Verhulst (1838) discovered that the leveling of population growth could
be represented by “the logistic equation.” The term “logistic” has a rare
meaning of “calculation by arithmetic,” which may explain the use of the
term. Data from animal populations showed that exponential growth was
not often observed but that an “S” curve of population growth was more
typical (i.e., it could be calculated from the data). But the logistic equation
was not adopted in the analysis of population growth until studies with
laboratory animals confirmed that a saturation point was typically attained.
Lotka (1924) expanded upon Verhulst’s work with the logistic equation to
come up with the formula that is still in use today.
Volterra (1926) expanded the use of the logistic equation to describe the
populations of competing species and developed the first published case
of an ecological model being used for resource management. His model
results were applied to explain changes in the proportion of fish in the
Mediterranean Sea that resulted from the suspension of commercial
fisheries during the war years of 1915 to 1918. Gause subsequently (1934)
provided experimental confirmation of these interactions.
A decade later, Nicholson and Bailey (1935) used finite difference models
to examine parasitism and predation (critical agricultural problems).
Difference models rely on discrete time steps rather than the continuous
time steps of differential equations. Therefore, difference equations are
closer to the data collected at regular intervals by biologists measuring
population changes. However, the mathematical properties of differential
equations are more easily solved by analytical techniques so they quickly
become more widely accepted. Today, both types of approaches can be
implemented in computer models.
Building upon earlier applications, Hutchinson (1954) constructed mathematical models of population regulation to argue for the importance of
feedback loops, which are integral to resource management. His insistence
on a rigorous approach to ecology led several of his students to invoke
mathematical techniques. Robert MacArthur added a quantitative analysis
to the field of community ecology in the development of the concept of
competitive exclusion (MacArthur 1958), which led him to the hypothesis
that competition determines relationships of species occupying the same
area (MacArthur 1960).
At about the same time Leslie (1945, 1948) developed a matrix approach
to examine changes in life stages over time, and that technique eventually
became a common tool in resource management. While working at the
Bureau of Animal Population at Oxford, Leslie used matrix algebra to
express age-specific relationships (Leslie 1945), explore logistic population
growth and predator–prey relations (Leslie 1948), and consider time lags
(Leslie 1959). Lefkovitch (1965) built upon Leslie’s ideas but classified individuals by development stage rather than age. This stage approach was also
used by Usher (1966) to classify trees. However, these matrix approaches
were not adopted by the broad ecological community for about 25 years.
10
Virginia H. Dale
be represented by “the logistic equation.” The term “logistic” has a rare
meaning of “calculation by arithmetic,” which may explain the use of the
term. Data from animal populations showed that exponential growth was
not often observed but that an “S” curve of population growth was more
typical (i.e., it could be calculated from the data). But the logistic equation
was not adopted in the analysis of population growth until studies with
laboratory animals confirmed that a saturation point was typically attained.
Lotka (1924) expanded upon Verhulst’s work with the logistic equation to
come up with the formula that is still in use today.
Volterra (1926) expanded the use of the logistic equation to describe the
populations of competing species and developed the first published case
of an ecological model being used for resource management. His model
results were applied to explain changes in the proportion of fish in the
Mediterranean Sea that resulted from the suspension of commercial
fisheries during the war years of 1915 to 1918. Gause subsequently (1934)
provided experimental confirmation of these interactions.
A decade later, Nicholson and Bailey (1935) used finite difference models
to examine parasitism and predation (critical agricultural problems).
Difference models rely on discrete time steps rather than the continuous
time steps of differential equations. Therefore, difference equations are
closer to the data collected at regular intervals by biologists measuring
population changes. However, the mathematical properties of differential
equations are more easily solved by analytical techniques so they quickly
become more widely accepted. Today, both types of approaches can be
implemented in computer models.
Building upon earlier applications, Hutchinson (1954) constructed mathematical models of population regulation to argue for the importance of
feedback loops, which are integral to resource management. His insistence
on a rigorous approach to ecology led several of his students to invoke
mathematical techniques. Robert MacArthur added a quantitative analysis
to the field of community ecology in the development of the concept of
competitive exclusion (MacArthur 1958), which led him to the hypothesis
that competition determines relationships of species occupying the same
area (MacArthur 1960).
At about the same time Leslie (1945, 1948) developed a matrix approach
to examine changes in life stages over time, and that technique eventually
became a common tool in resource management. While working at the
Bureau of Animal Population at Oxford, Leslie used matrix algebra to
express age-specific relationships (Leslie 1945), explore logistic population
growth and predator–prey relations (Leslie 1948), and consider time lags
(Leslie 1959). Lefkovitch (1965) built upon Leslie’s ideas but classified individuals by development stage rather than age. This stage approach was also
used by Usher (1966) to classify trees. However, these matrix approaches
were not adopted by the broad ecological community for about 25 years.
10
Virginia H. Dale