Cybernetics and Systems, Vol.30 (1999), No.6, pp. 571-585.

INTRODUCTION

     A theoretical description of any natural system includes two important aspects at least. First, one should construct a concrete mathematical model of both the admissible states of the system and its transitions between these states. Second, one should establish the choice rules (often in the form of an extremum principle) selecting, among the many theoretically admissible states of the system, only those states which are realized in nature under the given external conditions. In the present paper, we suggest that both these problems be solved by the methods of category theory.

     Apparently, the first to widely use set theory and topology for modelling the biological processes was Rashevsky (1954, 1955) laying the foundations of abstract biology. Later, in his works on relational biology, Rosen (1958, 1959) made the next step and applied category theory to the description of biological systems, in particular, a living cell. In subsequent years, an abstract category-theoretic approach to problems of mathematical biology was developed in various ways among which we would like to note primarily: the molecular set theory (Bartholomay, 1960, 1965), the organismic supercategories (Baianu, 1968, 1970), and the energy theory of abstract ecological systems (Leguizamon, 1975, 1993).

     In this work, we have attempted to construct a category-functor model of abstract natural systems (not necessarily biological) and to apply it to the description of ecological communities. The paper contains a development of ideas presented in Levich (1982).

     An analysis shows that, in all practically interesting cases, an arbitrary state or even the whole state space of a natural system can be identified with a set equipped with a certain mathematical structure: algebraic, topological, etc.

     For example, it is convenient to describe a state of an ecological community by means of a set with a partition into mutually disjoint classes where elements of the set are organisms forming the community, while partition classes are populations of biological species (Levich, 1982). Below, we will illustrate the basic statements of the general theory with the ecological community example.

     Returning to an arbitrary natural system, we note that a mathematical structure, specified on sets-states, models only those properties of the real system which are invariant under its transitions from one admissible state to any other. Precisely such a property is the existence of a species structure in an ecological community. Thus, in the general case, the state space of a natural system is simply a class of uniformly structured sets (such as groups, topological spaces, sets with partitions, etc.).

     Let A and B be two uniformly structured sets. Consider their direct product and fix a certain subset in it. Any subset is called a correspondence from A to B. In this connection, one uses the following notation: for the complete image of the element with respect to the correspondence and for the complete preimage of the element with respect to the same correspondence. Situations are possible when or/and for certain a and b. Let us list the most important types of correspondences from A to B:

1. Everywhere defined correspondences , such that for any ;

2. Surjective correspondences , such that for any ;

3. Functional correspondences , such that either is the empty set or consists of one element for any ;

4. Injective correspondences , such that either is the empty set or consists of one element for any

     Moreover, various combinations of types 1-4 are also admissible; in particular, everywhere defined functional correspondences which are identified with ordinary mappings. All of them will be needed below.

     Following Levich (1982), we call correspondences preserving a mathematical structure specified on the sets A and B structure morphisms and denote them by Greek letters: For example, structure morphisms of sets with partitions are correspondences transforming each partition class of one set as a whole into some partition class of another set, so that different classes are transformed into different ones. Other examples of structure morphisms are homomorphisms of groups and continuous mappings of topological spaces.

     According to the aforesaid, we will identify transitions between admissible states of an arbitrary natural system with corresponding structure morphisms. It is clear that each system is characterized by a specific class of structure morphisms reflecting its concrete “transitional” properties; even if two systems have the same state space, but distinct classes of structure morphisms, then one must distinguish such systems.

     Thus, we suggest to associate with each natural system a mathematical construction S consisting of two classes: and . Here is a fixed class of uniformly structured sets which will be also called objects; is a fixed class of structure morphisms between these sets. It is not hard to see the following:

1. With each ordered pair of objects one can associate the set of structure morphisms from A to B.

2. Each structure morphism of the class belongs to one and only one set , where .

3. For any three objects , one can introduce the multiplication of structure morphisms: , . Here is the product of structure morphisms and defined by the following condition: (, ) if and only if there exists an element such that and simultaneously.

4. The multiplication of structure morphisms is associative, i.e., for any , , , and .

5. For each , there exists a structure morphism called the unit morphism such that , for any , , and . This is the identity correspondence:

     Therefore, S is a category (Bucur & Deleanu, 1968; Goldblatt, 1979). Thus, the proposed model of natural systems has a simple category-theoretic interpretation according to which admissible states of a system are objects of the category S and permissible transitions between these states are morphisms of the category S.

     In what follows, in general considerations, S will designate a category such that the class consists of all uniformly structured sets, while the class consists of all structure morphisms between them.

     The next important step in describing a natural system is to establish an extremum principle selecting from the class of admissible states of the system only those states which are realized under the given external conditions. To this end, it is necessary to construct a function on the system state space taking values in a linearly ordered set (class) because the concept of “extreme state” is meaningful only in this case. The present section is devoted to a search for such a function defined on the class and having the most natural form from the viewpoint of category theory.

     First of all, let us recall some information about cardinal numbers of nonstructured sets. Consider the category Set whose objects are arbitrary sets and morphisms are arbitrary correspondences between these sets. Below, we will call subclasses of the direct product binary relations on the class .

