Chapter 12 Lecture 3 Flashcards
how do populations often grow
exponentially
for population growth, birth rates + immigration must be > than
the death rates+ emigration
what serves to regulate population growth
any factor that slows down the inputs relative to the outputs
factors that slow down the inputs relative to the outputs
predation, competition for scarce resources, herbivory, parasitism, disease, severe winter, drought
Raymond Pearl and L.J. Reed “On the Rate of Growth of the Population of the United States since 1790 and its Mathematical Representation”
Pop. growth in the U.S. declined - if the decline followed a regular pattern, it should be able to be described with a mathematical formula, and future population growth could be predicted
what did pearl and reed also reason
the rate of exponential
growth in the population would be related to population
size rather than time alone (since any time scale for any
population is arbitrary).
what did pearl and reed suggest
rather than using a
constant value for r which really represents unrestrained
population growth —- r should decrease as N increases
rather than using a
constant value for r which really represents unrestrained
population growth —- r should decrease as N increases according to the following relation:
dn/dt = rN r = r0(1 - N/K)
r0
intrinsic rate of growth when its size is close to 0
K
carrying capacity of the environment, the max number of individuals that the environment can sustain indefinitely
r=
r0(1 - N/K)
or
r0 - r0N/K
r = r0 - r0N/K
defines a straight line with slope -r0/K
y
dependent
x
independent
m
slope
b
y intercept
N
independent variable
r
dependent variable
slope
-r0/K
logistic equation (equation for restrained growth
dN/dt = r0N(1 - N/K)
what shape is the logistic curve
hump-shaped
what is the rate of growth of the logistic curve at two population densities
(dN/dt) = 0
when N = 0 and when K = N
when does the peak of the hump occur
N = K/2 (1/2 to carrying capacity)
Carrying capacity (K):
the maximum population size that can be supported by the environment.
Logistic growth model
a growth model that describes slowing growth of populations at high densities
what is the logistic growth model represented by
dN/dt = rN(1-N/K)
S-shaped curve
the shape of the curve when a population is graphed over time using the logistic growth model
Inflection point
the point on a sigmoidal growth curve at which the population has its highest growth rate
early growth is
exponential
inflection point
the point of the fastest growth after which growth begins to slow
As the population increases from a very small size, what happens to the rate of increase?
it grows until reaching 1/2 the carrying capacity (corresponding to the inflection point)
Rate of per capita increase can be modeled as:
(1/N)(dN/dt)
Individuals in the population continually decline in
their ability to contribute to population growth
where does maximum growth occur
N = K/2
as long as population size N doesn’t exceed the carrying capacity (N/K < 1), the population
continues to increase
when the value of N exceeds the value of K, the ratio N/K
exceeds 1 and population growth declines
the carrying capacity of a population is the eventual theoretical equilibrium size of
a population growing according to the logistic equation
When examining population growth over a period of time (the time course of population growth)…
a sigmoid or S-shaped curve is formed on the X axis over time
