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of N increases. The latter illustrates our principle (maximum of exergy, which is formulated as a maximum of the total biomass in this case). This principle is dual to Prigogine's one.

Fig. 8.3b can be interpreted analogously, but in this case the upper boundary (0a1b) is described by the expression D = ((1 + y)/2)N2 , and the lower boundary (0c2d) by the expression D = N2. Stable point 2 corresponds to the state with a single population, saddle point 1 to the community of two populations. The bifurcation diagram for system (5.2), i.e. the dependence of equilibrium values of D* and N* on the parameter y, is shown in Fig. 8.4. Since D* = N* = 2/(1 + y) and D* = N* = 1, we get two curves in the figure.

Within the interval 0 < y < 1 the values of D and N are greater on the stable branch of solution than on the unstable branch. At the bifurcation point y = 1 the type of solution changes, and for y > 1a "degenerated" community (single population) becomes stable but, as before, D and N are greater on the stable branch. At first sight we have obtained a contradiction with Prigogine's principle: the value of dissipation is higher on the stable branch. But this is not the case; this is a consequence of its incorrect extension to the parametric domain. Of course, we could save the situation keeping in mind that in our case we minimise the function of dissipation under certain constraints, but...the elegance of the original formulation disappears. At the same time the exergy principle is valid: On the stable brunch the exergy is higher than on the unstable one.

8.6. Phenomenological thermodynamics of interacting populations

It is necessary to say at the beginning that there were a lot of different attempts to apply the methods of statistical mechanics directly to the macroscopic description of ecosystems (Kerner, 1957, 1959; Polischuk, 1971; Alexeev, 1975), but they have not brought great success. However, as was shown in the brilliant Khinchin's work (1943), a statistical mechanics could not only describe purely physical systems, but the approach can also be used for the construction of certain phenomenological theories of biological communities.

It was shown above that the total biomass of community, N, could be considered as one of its macroscopic variables. It is also obvious that this single variable is not sufficient for the full macroscopic description of the system, since the knowledge about the value of N(t) does not give us any information about the further evolution of a community: it is necessary to know its temporal derivative. Therefore, we take into consideration the vector of Malthusian parameters or functions m = {11, .■•> 1n }.

We know that the concept of extensive and intensive variables plays a very important role in thermodynamics. It is obvious that for extensive values a concept of density or a specific value can be introduced, while this cannot be done for intensive values. In biological models, Ni and N are considered as some extensive variables, but the vector of composition (or structure) p = {p1;...,pn} is the intensive variable. Then a dynamic model of a community can be presented as

dt or, with the variables p and N as

It is obvious that the scalar product of vectors m and p, ¡1 = (m, p), is the mean Malthusian parameter (MMP). If, in addition, the Malthusian parameters (functions) depend only on pt then the system dynamics does not depend on its "total mass", i.e. we could increase (or decrease) the total biomass N by several times but the equation for N does not change. In other words, in this case two systems with the same structures but with different total biomasses will be dynamically equivalent. From this point of view, such systems are very "thermodynamic", since their dynamics depend neither on their size nor on their total biomasses. Moreover, since all thermodynamics identities (potentials) must be homogenous functions of the first order, the ik must depend only on pt. Thus, if we would like to remain within the thermodynamics framework we have to accept the assumption. In this case, if the equations for p and N are separated from each other, then the equilibriums with respect to p (we shall name them "structural" ones) and N can be established independently so that, for instance, in a structural equilibrium the total size can either change or remain constant. Later on we shall only consider evolution of the community structure, since knowing its dynamics we can always calculate the dynamics of its total size.

Formally, the dynamics description in terms of frequencies is equivalent to an assumption about the constancy of the total size, N = constant. Then the equations for the population sizes Nk are written in the form (6.2) where pk are replaced by Nk.

We postulate (without proofs) the following statement:

A community of interacting populations evolves in such a manner that its MMP always increases attaining its maximum at the stable structural equilibrium (Fisher's principle).

Before emerging from ecological problems, we shall perform a short but very fruitful digression into the similar field of evolutionary genetics (Svirezhev and Passekov, 1982).