     Let be the binary relation on defined by the rule:

Û there is an injective mapping of into .

     Here . It is obvious that has the following important properties:

1. for any (reflexivity);

2. if and , then

i.e., is a preorder relation on . We factorize this preorder by introducing the binary relation on the same class according to the rule:

Û and .

     It is not difficult to verify that the so defined relation is reflexive, transitive, and symmetric (the latter signifies that if , then as well). Therefore, is an equivalence relation on . One can show (Levich, 1982) that if and only if there exists a bijection between the sets and .

     For each , we denote by the class of all the sets which are -equivalent to the set , i.e., . One usually calls this class the cardinal number of the set and writes (if is a finite set, then is identified with the number of elements in ). In other words, cardinal numbers of nonstructured sets are elements of the factor class . We simultaneously have the canonical surjection , .

     It is natural to define an order relation on the factor class , namely:

Û .

     The reflexivity and transitivity of follow from the corresponding properties of the preorder . The antisymmetry of is established by the simple reasoning: and Û and Û Þ . Thus, is indeed an order relation on and, moreover, this order is linear (i.e., any two cardinal numbers are comparable) (Bourbaki, 1960).

Now, let us try to extend the above constructions to the case of the category S of uniformly structured sets. Let . By analogy with universal algebra, injective mappings of A into B (and vice versa) preserving a mathematical structure specified on these sets will be called structure monomorphisms.

Consider the binary relation defined on the class by the following rule:

Û there exists a structure monomorphism of A into B.

     Evidently, the relation is reflexive and transitive; hence is a preorder on . Factorization of this preorder creates the equivalence relation on the same class in the usual way:

Û and .

     For each , let us introduce the class of uniformly structured sets which are -equivalent to A. Following Levich (1982), we call this class the structural number of the set A and use the notation for it. Thus, by definition, structural numbers are elements of the factor class . We simultaneously have the canonical surjection , .

     With the aid of one can define the order relation on the factor class by setting

.

     The reflexivity, transitivity, and antisymmetry of are verified in an elementary way. However, unlike cardinal numbers, structural ones are ordered only partially. For example, there are no structure monomorphisms between two sets with the partitions: and (i.e., the corresponding structural numbers and are incomparable). Thus, functions defined on and taking values in cannot be used for formulating an extremum principle.

     Let us seek a way out of the above situation by constructing a suitable generalization of structural numbers. To this end, one should recall that S is a category and the factor class is ordered linearly.

     Given an arbitrary object A in the class , consider the mapping associating with each the set of structure morphisms from A to B and with each the everywhere defined functional correspondence from to for any . It is obvious that the conditions

1. if , then ;

2. if , then ;

3. for any ;

4. if and , then

are valid. Therefore, the mapping is a one-place covariant functor (Bucur & Deleanu, 1968) of the category S into the category Set. It is known as a representing functor.

     Following Levich (1982), we call the cardinal number the invariant of the object with respect to the object . It is not difficult to prove the proposition:

if then .

     Indeed, the inequality means that there exists a structure monomorphism of B into C. Let be such a monomorphism and be arbitrary structure morphisms from A to B. Since b is an everywhere defined functional injective correspondence, the equality implies .       Consequently, is an injection of into and . Q.E.D.

     In particular, if , then . This justifies the term “invariant” introduced above for . Since is a linearly ordered factor class, we have: or or for any . Thus, one can compare objects of the category S by comparing their invariants. The last circumstance forms the basis of the functor method of comparison of structured sets (Levich, 1982).

     Having fixed , we obtain the function , which is the composition of the injection , and the canonical surjection . The function is defined on the whole class and takes values in the linearly ordered factor class , i.e., it can be employed perfectly well in an extremum principle.

     Now, keeping in mind the subsequent applications to ecological communities, we concentrate attention on finite sets. Instead of S, consider the category whose objects are finite nonstructured sets and morphisms are correspondences between them having the type “a” (the multi-index a takes one of sixteen values which are subsets of the set of the main types of correspondences listed in the previous section). Assume , , and , where are nonnegative integers. Let us denote by the invariant of the object with respect to the object in the category . It is obvious that coincides with the number of morphisms from to and is a function of m and n. The following is a list of the results of calculating the invariants for all the sixteen values of the multi-index a:

Here is the system of all coverings of the set , , and is an arbitrary k-element set.

     As a special case of S, consider the category such that the class consists of finite uniformly structured sets and the class consists of structure morphisms which have the type “a” as correspondences between sets (see the previous paragraph). The category already gives sufficient possibilities for modeling a broad spectrum of natural systems.

     Let and be the invariant of the object B with respect to the object A in the category , i.e., . (For example, if is the category of finite sets with partitions, then , where is the i-th partition class of the set A and is the partition class of the set B into which the class passes as a result of morphisms from (Levich, 1982)). Moreover, let be the same sets A and B, but only deprived of the mathematical structure on them (also known as supports of the structure). Consider the quantity called the specific invariant of with respect to ; it has the meaning of the number of correspondences from per one structure morphism from . It is convenient to take the specific invariant as a quantitative measure of deviation of the structured sets A and B from the corresponding supports and of the mathematical structure. Applied to natural systems, this means that the states of a system with greater values of the specific invariant are “stronger” structured than the states with smaller ones.

