Deterministic and Stochastic Models of Biological Systems
51
III. The General Notion of Mathematical Model 9J1
of a Biological System (5
Corresponding to the biomathematical formulation of biosystem (5 just
given, by the "mathematical model" 9J1 of a biosystem (5 will be meant an
associated mathematical construct or structure such that: 1. the basic mathematical elements of that structure are mathematical symbols made to correspond with pertinent elements or properties of the biosystem (5; and 2. the
mathematical operations on these symbols, permitted by the identification
of this construct with the appropriate domains of mathematics, accompany
and formalize in some meaningful fashion the underlying biosystemic
relations and transformations. It follows that in setting up a mathematical
model, unless mere symbolic description is the sought-for end itself, it is
necessary to enlarge the primary correspondence into a workable, deductive
mathematical form. The utilization of such a representation will be discussed in greater detail presently, but first the concept itself will be
examined more closely.
The relation between the model and the original system may be visualized, at first, simply as a mapping or association f1 between the objects or
elements in the biological domain defined by (5 and the symbolic elements
of we in the mathematical domain:
(4)
But, as already noted, more is required of f1 than the fact that it establishes a
correspondence or translating device between a biological real system and
an abstract mathematical system: it must also set up a correspondence between inherent biological interrelationships and the formal mathematical
operations allowed by the mathematical domains containing the symbolic
system 9J1. This kind of association may be compared to, say, a "homomorphism" f1: G -+ H of an abstract mathematical group G onto another
group H. Homomorphisms allow the transfer of a problem in one mathematical structure to another, perhaps more computable structure. Similarly,
the purpose of a mathematical model of a biological system is the translation of its quantitative or relational aspects into a mathematical form
which then allows such properties to be pursued further along the lines of
formal mathematical procedures, in much the same manner that one solves
a mathematical problem or establishes a new theorem.
One cannot prescribe algorithms or operational procedures for passing
formally from the biosystem to an appropriate mathematical model. First
of all, in the description just stated of the latter concept, there was no
implication of a unique 1: 1 correspondence between biosystem and model 4 •
4 In fact, subsequent examples in this paper of deterministic vs. stochastic
models as alternatives for a given biosystem will be seen to demonstrate this very
point.
4*
51
III. The General Notion of Mathematical Model 9J1
of a Biological System (5
Corresponding to the biomathematical formulation of biosystem (5 just
given, by the "mathematical model" 9J1 of a biosystem (5 will be meant an
associated mathematical construct or structure such that: 1. the basic mathematical elements of that structure are mathematical symbols made to correspond with pertinent elements or properties of the biosystem (5; and 2. the
mathematical operations on these symbols, permitted by the identification
of this construct with the appropriate domains of mathematics, accompany
and formalize in some meaningful fashion the underlying biosystemic
relations and transformations. It follows that in setting up a mathematical
model, unless mere symbolic description is the sought-for end itself, it is
necessary to enlarge the primary correspondence into a workable, deductive
mathematical form. The utilization of such a representation will be discussed in greater detail presently, but first the concept itself will be
examined more closely.
The relation between the model and the original system may be visualized, at first, simply as a mapping or association f1 between the objects or
elements in the biological domain defined by (5 and the symbolic elements
of we in the mathematical domain:
(4)
But, as already noted, more is required of f1 than the fact that it establishes a
correspondence or translating device between a biological real system and
an abstract mathematical system: it must also set up a correspondence between inherent biological interrelationships and the formal mathematical
operations allowed by the mathematical domains containing the symbolic
system 9J1. This kind of association may be compared to, say, a "homomorphism" f1: G -+ H of an abstract mathematical group G onto another
group H. Homomorphisms allow the transfer of a problem in one mathematical structure to another, perhaps more computable structure. Similarly,
the purpose of a mathematical model of a biological system is the translation of its quantitative or relational aspects into a mathematical form
which then allows such properties to be pursued further along the lines of
formal mathematical procedures, in much the same manner that one solves
a mathematical problem or establishes a new theorem.
One cannot prescribe algorithms or operational procedures for passing
formally from the biosystem to an appropriate mathematical model. First
of all, in the description just stated of the latter concept, there was no
implication of a unique 1: 1 correspondence between biosystem and model 4 •
4 In fact, subsequent examples in this paper of deterministic vs. stochastic
models as alternatives for a given biosystem will be seen to demonstrate this very
point.
4*