The main subject is the population, which can be considered as the community consisting of n2 sub-populations of "genotypes". Every one of them is denoted by the pair {k,j}; k,j = 1,..., n; the size of the corresponding sub-population is equal to Nkj. In evolutionary genetics the Malthusian parameters ikj are named as fitness of genotypes. Assume that the genotypes {i, j} and {j, i} are identical so that ij = 1ji and Nij = Nji. The MMP (or the mean fitness of population) will be equal to 1 = Yij=1 Mijuij where uij = Nij= Yjj=1 Nij are the genotype frequencies. The analogue of Eq. (6.2) is written in the form dukj

kj=1

The principle of the MMP maximum in evolutionary genetics is called the "Fundamental Theorem of Natural Selection"; one of its formulations states that "under the pressure of natural selection the mean fitness of population increases, attaining its maximum at the stable equilibrium". The theorem has been proved by Sir Robert Fisher.

Let ikj be constant and ukj be independent, then di ^ ■ ^ i dukj v * v * 2

dt k=i j=i dt k=i j=i i.e. the value of 1 is always increasing, tending to its maximum in the space of genotype frequencies. Since 1 is a linear form of ukj the maximum of 1 is attained at one of the cones of simplex £ij=i 1ijuij = 1: the population with maximal ikj survives, the others are eliminated during the process of natural selection. We see that, in this case, the principle of maximum of the MMP, which is even proved here (not only postulated), leads to the trivial result; in order to obtain some more or less meaningful results we must assume that there are some kind of constraints or connections in the system: either within the vector space of m or within the vector space of u (or p) or between the first and the second spaces.

Such a type of connection in evolutionary genetics is described by the Mendel laws connecting the frequencies of genotypes in the parents and off-spring populations so that ukj=^ XX uk^j (X usjj. (66)

Thus, the genotype frequencies are not independent; the genotype structure of population is changed not only by natural selection, but also as a result of Mendelian splitting, which generates structural constraints within the space of genotypes. Then maxu 1 can be attained within the simplex, i.e. a polymorphous state of population may exist.

The principle of maximum for the MMP is a natural generalisation of the Fisher Theorem of Natural Selection. Note that this principle is a hypothesis, postulate, credo;

in general, everything is possible. But this is not a theorem, requiring proof, although its variety in evolutionary genetics is a theorem.

If we follow the principle then among all virtual structures we can select a single one which will satisfy this principle; so we announce that this structure really exists. In this case it is not necessary to know the equations describing the local dynamics of the system; and the biological laws, which define the dynamics, are expressed through some geometrical properties of the spaces of m and p and extreme properties of some functions, describing global characteristics of the community.

It is well known that Onsager's reciprocal relations are the basis of any phenomenological thermodynamics. In this case these relations have introduced the linear relation between m and p:

Mk = X MkjPj, Mkj = Mjk; j = 1; • ■ •,n- (6-7)

Substituting Eq. (6.7) into Eq. (6.2), we get dpjL = Pk(Mk 2 M); (6-8)

Formally Eqs. (6.8) coincide with the classic Fisher-Haldane-Wright equations of evolutionary genetics. What is the reason for this coincidence? What is the biological sense of Onsager's linear relations (6.7)? Note that Mendel's relations (6.6) are valid if the population heredity is determined by a single gene with n multiple alleles. Then, instead of the n2 sub-populations of genotypes with frequencies ukj, we can consider the n subpopulations of alleles with frequencies Pk = 5j=1 ukj- In this case the relation (6.6) is represented as ukj = PkPj- By substituting this relation into Eq. (6.4), after simple transformations we obtain Eq. (6.8), i.e. the Fisher-Haldane-Wright equations. The genetic sense of Onsager's hypothesis is: if the values of ¡k = 1 MkjPj are the mean fitness of kth alleles then these equations do not differ from the general equations for the community of n competing populations that are the populations of alleles. The genotype is interpreted as a collision of two alleles, and its fitness is interpreted as a result of the collision.

