52
A. F. BARTHOLOMAY
However, a careful biomathematical formulation of a given biosystem under
study can greatly facilitate the process of model construction, referring
now to the scientific truism that careful formulation of a mathematical
problem leads one 50 % of the way toward its solution. An additional guide
that may be offered is the delineation between two broad approaches to the
construction of such models, as follows:
a. The direct approach refers to the direct search de novo for a mathematical
system or discipline which matches the system ~. It is particularly in this
approach that the biomathematical formulation should be suggestive.
For example, if the biological problem or system is one of kinetics
or, more specifically, of the determination of an index for measuring
rate of transformation of some material (s), then an obvious choice
of corresponding mathematical domain would be the Calculus and the
Theory of Differential Equations since these mathematical areas have
to do essentially with the study of rates of change of mathematical
functions (v.i).
b. The indirect or analogue approach. The indirect approach amounts to
finding another system, say, ~ (C, S', F', E', /,), either in biology itself or
in some other discipline which bears some resemblance to those aspects of
the system under study and for which a tested and "worked out" mathematical model \ill' has already been constructed. Such analogues for biosystems have been found, for example, in physics, chemistry and, more
recently, in the engineering sciences, and even in economics and the social
sciences-wherever "systems" are studied from a mathematical point of
view. In fact, it would appear that the majority of mathematical models in
biology may have arisen in this way.
The use of such analogues, however, requires great caution, for there is
the danger that the systems which they represent may become misleadingly
identified with the biological systems, without of course being completely
equivalent to them. In other words, while the analogous model \ill' may
suggest a model \ill for the biosystem ~, it must be kept in mind that \ill'
itself represents a measure on a system whose composition, structure, and
function are not in one-to-one correspondence with the system under
study. If, on the other hand, the system can be redefined legitimately in a
manner which establishes such a correspondence, then \ill' may be used with
greater validity. However, this would depend on defining new concepts
or redefining old biological concepts and would thus carry the danger of
imposing an artificial pattern on the biological system. Dangers of this type
possibly may be avoided by using the analogue and its associated model \ill'
simply as an indicator of a discipline of mathematics which is useful in
quantifying the type of system under study, and then proceeding to construct \ill independently of \ill' (as though it were the first step in the direct
approach just discussed).
A. F. BARTHOLOMAY
However, a careful biomathematical formulation of a given biosystem under
study can greatly facilitate the process of model construction, referring
now to the scientific truism that careful formulation of a mathematical
problem leads one 50 % of the way toward its solution. An additional guide
that may be offered is the delineation between two broad approaches to the
construction of such models, as follows:
a. The direct approach refers to the direct search de novo for a mathematical
system or discipline which matches the system ~. It is particularly in this
approach that the biomathematical formulation should be suggestive.
For example, if the biological problem or system is one of kinetics
or, more specifically, of the determination of an index for measuring
rate of transformation of some material (s), then an obvious choice
of corresponding mathematical domain would be the Calculus and the
Theory of Differential Equations since these mathematical areas have
to do essentially with the study of rates of change of mathematical
functions (v.i).
b. The indirect or analogue approach. The indirect approach amounts to
finding another system, say, ~ (C, S', F', E', /,), either in biology itself or
in some other discipline which bears some resemblance to those aspects of
the system under study and for which a tested and "worked out" mathematical model \ill' has already been constructed. Such analogues for biosystems have been found, for example, in physics, chemistry and, more
recently, in the engineering sciences, and even in economics and the social
sciences-wherever "systems" are studied from a mathematical point of
view. In fact, it would appear that the majority of mathematical models in
biology may have arisen in this way.
The use of such analogues, however, requires great caution, for there is
the danger that the systems which they represent may become misleadingly
identified with the biological systems, without of course being completely
equivalent to them. In other words, while the analogous model \ill' may
suggest a model \ill for the biosystem ~, it must be kept in mind that \ill'
itself represents a measure on a system whose composition, structure, and
function are not in one-to-one correspondence with the system under
study. If, on the other hand, the system can be redefined legitimately in a
manner which establishes such a correspondence, then \ill' may be used with
greater validity. However, this would depend on defining new concepts
or redefining old biological concepts and would thus carry the danger of
imposing an artificial pattern on the biological system. Dangers of this type
possibly may be avoided by using the analogue and its associated model \ill'
simply as an indicator of a discipline of mathematics which is useful in
quantifying the type of system under study, and then proceeding to construct \ill independently of \ill' (as though it were the first step in the direct
approach just discussed).