal to 0. Therefore, they can be represented in the following forms:

q^(Nk, R)—Nkfkn (Nk, r), q°out(Nk, St) — Nkfk0ut(Nk, St), Mk (Nk) — Nkmk (Nk), Dk (Nk) — Nkdk(Nk).

All the values which are denominated by small letters may be interpreted as specific ones, i.e. they are values per one individual.

In order to concretise the competition function Fk(N1,...,Nn) we need a model of competition. For this we use the so-called collision concept, which is very popular both in the mathematical ecology (Lotka-Volterra equations) and in chemical kinetics. We assume that the energy expenditures for competition are the results of energy dissipation during the act of collision between one pair of individuals. If one individual belongs to the kth species and another belongs to the jth species then the amount of dissipated energy is equal to 8kj. This interaction is symmetric, i.e. 8kj — 8jk. Collisions of three and more individuals are neglected. We also assume that the system considered (community of the same trophic level) is well mixed, i.e. any collision is a result of random pair choice from a large ensemble of N — £JL1 Nk individuals. (Here we temporarily return to the primary interpretation of Nk as a number of particles or individuals that will be denominated by " ~ ".) In this case the number of collision between kth and jth individuals is equal to N,Nj. The process of choice takes place once during one temporal unit. Since the pair collision of kth and jth individuals is accompanied by the dissipation of 8kj units of energy then the total dissipation for kth population will be equal to n n

By returning to energy units, Nk = ekNk we get n d n

where ykj = 8kj/ekej. Since 8kj = 8jk , ykj = yjk. By substituting all these expressions into Eq. (2.1) we immediately get the kinetic equation for the population number:

-N = Nk : mNk ,R) f N, St)] - [mk (Nk ) + dk(Nk )] - X JkjNj\,

In order to complete the system we have either to supplement equations for the new variables R and St or to consider them as parameters. In the latter case we have to formulate additional hypotheses about the connection between the system and its environment (for instance, other external systems can be considered as the environment). However, if the connection is very tight, i.e. if not only the state of the system depends on the state of environment but also vice versa, then a consideration of the trophic level as some isolated system becomes very problematic: we have to include both the level and its environment (the resource or previous level and the next level) into the whole system. Nevertheless, we can separate our level if we assume that

1. A resource "pool" is much larger than the level, so that the consumption of resource by the level does not practically change the amount of resource in the pool. Such a situation is typical for the autotrophic level with respect to solar energy. In other words, the system interacts with an infinite resource pool. There is a "bon mot" for this: an interaction "continent-island". It is obvious that in this case R < R* = constant.

2. The state of a given level does not practically influence the state of other systems, then /k°ut = fkUt(Nk); or, the reversed influence is so low that we can neglect it (fkout = 0). We shall use the latter assumption in our further deliberations.

By denoting qii(Ni, R*) = fk(Nk), mk (Nk) + dk (Nk) = hk (Nk) system (2.12) is re-written as dNk < n )

~dNL = Nk j[ fk (Nk ) - hk(Nk )] - X ykjNj\, k = 1, ...,n. (2.13)

The system is very similar (at least, in the part which describes competition between species) to the Lotka-Volterra equations. This is not surprising since we also used the collision concept.

8.3. Community trajectory as a trajectory of steepest ascent

In this section we describe one method (the method of steepest ascent), which allows us to represent the trajectories of kinetic equation (2.13) as some extremals.

Let us assume that the continuous and differentiable function W(N1,...,Nn) belonging to the positive orthant Pn is introduced in such a way that:

dNk 9W

Calculating the derivative of W with respect to time along the trajectories of Eq. (3.1), N(t) = {N1(t),...,Nn(t)}, we have dw n >w dNk n i >W\2 „