By calculating (dM/dt) along trajectories of Eq. (6.8) we get

i.e. the mean fitness of population tends to maximum in the process of evolution. This is again the well-known Fisher's Fundamental Theorem of Natural Selection. How can the result be interpreted from the thermodynamic viewpoint?

We represent the fitness coefficients in the form ¡kj = F(1 — skj) where F is the mean number of off-spring per one individual, and skj is the probability for the genotype {k, j} to survive until its reproductive age and to produce the vital posterity. Then the number of off-spring, equal to FNYn=1 1 skjPPj = N(F — ¡M), must perish during one generation in order for the process of genetic evolution to continue. This is the necessary condition for the continuity of genetic evolution and its cost! The lost of individuals is a typical irreversible process with a biomass dissipation, and the value Diss = N(F — ¡1) can be considered entirely as the function of dissipation or the entropy produced within the system. In accordance with Prigogine's theorem, Diss = (d^/dt) ! minp (d^/dt) > 0 , and the gene structure of population, p, tends to the stable equilibrium, p*. Since the evolutionary dynamics does not depend on N, and F can be always chosen sufficiently large (so that F > ¡1*), the statement about minp (d^/dt) is equivalent to the statement about maxp ¡1. This means that in the thermodynamic theory of evolution Prigogine's theorem about the minimum of entropy production is equivalent to Fisher's Fundamental Theorem of Natural Selection in the classic evolutionary genetics.

8.7. Community in the random environment and variations of Malthusian parameters

Assume that the community is embedded into the random environment, whose random fluctuations cause fluctuations of Malthusian parameters, without affecting numbers or frequencies (Svirezhev, 1991). This is a natural assumption, if it is taken into account that in reality the fluctuations of such environmental factors as temperature, humidity, salinity, etc. affect in the first place the intrinsic growth rates of populations, i.e. their Malthusian parameters. Then, implicitly, by means of changing their growth rates, they change the population sizes or biomasses. When looking at models used in ecological modelling, we can see that they all describe the dependence on the environment in the form of dependence of the relative growth rates on the environmental factors.

Let ¡k = + Jk where Jk is the normally distributed random value with the zero arithmetic mean and the variance rkk. Note that the hypothesis of normality is one of the most popular hypotheses about the nature of randomness in natural sciences. It works very well if the random fluctuations are the result of the impact of a lot of weakly correlated random factors. Arbitrarily this is valid for ecosystems, therefore we accept the hypothesis.

If all ¡k would be statistically independent, then for the description of the values Jk the knowledge about the variances rkk would be enough, but we assume that they are statistically dependent, i.e. they are connected to each other by certain influences of the environment on the community. Then we already have to consider the random vector J = {Jj,... , Jn}, about which we assume that it is again normally distributed with zero mean but with covariation matrix T = \\rkj\\. Since the MMP ¡1 is a linear function of random vector m , the ¡1 is also normally distributed with the mean ¡10 = (mo , p) and the variance T =(p , Tp) = Xn=1 5j=1 TkjPkPj. In this case the following probabilistic statement is valid:

the inequality ¡1 $ 0 holds with the probability equal toF(10/VT) , where F(a) is the probability integral F(a) = (1/V2P) J"a_x e^^dx.

If we are now given, as is usually done in statistics, some confidence level of probability 8* (for instance, 8* = 95%), to which the certain quantile a* of normal distribution (so that 8* = F(a*)) corresponds, then the MMP 1 $ 10 — a*^ff with the probability 8*.

If 1 $ 0 then the total size of a community will never decrease. If the community structure (the vector p) and 10 are fixed then the fulfilment of relation 10 = a* V? means those random fluctuations of the species Malthusian parameters do not bring about a decrease in the total community size. However, this statement is only fulfilled with a certain probability 8* = <F(10/V?). The probability can be considered as a measure of the community stability. Since the values a* and 8* are connected one-to-one with each other, later on we shall also use ap as a measure of stability.