     According to the aforesaid, we introduce the following definition:

 For a natural system admitting a description within the framework of the category , the logarithm is called the entropy of the state with respect to the state .

     If a class of states between which transitions are permitted from the viewpoint of some substantial reasons, for example, due to conservation of macroscopic parameters of the system, is interpreted as a macrostate of the system and a result of an arbitrary transformation of the system is interpreted as its microstate, then is a generalization of Boltzmann’s entropy which is conventionally defined as the logarithm of the number of distinct microstates corresponding to a given macrostate. The possibility of the above interpretation and the coincidence of special cases of the quantity with the traditional formulae for Boltzmann’s entropy justify the term “entropy” applied to . Note, however, that, in the present approach, the entropy appears without any probabilistic considerations.

     For fixed , the entropy is a function of taking values in the positive real semiaxis . Indeed, this follows from the obvious inequalities and , where the supports of the structure correspond to the objects A, B and the variable .

It is now clear that the desired extremum principle can be formulated in the form of the postulate:

     In reality, a natural system passes from a given state to the state for which the entropy is maximal within the bounds defined by the external conditions (for example, available power and other resources).

     The suggested principle admits the following interpretations:

1. Since, for fixed , the specific invariant is regarded to be the measure of deviation of the structured set from the support of the structure, the extremum principle selects those states of the natural system which are deviated most strongly from their nonstructured analogs , i.e., are “maximally” structured.

2. For fixed , the invariant depends only on the number of elements in the set . Therefore, if is also fixed, then the entropy will be the greatest when the invariant is the smallest. However, we can treat a small number of structure morphisms from A to B as the high “stability” of the state . Thus, the extremum principle realizes the states of the natural system with both the maximum number of elements in the set-support of the mathematical structure and the maximum “stability” with respect to morphisms of the structure.

3. It is also possible to interpret the entropy

     In the next section we will discuss an ecological application of the above extremum principle.

     Consider an ecological community consisting of organisms of the same trophic level without an age structure (for example, cells of phytoplankton) which belong to w species and consume m mutually irreplaceable resources. Assume that in this community fission of cells and their death are admissible but absorption of one organism by another and introduction of organisms from outside are inadmissible. Since, in the adopted category-theoretic model of natural systems, states of the ecological community are described by means of sets with partitions, transitions between states of the community satisfying the above requirements are injective surjective structure morphisms of sets with partitions.

     We will seek the stationary final state of the community. The following is a modification of the extremum principle for this special case:

     Let be the number of organisms of the species i in the community, be the vector of the species sizes, be the total number of organisms, be the amount of the k-th resource in the environment, be the requirement of an organism of the species i for the k-th resource. Evidently, any admissible state of the community is characterized completely by the vector. One can show (Levich, 1982) that where A corresponds to . On the other hand, we know from the previous section that . Thus, a conditional extremum problem appears (Levich, Alekseev, and Nikulin, 1994):

A solution of this problem is yielded by the species structure formula (Levich, 1980)

which connects the species sizes in the ecological community with the resources on which n and depend. Levich (1980) and Levich,. Zamolodchikov, and Rybakova (1993) have demonstrated the adequacy of the species structure formula to the empirical data (see also Lurie, Valls, and Vagensberg (1983)).

     It has been shown as well that:

     A solution of the extremum problem exists and is unique under any () and realizes a maximum of the functional (the existence and uniqueness theorems) (Levich, Alekseev, and Nikulin, 1994).

     The space of environmental resources consumed by the community splits (stratifies) into non-intersecting subsets, any of which corresponds to a unique collection of resources consumed completely; and in these subsets the state of the community depends on the same resources as arguments and only on them (the stratification theorem) (Levich, Alekseev, and Nikulin, 1994).

     The fractional sizes of the species depend only on the resource amount ratios in the environment and take the greatest values under the resource ratios which are equal to the given species' ratio of demands for them (the optimization theorem) (Levich, Alekseev, and Rybakova, 1993; Alexeyev & Levich, 1997). This optimization theorem creates an efficient method of management for the species structure of ecological communities with the aid of environmental resource factor flows (Levich & Bulgakov, 1993; Levich, Khudoyan, Bulgakov, and Artiukhova, 1992; Levich & Bulgakov, 1992; Levich, Maximov, and Bulgakov, 1997).

     The variational problem of finding the maximum entropy under restricted (from above) environmental resource consumption turns out to be equivalent to the variational problem of finding the minimum environmental resource consumption by the community under restricted (from below) “structure entropy” of the community (“the Gibbs theorem”). This allows the suggested extremum principle to be interpreted as the principle of minimum environmental resource consumption (providing a sufficient degree of system organization) (Levich & Alexeyev, 1997).