Deterministic and Stochastic Models of Biological Systems
53
It has been said with some justification that virtually every branch of
mathematics will turn out to be of use in mathematical biology. And indeed,
one can delineate classes of topological or "relational" or set theoretic
models, geometric models, symbolic logic models, statistical (including
information- and decision-theoretic) models, probabilistic models, algebraic
models, analytical or deterministic models, stochastic models, etc., and
mixtures of these. In the time remaining, the latter two classes of models
will be discussed and their interrelationships stressed. In some cases, these
models can lead to closelv related results; in other cases to differing predictions. In general, they are unfortunately regarded as competing classes,
whereas it would be more scientific to regard them as coextensive.
IV. Deterministic or Analytic Models
A deterministic mathematical model is characterized by the representation of certain quantifiable aspects of a biosystem in the form of ordinary
algebraic variables, or analytic functions of time. The actual construction
of such a model generally involves the application of a direct association
procedure ft between the biosystem and the mathematical domain of
ordinary algebra: algebraic variables x are identified with the pertinent
components of~, and then known biological or other scientific principles
are applied to these algebraic variables to obtain, usually, algebraic or
differential equations that constitute the model.
Implied by such a representation is the expectation that, given the initial
value Xo =/ (0) of such a variable x =1 (t), the entire future course is determinable from the mathematical expression forI (t); i.e. once 1 (t) is obtained, the
value of x at any subsequent time t = tj is obtained by substituting tj into 1 (t).
Most often such a model involves an approximation of a discrete quantity such as the cardinality of a compositional set by a continuous variableit is therefore an idealization to this extent, at least. If one uses the model in a
purely theoretical sense, i.e. to make theoretical predictions about the
states of the system on the basis of formal mathematical deductions, then
some interesting und useful new predictions about the behavior may be
obtained as analytical hypotheses. On the other hand, as soon as one begins
to test such hypotheses by in vitro or in vivo experiments, say or to use constants appearing in the algebraic or analytic expressions of the x's as convenient indices for describing the behavior of the system, then problems of
estimation using pertinent experimental data are encountered. And whether
one turns to simple curve-fitting-by-eye procedures or statistical estimation
procedures, one requires a statistical or probabilistic context. This does not
violate the original deterministic assumption-which may be feasible in
certain circumstances-but it does point to the inadequacy of the analytical
model as a framework for data analysis.
53
It has been said with some justification that virtually every branch of
mathematics will turn out to be of use in mathematical biology. And indeed,
one can delineate classes of topological or "relational" or set theoretic
models, geometric models, symbolic logic models, statistical (including
information- and decision-theoretic) models, probabilistic models, algebraic
models, analytical or deterministic models, stochastic models, etc., and
mixtures of these. In the time remaining, the latter two classes of models
will be discussed and their interrelationships stressed. In some cases, these
models can lead to closelv related results; in other cases to differing predictions. In general, they are unfortunately regarded as competing classes,
whereas it would be more scientific to regard them as coextensive.
IV. Deterministic or Analytic Models
A deterministic mathematical model is characterized by the representation of certain quantifiable aspects of a biosystem in the form of ordinary
algebraic variables, or analytic functions of time. The actual construction
of such a model generally involves the application of a direct association
procedure ft between the biosystem and the mathematical domain of
ordinary algebra: algebraic variables x are identified with the pertinent
components of~, and then known biological or other scientific principles
are applied to these algebraic variables to obtain, usually, algebraic or
differential equations that constitute the model.
Implied by such a representation is the expectation that, given the initial
value Xo =/ (0) of such a variable x =1 (t), the entire future course is determinable from the mathematical expression forI (t); i.e. once 1 (t) is obtained, the
value of x at any subsequent time t = tj is obtained by substituting tj into 1 (t).
Most often such a model involves an approximation of a discrete quantity such as the cardinality of a compositional set by a continuous variableit is therefore an idealization to this extent, at least. If one uses the model in a
purely theoretical sense, i.e. to make theoretical predictions about the
states of the system on the basis of formal mathematical deductions, then
some interesting und useful new predictions about the behavior may be
obtained as analytical hypotheses. On the other hand, as soon as one begins
to test such hypotheses by in vitro or in vivo experiments, say or to use constants appearing in the algebraic or analytic expressions of the x's as convenient indices for describing the behavior of the system, then problems of
estimation using pertinent experimental data are encountered. And whether
one turns to simple curve-fitting-by-eye procedures or statistical estimation
procedures, one requires a statistical or probabilistic context. This does not
violate the original deterministic assumption-which may be feasible in
certain circumstances-but it does point to the inadequacy of the analytical
model as a framework for data analysis.