STABILITY AND EVOLUTION IN THE THEORY OF MANY-POPULATION CORRELATION FUNCTIONS

 

Andrei Dumitru Iacobas

Eastern & Central Society of mathematical Ecology

Ovidius State University, Dept. Biophysics & Biomathematics

58 Ion Voda Street, 8700 Constanta, Romania

E-mail: aiacobas@rtns.ro, iacobas@medcon.ro, Fax. (+40-41) 672899, 618372, 673500

Here the population is the set of all individuals that could be permuted without changing the biocoenotic features and the individual is the average reproduction unit. Each many-population correlation function defines a pyramid of correlated populations based on predation, symbiosis, parasitism, competition etc., while every many-individual correlation function defines an average group. New concepts like the: many-population temperature, many-population potential, many-population entropy, pyramole, pyraday a.s.o., are defined and some selection rules of long-term stable pyramids and biocoenoses, known as Dracula’s postulates, are advanced. The postulates, somehow similar to Dalton's laws in chemistry, are actually steady-state conditions.

Key words: stability, evolution, virtuality, many-population temperature, many-population entropy.

STABILITY AND EVOLUTION IN THE THEORY OF MANY-POPULATION CORRELATION FUNCTIONS

Andrei Dumitru Iacobas

Ovidius State University, Dept. Biophysics & Biomathematics

58 Ion Voda Street, 8700 Constanta, Romania

E-mail: aiacobas@rtns.ro , iacobas@medcon.ro, Fax. (+40-41) 672899, 618372, 673500

Bogolubov, Born, Green, Kirkwood and Yvon developed the theory of correlation functions to describe the structure and properties of simple liquids. (e.g. Rice & Gray, 1965). Rather than attempt the total configuration distribution of the system of volume V, composed of N identical particles,

(1)

it describes the probability dW of configurational grouping of s (= 2, 3, …) particles irrespective of the positions of the remaining (N - s) particles in the assembly. Therefore, the probability to find the grouped particles in the volume {dq_s} = dq₁ ... dq_s is:

(2)

For an ideal gas (i.e. no correlation among the particles):

(3)

By definition, the functions g_s have the norm 1. They satisfy the recurrence relation:

(4)

for any s > p $ 1, and depends on the configuration integral Q_N and the potential U_N ({q}):

(5)

The key problem in the theory is to determine the many-particle correlation function. To do this, one needs to know the pair-correlation function and the potential. Usually, the potential is postulated (e.g. as Lennard-Jones type).

The theory proved to be effective in describing the noble liquids (Croxton, 1974) and even complex liquid films, like the vicinal water (Spataru & Iacobas, 1987).

There are many types of correlations inside the biocoenose, either among individuals, as among populations, the latters based on competition, predation, parasitism, symbiosis etc., consistently described by many authors (e.g. Aikman, 1992, Benjamin & Sutherland, 1992, Cushing 1994, Hallam 1986, Palmer, 1992). The correlations strongly influence the physiological features of the individuals and the population size. The aim of this contribution is to find criteria of long term stability in constant external conditions. To this end, some concepts of thermodynamics and theory of correlation functions have to be adapted and the formalism has to be generalized to account the ecosystem complexity. As Lotka did (Lotka, 1956), I consider my right to define the main notions to be used in the theoretical approach.

Individual = the average reproduction unit. Because of this simplifying definition, there are no age classes within the populations.

Population = the set of all individuals from the biocoenose that could be permuted without any change of biocoenotic macroscopic features.

Many-Individual Correlation (MIC) = any kind of relation among individuals that could affect their physiological features. It could be homogeneous (the individuals belonging to the same population) or heterogeneous (else).

Many-Population Correlation (MPC) = any kind of relation among populations that could affect the population size.

Group = the correlated individuals by the same MIC.

Pyramid = the set of all correlated populations by the same MPC.

Pyraday (PD) = conventional unit of the average lifetime (the shortest lifetime of the correlated populations in a stable pyramid).

Pyralife (PL) = the least common multiple of the average lifetimes within the correlated populations in a stable pyramid, expressed in pyradays.

Pyramole (PM) = conventional unit for the population size. For a pyramid containing only two correlated populations in long term stability, the simplest rule is 1:1, meaning for predation that 1 pyramole of predator eats 1 pyramole of prey, every pyralife. By evidence, the pyramole does not contain the same number of individuals for distinct populations.

Many-population temperature = intensive measure of chaos in pyramid.

Many-population entropy = extensive measure of chaos in the pyramid.