Obviously, dW/dt = 0 only at a point where either all 9W/>Nk = 0, or all Nk = 0, or, in the intermediate case, when several 9W/>Nk = 0 and for other indices Ns = 0, i.e. at stationary points (equilibriums) of system (3.1). From Eq. (3.2) it follows that the function W [N(t)] always increases along the trajectories of Eq. (3.1) and reaches its local maximum at the equilibrium Np only if Np is asymptotically stable. In fact, since trajectories of the dynamical system (3.1) are dense within some domain V, to which the point N* belongs, for each point N neighbouring N*, a trajectory N(t) ! N* going through N could be found. The function W is continuous and steadily increasing, and it follows that W(N*) > W(N) along the trajectory N(t). Hence,

t nev

On the other hand, if the function W(N1,...,Nn) has an isolated maximum at some stationary point N* = {N*,..., N*} E Pn then the state N* is asymptotically stable, since the function L(N) = W(N*) — W(N) is the Lyapunov function for system (3.1) in this case. This fact is obvious enough, since L $ 0 in a domain which contains N (L = 0 only at N*) and its time derivative along system trajectories is dL/dt = —dW/dt # 0 with dL/dt = 0 only at the point Np .

The sufficient condition for the maximum of W is the following: the quadratic form

(the second differential of W) is positive definite. In this case W is strictly concave (convex upwards) at a neighbourhood of Np. If the second differential is positive definite for any N e Pn then the local property of strict concavity becomes global, i.e. it takes place for the whole positive orthant. And as far as the orthant Pn also represents a convex set, W has a single isolated maximum in this set (on compact subsets or on the boundary). Hence, its local maximum is at the same time global, and system (4.1) has a single stable equilibrium. Moreover, if lWl n as iNl n, this equilibrium is stable for any initial trajectory displacement within Pn (globally stable, absolutely stable), thus suggesting that any system trajectory initiated within Pn approaches the equilibrium as t n.

If the maximum of W is not attained inside the positive orthant Pn, it will be attained on its boundary, in the appropriate coordinate hyperplanes. This implies that one or several equilibrium coordinates will be zero. In the ecological interpretation it means that in the process of ecosystem evolution one or several species are to be eliminated, since the new equilibrium state with a fewer number of species is stable again.

We see that, generally speaking, many characters of dynamic behaviour of the system are determined by the landscape of the function W. For instance, such general properties as the existence of equilibriums, their coordinate and their stability can be defined in terms of its landscape topography: the existence of peaks and valleys, and their locations. However, we do not know what kind of path in the landscape corresponds to the trajectory of the system movement? In order to answer this question, first we have to introduce some definitions.

A trajectory which goes along a gradient of the surface W(N) is called a trajectory of steepest ascent. If this trajectory initiates at the point N0, then it is the shortest way from this point to the top of a peak. The standard equations of steepest ascent are represented in a vector form as dN/dt ~ grad N, and they differ from Eq. (3.1). By applying the Svirezhev-Shahshahani transformation zk = ±2-jN~k; k = 1,..., n, which transfers the positive orthant Pn into the full coordinate space Rn, we now consider the movement in Rn. In the new coordinates W = W(z) = W(zi,..., zn) system (3.1) is written as dzk/dt = 9W/>zk, k = 1,..., n, or, in a vector form, dz/dt = grad W, i.e. trajectories of Eq. (3.1) will be trajectories of steepest ascent in the special space Rzn. Since in Rzn dW/dt = 22=1 (>W/9zk)(dzk/dt) = Y.n=1 (>W/>zk)2, the square velocity of movement along trajectories of steepest ascent may be defined as n n n v2 = dW/dt = £ (>W/>zk)2 = X [(>W/>Nk)(>Nk/Szk)]2 = X [(>W/>Nk)(±Nk)]2. k=1 k=1 k=1

From this expression it could be seen that, as we approach equilibrium, in which either d W/ dNk = 0 or Nk = 0, the velocity is permanently reduced although it can be sufficiently high when far from this equilibrium.