Let the principle of the maximum MMP be valid, so that it can be realised with the help of the structured change. In our case the maximum of the MMP was attained at such p = p*, for which the lower bound of 1 has attained its maximum with the fixed probability 8*, so that maxp (inf 1) = 1 — a We assume that the transition p ! p*, which occurs at the fixed m0 and T, is the system's adaptation to the given environmental conditions. At this time, the total size may be either changeable or constant. Generally speaking, the latter statement is not completely correct but this is a reasonable approximation if the total number of species is large. In this case the total size varies insignificantly, even if variations of several pk are very large. We shall name such an adaptive change of the structure as fast system evolution. In the course of the fast evolution the system quickly passes from any state to the state with maxp (inf 1) where p = p*, by the same token adapting to the environment. Note that the lower bound may be both positive and negative depending on ap, i.e. the total size, even at the optimal state, may both increase and decrease under the influence of random perturbations. Certainly, we can always select ap in such a way that 1p $ 0 (do not forget about the probabilistic origin of this inequality!). If 1* $ 0 then we can always state that the system is not degenerated, i.e. the total size of the community does not decrease. Therefore, it is natural to require: inf 1* = 0. The value of probability, equal to 8* = F(a*), corresponds to the condition; it also determines the stability reserve.

All these statements allow us to formulate the following concept of stable structural equilibrium and stability reserve.

At the stable structural equilibrium:

Here 1* = 10 — a*Vf*, where the notation ( * ) points to the fact that the value of t is taken at the point p*, i.e. at the point where inf(1) reaches its maximum. Therefore, the community at the stable structural equilibrium is described by the vector pp (microscopic variables) and the value of 8* = F(a*) (a macroscopic variable), which could be interpreted as a probability of existence of this equilibrium. Note once again that the probability does not depend on the total size of the community.

Consider a problem that is dual to problems (7.1) and (7.2): find maxp 8* under constraint inf(1*) = 0, i.e. in that way we find the most probable structure. The problem is equivalent to the problem of maxp ap by virtue of the monotonicity of the probability integral. By virtue of (7.2) a* = 1k/V?, and we can postulate the original problem as: how to find the maxp (lo/V?). These dual problems have the same solution, whence a very important conclusion follows: The structural equilibrium with maximal MMP is the most probable.

In order to find the distribution of species frequencies corresponding to the stable structural equilibrium we have to solve the problem of maximisation defined in (7.1) and (7.2). At the first stage we do not take into account the condition of non-negativeness of frequencies. By applying the Lagrange multipliers method we shall find a maximum of the function n c' V 0 *

or, in the vector-matrix notations (e = {1,..., 1} is a unique vector)

We have already used this method, so the procedure is standard, and therefore without unnecessary details we get the final results at once.

The community structure at the stable structural equilibrium is defined by the vector p* = T2V° . (7.4)

The probability of this equilibrium is equal to

Here T21 = \\rkj!ll is the matrix reciprocal to the covariance matrix T = Hr^ll. Both are positive definite.

Generally speaking, the value (a*)2 = (T21ik, ik) can be used as a measure of probabilistic stability of the equilibrium. In the space {1°,..., 1} the condition (a*)2 = (T21io, ik) = constant defines an ellipsoid, and if we now assume that the Malthusian parameters change in a quasi-stationary way and their evolution is such that their trajectories always belong to the ellipsoid, then the probability 8* does not change. We shall name such types of processes as isostable.

Generally speaking, as is seen from formula (7.4) several frequencies (let p* < k, s [ [1, n]) can be negative. In this case, since the function inf 1 and the simplex S : Pk $ k,Ypk = 1 are convex, the maximum must be situated on the border and is unique. Then the optimal solution (pp)0 is defined as:

2. In the matrix T we cross out the corresponding sth rows and columns by reducing it to the matrix T0 .

3. In the vector m0 we cross out the corresponding sth elements reducing it to the (mo)'-

4. For these new matrix and vector we calculate the reduced new optimal vector (p*)'-

If several components of the new optimal vector are negative again, then the process is repeated.