First superposition postulate: Any real biocoenose could be approximated with the weighted superposition of virtual ones, having the same composition, but different correlations among populations.

The set of the virtual biocoenoses form an orthonormal basis. Because of this, the distribution G of the system, can be expressed as:

(6)

Here, N is the total number of individuals, N_i the number in the i^th population (i = 1, 2, … m), A, B, C, … are the weights: A_i of virtual biocoenose with maximum i correlated populations, B_j the sub-weight of the virtual systems containing j pair-populations, … The G_...’s are the distributions of the virtual many-population pyramids. The subscripts denote the populations involved while the superscripts the number of individuals in every population. By evidence, the G_...’s are invariant for any permutation of populations inside the pyramid.

The above expansion contains all theoretically possible correlations. Most of them are out of sense, so that their weights vanish. The weights have cyclic variations. Every many-population correlation function can be further expanded in may-individual correlation functions, accounting for group formation. By evident reasons only some group structures are significant, while the other terms of the double expansion could be neglected. The advantage of this method arises in the possibility to incorporate separate results of any kind of relation among individuals as among populations to form a more realistic picture.

One could approximate in the same manner (identical weights) the total potential U with the weighted superpositions of the pyramid (i.e. many-population) potentials:

(7)

The pyramid potentials are invariant for any permutation of populations.

Due to the orthogonality of the virtual systems, the scalar product of any two intensive measures, F and H, is:

(8)

with the same conditions of normality as above. This is a very important property, since it considerably simplifies the formalism when computing the thermodynamic features.

Recurrence relations relate the many-population correlation functions. For three populations, the simplest one is similar to that proposed by Kirkwood for liquid argon (Croxton, 1974):

(9)

involving the following relation among the many-population potentatials:

(9’)

Since the potential is an extensive measure, the products of virtual distributions are turned into sums of their potentials.

The constant entropy and the null chemical potential gradient, defining the steady-state of the pyramid, impose to the correlated populations some constraints. They act as selection rules. In the quantum physics, as well as in the ancient Pythagoras’ theory of music, the selection rules are usually expressed by sequences of integers.

First pyramid postulate: within a stable pyramid, the set of the population’s average lifetimes, expressed in pyradays, is the image of a sequence of natural numbers.

The postulate, actually a resonance principle, is quite natural since it ensures the most effective correlation, for instance between the maturation of preys and that of predators. It is justified by the biomass conversion in a stable predation pyramid. (Kooijman, 1993). In some of my books (Iacobas, 1995; Iacobas, 1996, Iacobas, 1998), I defined the social time and the ecological time by the speed of social/ecological transformation in the pre-Hilbert space of steady-states. Within the pyraday, time is not flowing uniformly, as compared with the astronomic one, but the number of pyradays of each correlated population remains constant. The postulate imposes physiological modifications to the partners to fit the correlation requirements. (Dobson, 1988; Dobson & Lyles, 1989). The artificial as the cooked food, the chemical supply in agriculture (even by modifying the normal occurrence of some isotopes), the effects of the imposed died and recommended menus, all together alter the normal predation and the above postulate could not be verified in the early stages of the pyramid formation.

First superposition lemma: within a stable biocoenose, the set of pyramid life cycles is the image of a sequence of natural numbers.

This lemma extends the philosophy of pyramid stability to the entire ecosystem.

The chemist prefers to express the quantities in moles rather than in grams since the former makes clear the combination rule of atoms to form the molecules. I think the pyramole brings the same advantage. The "chemical" formula of a stable group within a stable pyramid I (0 C = the multitude of all possible combinations of the m populations), composed of s # m populations, in a particular pyraday t, is: [n₁n₂…n_s]_I;t, where n_i (I; t) (i = 1, 2, …, s) is the number of the i-type individuals involved in the correlation I at the moment t. By evidence, for all indexes i: N_i(I)/n_i(I) = const = X (I), which is the pyramid equivalent of the Avogadro’s number. It depends only on the set I of correlated population and represents the number of identical groups in the biocoenose. Every population in a pyramid composed of stable groups have n_i(I)X(I) individuals. Since n_i(I) varies cyclically, with the pyramid life cycle as period, N_i(I) follows the same dynamics. The energy constant, similar to Boltzmann’s constant is R/X(I). Of course, it differs from pyramid to pyramid.

Second pyramid postulate: within a stable pyramid, the set of population’s sizes, expressed in pyramoles, is the image of a sequence of natural numbers.

