376
ference in the birth rate constant and the death rate
constant (r = b - d). This implies that the rate of
population birth and death is a fixed proportion of
the number of individuals in the population. As
populations get large with respect to space or resources available to them, one would expect the
proportion of individuals dying to increase due to
such effects as stress, shortages of food or space,
etc. One might also expect a depression in the proportion of females in the population giving birth to
diminish for much the same reasons. These factors
are termed density-dependent factors in that they
are conditions that alter the rate of popUlation
growth as a function of the population density.
A Simple Population Model
A simple case of how the births and deaths in a
population might be effected by density of the
population would be to assume that the decrease in
birth rate is a linear function of population density.
A similar assumption could be made for the population death rate. The population growth as a function of time under these assumptions would be:
dN
dt = {(b - bddN) - (d + dddN)}N (25.6)
where b is the proportion of the population giving
birth, b dd is the density related change in the proportion of the population giving birth, d is the proportion of the population dying, and ddd is the density related change in the proportion of the
population dying.
One can rearrange the terms of this equation to
obtain:
dN
dt
{(b - d) - (bdd + ddd)N}N (25.7)
If the expression bdd + ddd is replaced by a new
expression so that:
Then the differential equation describing
change of the population becomes:
the
dN
dt
dN
dt
(r - i) Nor (25.8)
(25.9)
Herman H. Shugart
Population ecologists refer to this equation as the
logistic equation or the Verhulst-Pearl equation
(Fig. 25.1). The dynamics of the equation are considerably different from the equation for the exponential increase of a population (Equations 25.4 and
25.5). A population behaving as described by the
logistic equation will rise or fall to K over time (see
Fig. 25.1). K is called the carrying capacity and is
associated with concepts that are well ingrained in
the vocabulary of population biology and wildlife
management-so much so that it has taken to be a
tangible property of a given habitat or location. For
example, wildlife managers are involved with
knowing the carrying capacity of a given area with
regard to a deer herd or other managed animal
population. The carrying capacity concept applied
in wildlife biology is derived directly from the logistic equation (Equation 25.9). There are important limitations in the behavior of populations implied by the logistic equation. Important among
these is that the expected condition of the population over time is an unchanging popUlation at the
carrying capacity (see Fig. 25.1).
This simple example illustrates several aspects
that are found in other, more elaborate developments of ecological modeling. The lexical phase
would involve the identification of the numbers of
individuals in the population as the system's parts
of interest and not considering sex ratios, individual
maturity, age structure of the population, etc. as
germaine to this particular modeling exercise. The
parsing phase would identify population growth
and limitations on growth as the phenomena of interest. The modeling phase would involve the specification of the r and K parameters in the models.
Compartment Models and Material Flow
Food chains and food webs initially emphasized the
energy-related aspects of feeding relations in ecological systems. The understanding of transfers of
energy in and out of plants and animals was (and
is) an important topic for physiologists and physiological ecologists. There was an important crossseeding of techniques from physiological researchers developed in the 1930s. These mathematical
techniques were intended to quantify the dynamics
of various substances moving through the vertebrate body (Shugart and O'Neill 1979). These
systems-oriented methods evolved from laboratory
ference in the birth rate constant and the death rate
constant (r = b - d). This implies that the rate of
population birth and death is a fixed proportion of
the number of individuals in the population. As
populations get large with respect to space or resources available to them, one would expect the
proportion of individuals dying to increase due to
such effects as stress, shortages of food or space,
etc. One might also expect a depression in the proportion of females in the population giving birth to
diminish for much the same reasons. These factors
are termed density-dependent factors in that they
are conditions that alter the rate of popUlation
growth as a function of the population density.
A Simple Population Model
A simple case of how the births and deaths in a
population might be effected by density of the
population would be to assume that the decrease in
birth rate is a linear function of population density.
A similar assumption could be made for the population death rate. The population growth as a function of time under these assumptions would be:
dN
dt = {(b - bddN) - (d + dddN)}N (25.6)
where b is the proportion of the population giving
birth, b dd is the density related change in the proportion of the population giving birth, d is the proportion of the population dying, and ddd is the density related change in the proportion of the
population dying.
One can rearrange the terms of this equation to
obtain:
dN
dt
{(b - d) - (bdd + ddd)N}N (25.7)
If the expression bdd + ddd is replaced by a new
expression so that:
Then the differential equation describing
change of the population becomes:
the
dN
dt
dN
dt
(r - i) Nor (25.8)
(25.9)
Herman H. Shugart
Population ecologists refer to this equation as the
logistic equation or the Verhulst-Pearl equation
(Fig. 25.1). The dynamics of the equation are considerably different from the equation for the exponential increase of a population (Equations 25.4 and
25.5). A population behaving as described by the
logistic equation will rise or fall to K over time (see
Fig. 25.1). K is called the carrying capacity and is
associated with concepts that are well ingrained in
the vocabulary of population biology and wildlife
management-so much so that it has taken to be a
tangible property of a given habitat or location. For
example, wildlife managers are involved with
knowing the carrying capacity of a given area with
regard to a deer herd or other managed animal
population. The carrying capacity concept applied
in wildlife biology is derived directly from the logistic equation (Equation 25.9). There are important limitations in the behavior of populations implied by the logistic equation. Important among
these is that the expected condition of the population over time is an unchanging popUlation at the
carrying capacity (see Fig. 25.1).
This simple example illustrates several aspects
that are found in other, more elaborate developments of ecological modeling. The lexical phase
would involve the identification of the numbers of
individuals in the population as the system's parts
of interest and not considering sex ratios, individual
maturity, age structure of the population, etc. as
germaine to this particular modeling exercise. The
parsing phase would identify population growth
and limitations on growth as the phenomena of interest. The modeling phase would involve the specification of the r and K parameters in the models.
Compartment Models and Material Flow
Food chains and food webs initially emphasized the
energy-related aspects of feeding relations in ecological systems. The understanding of transfers of
energy in and out of plants and animals was (and
is) an important topic for physiologists and physiological ecologists. There was an important crossseeding of techniques from physiological researchers developed in the 1930s. These mathematical
techniques were intended to quantify the dynamics
of various substances moving through the vertebrate body (Shugart and O'Neill 1979). These
systems-oriented methods evolved from laboratory