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WILLIAM STREIFER
e.g. the probability of encountering a mate or predator. In contrast, the
models discussed herein are primarily deterministic and such random
events are taken into account as averages. Furthermore, in such models
climate may be considered to vary according to some statistical
distribution in its influence on births, deaths etc.; however, the birth
rate itself is not considered to be a random variable.
In general, the approach employed here is based on partial and
ordinary differential equations. Many algebraic models are in fact
discrete approximations to partial differential equations (Sinko and
Streifer, 1967), and certain integral formulations and branching processes
can also be shown to be equivalent to partial differential equations
(Martinez, 1966). Currently, digital computers are routinely employed
to obtain solutions to mathematical models. There also exist models,
formulated directly for computers, in the form of flow charts. Often
these models too are discrete approximations to partial differential
equations. The results of some digital computer models will be cited in
this paper, but the models themselves will not be discussed in detail.
Also, although reference to experimental results and observations will
be made as needed, no discussion of experimental procedures or data
acquisition will be included.
11. TOTAL POPULATION MODELS
The earliest population models attempted to describe the population
dynamics in terms of only one variable for each species; the total
population of that species. For single species it was postulated that the
rate of change of the total population, N, is proportional to N, viz.
dN
- = TN
at
(Malthus, 1798; Pielou, 1969). The net effect of the birth rate less the
death rate is incorporated in the single constant T and the resulting
population either increases or decreases exponentially for positive or
negative T (respectively). The limitations of this model are obvious;
populations either grow without limit or decay to extinction. Verhulst
(1838) (see also Pearl and Reed, 1920; Lotka, 1925; Gause, 1934;
Pielou, 1969) modified the equation by adding a nonlinear term,
dN
- = r N ( I - N / K )
at
The resulting logistic equation is also inadequate for many populations
since it always predicts a monotonic increase in the population to its
equilibrium value N , = K.
WILLIAM STREIFER
e.g. the probability of encountering a mate or predator. In contrast, the
models discussed herein are primarily deterministic and such random
events are taken into account as averages. Furthermore, in such models
climate may be considered to vary according to some statistical
distribution in its influence on births, deaths etc.; however, the birth
rate itself is not considered to be a random variable.
In general, the approach employed here is based on partial and
ordinary differential equations. Many algebraic models are in fact
discrete approximations to partial differential equations (Sinko and
Streifer, 1967), and certain integral formulations and branching processes
can also be shown to be equivalent to partial differential equations
(Martinez, 1966). Currently, digital computers are routinely employed
to obtain solutions to mathematical models. There also exist models,
formulated directly for computers, in the form of flow charts. Often
these models too are discrete approximations to partial differential
equations. The results of some digital computer models will be cited in
this paper, but the models themselves will not be discussed in detail.
Also, although reference to experimental results and observations will
be made as needed, no discussion of experimental procedures or data
acquisition will be included.
11. TOTAL POPULATION MODELS
The earliest population models attempted to describe the population
dynamics in terms of only one variable for each species; the total
population of that species. For single species it was postulated that the
rate of change of the total population, N, is proportional to N, viz.
dN
- = TN
at
(Malthus, 1798; Pielou, 1969). The net effect of the birth rate less the
death rate is incorporated in the single constant T and the resulting
population either increases or decreases exponentially for positive or
negative T (respectively). The limitations of this model are obvious;
populations either grow without limit or decay to extinction. Verhulst
(1838) (see also Pearl and Reed, 1920; Lotka, 1925; Gause, 1934;
Pielou, 1969) modified the equation by adding a nonlinear term,
dN
- = r N ( I - N / K )
at
The resulting logistic equation is also inadequate for many populations
since it always predicts a monotonic increase in the population to its
equilibrium value N , = K.