The total number of individuals belonging to the same population i, at a particular time t, is by evidence the weighted sum of individuals involved in the distinct pyramids.

(10)

The total number of individuals in the pyramid I is:

(11)

Second superposition lemma: within a stable biocoenosis, the set of pyramid’ sizes, expressed in pyramoles, is the image of a sequence of natural numbers.

To simplify the mathematics, we shall consider the pyramid composed of identical groups. Therefore, every pyramid term in the many-population expansion can be replaced by the corresponding sum of groups. For instance:

(12)

By this procedure the statistical calculus (using Gibbs distribution) in the space of configurations of all individuals from the system is reduced to the process within a group.

Any physical characteristic of the system, depending on a particular type of pyramid:

(13)

can be averaged by:

(14)

Every many-population (pyramid) potential can be developed in terms of many-individual (group) potentials. Since the groups are acting independently:

(15)

Here we need new assumptions about the inter-individual relations.

We can also express the conditioned correlation of a group of individuals from s populations, when a group from other p populations (1 < s + p < m) is correlated.

(16)

Therefore, one can model the formation of bigger pyramids from smaller ones. If p=1 and n_s+1 =1, then the addition of a new population (here one pyramole of the top predator) means:

(17)

The introduction of a new predator modifies the predation relations among the other correlated populations in the former pyramid.

Let it be a biocoenose composed of two populations: the prey (N individuals) and the predator (P individuals), the steady-state predation formula being: n : p (n preys supports p predators). We shall consider the following assumptions:

1. Immigration equals emigration => the number of distinct populations is constant and their sizes are affected only by internal factors;

2. The biotopos is uniform and constant => only the correlations are disturbing the uniform distribution of the individuals;

3. The system is large enough => no boundary contribution & no density limitations => the intrinsic growth rate is the difference between the born rate and the death rate;

4. Every population consists on identical individuals;

5. Every individual type is constant => no mutation;

6. Every correlation type is constant => no adaptation;

7. The ecosystem is at steady-state.

If g_n⁽ⁿ⁾ and g_p^(p) are the densities of separated prey and predators in the spaces {q ⁿ} = Vⁿ and {q ^p} = V^p, and g_n,p^(n+p) is the density of the correlated populations in the space {q^n+p}= V^n+p then, their time derivatives depend on their intrinsic growth rate r_n = b_n - d_n , the starvation rate of the predator s, the predation efficiency e, and the prey-predator conversion rate c. Therefore:

(18)

with the solution:

(19)

The solution is also a recurrence relation to get, under the above approximations, the two-population correlation g_n,p from the uncorrelated population features g_n and g_p. Considering only constant rates, the steady-state populations are:

(20)

The above equations and equilibrium populations look like the results of the classical Lotka-Volterra predator-prey formalism used in many textbooks (e.g.: Ricklefs, 1990) and simulated by the well-known Populus package, very popular among the ecologists and ecological students. In a very similar manner one can introduce other constraints, like the density-dependent growth (Allen, 1989), or the theta-logistic predator-prey relationship (Gilpin & Ayala, 1973; Holling 1965). The constraint can be imposed to all virtual systems or only to some of them. Even there are many mathematical complications, the formalism of many-population correlations makes nothing less the classical approach, but brings more realism.

Considering the many-population expansion (6) and the property (8), the local balance equation of the many-population entropy density is:

Re-arranging the equation (21) upon the weights, one obtains:

Since the equation (22) must be valid whatever be the set of weights (that is, for any possible approximation with virtual systems), the factors of weights should vanish. This means that each virtual biocoenosis satisfies the condition of entropy local balance. At steady-state, the virtual conditions are:

(23)

 

…

 

 

Lemma: the real biocoenosis steady-state is the result of all virtual systems steady-state.

 

8. Condition of evolutionary jumps

To understand the idea of the theory, we shall consider at the very beginning the simplest possible situation. Let there be a biocenose, composed of only two populations, a prey and a predator, completely correlated by predation. That is, the predator has nothing else to eat and the prey has no other predator. The biocoenose is confined in the limited volume V, with constant and uniform boundary conditions.

Let then N (= N₁(2) + N₂(2)) be the total number of individuals. Because of the complete correlation, all the weights from the expansions (6) and (7) are zero, except: A₂ = B₁ = C_1,2 = 1. The pyramid formula is: n₁ (2) : n₂(2), meaning that n₁ (2) preys supports, in a long term stability, n₂(2) predators. According the Second Pyramid Postulate, there are N₁(2)/n₁(2) = N₂(2)/n₂(2) = X(2) identical groups in the pyramid. The distribution function is therefore:

 

(24)

while the potential is:

 

(25)

The above approximation of the two-population potential is possible because the groups are practically independent in their work.