Principally the process has to be converged to the situation with positive component. The measure of stability in this case is equal to [(a*)2]' = ((T"1)'^)', (m0)'), where the index (') corresponds to the finish of the process.

Let (7-2% = j 7M then p* = (j1 V/EU 72%, as follows from Eq. (7.4). The necessary and sufficient conditions, that the maximum lies within the simplex S, are n

Note that the appearance of negative frequencies points to the fact that the optimisation of the structure of community, i.e. its adaptation, requires the elimination of corresponding species out of the community.

Until now we did not assume any constraints for the MMPs of different species, m0. In all the previous considerations we have also implicitly assumed that a* > 0, i.e. the probability of existence of the stable structural equilibrium 8* > 1/2. But it could be that M00 < 0, then in order to satisfy the equality inf(M0) = 0, i.e. Mo — a*—T = 0, we have to assume that a* < 0. Theoretically it is possible, since a* = ±V(T—Mo, m0), we can choose the negative sign (—). Then the probability 8* < 1/2, and it is minimal, so that the corresponding structures are naturally named low-probability ones. Any other structure will be more probable, therefore the low-probability structural equilibrium must be unstable and when necessary it must be disintegrated with elimination of some species (since the minimum is reached only within the simplex and the maximum can be reached only at the borders). From Eq. (7.4) it follows that

and the value of Mo will be negative only if (e, T—1M0) < 0, or n n n

Thus, the condition defines the border between structures with high and low probabilities of existence. If we use the geometric interpretation within the space M..., m«}, then relation (7.8) defines the plane which passes through the coordinate origin and divides the entire set of m0 into two subsets corresponding to probable and low-probability structures.

It is interesting that if the function inf(M) has a minimum then its maximum can be situated only in one of the simplex S corners, where the measure a* will be maximal. In other words, the stable structural equilibrium for low-probability structures is the state with single sth species, for which a* = maxk (M°t/p——kk—). Hence, in the low-probability structure all species (except one) are eliminated.

If we turn again to the geometric interpretation, then the conditions n

give the family of n planes passing through an origin of coordinates. They cut such a cone out of the set of m0 that if the vector m0 lies within the cone then there is no species which would be eliminated out of community.

In order to illustrate visibly all these results we consider the elementary community consisting of a minimal number of species making it still possible to speak about interaction, i.e. about two species.

So, let the two-species community be with p1 = p, p2 = 1 — p, M0 = Mi, M0 = Mi and the covariation matrix

t11

t12

t12

t22

The value p* given the community structure is (see Eq. (7.4)):

r22M1 + T11M2 2 r12(M1 + Mi) Using conditions (7.6), which in our case are written as t22M1 2 r12M2 > 0; r11M2 2 r12M1 > 0; (7.11)

we immediately obtain in the plane { m1 , M2} a domain of co-existence of the two interacting populations for given statistical characteristics. For a more visible interpretation we pass from a covariation to a correlation matrix where r11 = s2, r22 = s2 ; r1llslr11 r22 = P, and where and sf are variances of m1 and m2, and p(-1 # p # +1) is a coefficient of correlation between them. Then the conditions of co-existence are represented in the form:

In Fig. 8.5a and b the domains defined by these inequalities are shown for the cases of positive and negative correlation. One can see that the negative correlation between the Malthusian parameters of species significantly increases the possibility of their co-existence.

a

Fig. 8.5. Domains of co-existence (1, domain of ecological stability): (a) p > 0, (b) p < 0

As shown above, the condition £\=1 {rk 1 )m > 0, which in our case is written as T22ml + t11m2 > t12(mi + M2) or is sufficient for existence of the maximum of inf(M0)• Therefore, if m1 < p(s1 / s2)m2 then the maximum is attained at the point p* = 0, if m2 < p(s2/s1 )m1, then at the point p* = 1. The equation ((s2/s1 ) — r)m1 + ((s1 / s2) — r)m2 = 0 in the plane {m1 , M2} defines the line which divides the entire set of m1 and m2 into two subsets corresponding to the maximum and minimum of inf(M) (Fig. 8.6).

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