It is interesting that W(z) in Rn, where 9W(z)/9zk = 0, corresponds to the stationary points of W(N) with Nk = 0 in Pn. But it is not necessary that 9W(N)/9Nk = 0; more often these derivatives are negative. In other words, the stationary points of W(N) situated at the boundary of Pn correspond to the internal stationary points of W(z) in Rzn. It is obvious that all these points are at the same time the equilibriums of system (3.1) and dzk/dt = >W/>zk, k = 1,..., n. Notice, as well, that all these points are defined by the necessary conditions of the extremum for the functions W(N) and W(z). But the number of equilibriums of this system exceeds the number of equilibriums of the system (3.1)! For instance, if the internal non-trivial equilibriums for both the systems are single then the system (3.1) has the maximal number equal to (2n — 1) of all possible stationary points with n, n — 1,..., 2,1 non-zero coordinates. On the other hand, any point which differs from these only in signs of non-zero components will be stationary for the system dzk/dt = 9W/>zk, k = 1,..., n. The number of such points is, evidently, equal to 2n + n2n—1 + ••• + Ckn2k + ••• + 2n = 3n — 1. Thus, the stationary points of the system dzk/dt = 8W/>zk are derived from the stationary points of system (3.1) as their symmetric reflections with respect to all sorts of coordinate hyperplanes and the origin, while the values of the function W(z), are identical in symmetric points. We illustrate all the reasoning by the following example (see also Fig. 8.1).

Let W = a — bN. In this case the positive orthant coincides with the positive semi-axis [0, i], and the maximum W is attained at the point N* = 0. At this point >W/>N = —b < 0. Introducing the new variable z = ±2yJN we get W = a — (b/4)z2, which is now presented for the entire axis [—i, i]. This function has a maximum at the same zero point, but the derivative >W/>z = — (b/2)z at this point is equal to zero.

In order to illustrate the symmetry of stationary points we consider the second example (Fig. 8.2). Let W = N(1 — N). It is obvious that W has an internal maximum at the point N* = 0.5, which is equal to 0.25. By going over to the variable z = ±2^N we get W = z2(4 — z2)/16. This function is symmetrical with respect to the origin z = 0, and it has two symmetrical maximums at the points z*,2 = ±V2 (that correspond to Np = 0.5). Both maximums are equal to 0.25. It is easy to see that both dW/>N = 1 — 2N at the point N* = 0.5 and >W/>z = z(2 — z2)/4 at the two points zp,2 = ±V2 are equal to zero, i.e. the extremum necessary conditions are fulfilled for both these cases.

Revenons a nos moutons and try to find the function W, which will be a function of steepest ascent for kinetic equations (2.13). As one of many possibilities let us consider the following function

Fig. 8.1. The function W = a — bN in the natural (a) and transformed (b) coordinates.

Fig. 8.1. The function W = a — bN in the natural (a) and transformed (b) coordinates.

Calculating the partial derivatives of W with respect to Nk and taking into account that

It is easy to see that the system of kinetic equations (2.13) describing the thermodynamics of trophic level is represented in the form of Eq. (3.1). It means that the function W given by Eq. (3.5) grows along the trajectories of Eq. (2.13) and reaches its maximum at the stable equilibrium of system (2.13).

8.4. Extreme properties of the potential W and other potential functions. Entropy production and Prigogine-like theorem

All these results can be interpreted from the thermodynamic point of view. Since all biomasses are measured in energy (enthalpy) units, the concepts of biomass and energy are equivalent. The existence of a competing community is determined by the resource inflow, f — h = ik=1 Nk [fk(Nk) — hk(Nt)], which is accumulated by the system, increasing its internal energy. Kinetic equations (2.13) are the consequence of the First Law, while they can be used as a dynamic model of the competing community. This allows us to study such a general property of it, such as stability, by establishing the analogies between the dynamic and thermodynamic criteria. In addition, in order to judge a direction of the system's evolution and its stability (and hence, its possible evolution) it is desirable to have energy and entropy criteria of evolution. It is natural that the first thought would be to apply methods of thermodynamics of irreversible processes.

In our case the potential function W could be used as some analogue of the thermodynamic potential G for such an open system as the competing community or trophic level. In fact, as for ykj = jjk, n

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Getting Started With Solar

Getting Started With Solar

Do we really want the one thing that gives us its resources unconditionally to suffer even more than it is suffering now? Nature, is a part of our being from the earliest human days. We respect Nature and it gives us its bounty, but in the recent past greedy money hungry corporations have made us all so destructive, so wasteful.

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