Which are the conditions to subdivide the individuals in more than 2 populations? That is, how is possible the evolutionary jump of some of them, to form a new population? Maybe a predator of the former two? Or a new prey, better adapted to fight against predation or to find new resources?

Let us suppose the potential existence of m > 2 distinct populations, related by predation, obtainable by the redistribution of the individuals. For the moment, there are only two populations, but let us consider m = 3, that is two real populations and a virtual one. We can use the following relations (adapted from the previous contribution):

(26)

with the evident initial condition:

(27)

The populations have been numbered in the order of predation. The top predator is number 3 and we are concerned about the conditions of its occurrence. All the correlation functions are by evidence positively defined.

In the followings, the superscripts, denoting the number of individuals in the group and the type of correlation will be omitted (when understandable) to simplify the writing. The correlation functions and the potentials are invariant to any permutation of indexes.

Because of the independence of the groups, the new (yet virtual) distribution and potential are:

 

(24’)

 

 

(25’)

 

We know from the definition of correlation functions:

 

(28)

 

where Q_N is the configuration integral (norm factor) and b_1,2 = 1/k_1,2T_1,2, b_1,2,3 = 1/k_1,2,3T_1,2,3 are the thermal coefficients of the two and three correlated populations (k_... is the many-population Boltzmann constant and T_... is the many-population temperature).

The conditioned correlation function is:

(29)

It shows how the correlation between the populations 1 and 2 depends on the presence of the 3^rd population in the biota. If the new pyramid has the formula n₁n₂1, that is: n₃ = 1, then:

 

(29’)

since, by the norm condition:

 

The three-population potential can be approximated by:

(30)

 

because the single-population potentials are senseless if only predation is considered. Every two-population potential in (30) is positive and depends implicitly on the existence of the third population. Taking the gradient of the equation (28) as respect with the co-ordinates of mass center of the individuals belonging to the first population in the group, then to the second, and considering max (n₁(3), n₂(3), n₃(3)) negligible as compared to X(3), one obtains from the approximation (30):

 

or:

(31)

 

since the law of reciprocal actions makes:

(32)

 

Generally, the potential gradient as respect to the prey is positive, while to the predator is negative. Additionally, where the prey is more abundant, the predator has a higher density. Therefore,

 

 

(33)

Since the other constants: k_1,2,3 , X(3) and V are all positive, the (pre)condition to appear a new predator by evolutionary jump is:

T_1,2,3 < 0 (34)

I have to remind the discussion is about a virtual correlation and, by consequence, about a virtual temperature. Do not forget also that the temperature is the intensive measure of chaos in a thermodynamic system. The many-population temperature measures the intensity of chaotic "coupling" among individuals from the correlated populations.

9. About the many-population temperature

The many-population temperature is normally defined by:

 

(35)

 

where:

 

(36)

is the many-population group mass (M_1,…,s is the pyramolar mass), f_1,…,s the number of degrees of freedom within the group, and <v²_1,…,s> is the expected value of the random walk speed of the group. By this definition, the many-population temperature of the real pyramid is always positive.

The ratio between the successive correlation functions is:

 

 

 

(37)

 

 

 

 

 

 

One could say nothing for the moment about the sign of the exponent in normal circumstances, but could expect it is negative and the inferior correlation dominant. By evidence, there are the following inequalities:

 

 

(38)

 

 

For the negative three-population temperature (virtual pyramid), the exponent is for sure positive, and the superior correlation (yet virtual) is dominant. Once the evolutionary jump performed, involving the occurrence of the new population, the new many-population temperature switches to positive values and the fight for the existence starts on.

Concluding remarks

1. The classical formalism of many-particle correlation functions have been generalized and adapted to fit the mathematical ecology requirements. To these end some concepts had to be redefined.

2. The real biocoenose is approximated with the weighted superposition of some virtual ones, having the same composition but different correlation degrees among individuals and populations.

3. Any feature of the ecosystem, depending on a particular type of correlation can be averaged using the proposed formalism.

4. Two Dracula’s postulates and one lemma express the pyramid stability.

5.The evolutionary jump appears when the many-population temperature becomes negative.

6. The work is a provocation for theoreticians to reconsider the fundamental ideas of ecology and for experimentalists to verify its results.

 

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