Biol 201 2
nd
Summer Session Review for 2
nd
Exam
Population Growth
Exponential growth equation: N
t
= N
0
e
rt
calculates population size
•
Or as a differential equation
dn/dt = rN (expresses the rate of population change as the
product of r and N)
•
N
0
= number of individuals at time 0
•
N
t
= number of individuals at time > 0
•
r = intrinsic rate of natural increase (per capita rate of increase)
•
e = base of natural logs (2.7183)
•
t = number of time intervals in hours, days, years, etc.
Populations grow slower than the exponential model predicts
but WHY?
•
All populations live in the real world
•
The environment is not infinite
•
Resources are FINITE!
•
Basically, there are constraints from the environment!!
Logistic Equation of population growth (Sigmoidal Growth
)
gives the rate of population change
as a function of r
m
, n and k
dn/dt = rN (1 (N/K))
as the ration of N/K increases, population growth slows!!
•
r = per capita rate of increase
•
N = population size
•
K = carrying capacity of the environment for that species (i.e. the maximum # that can be
supported)
•
The maximum rate of increase, r, occurs at a very low population size
•
r decreases as N increases
•
if N < K
, r is positive and the population grows
•
if N = K
, r = 0 and population growth stops
•
if N > K
, r is negative and the population declines
Liebig’s Law of the Minimum:
the resource in shortest supply relative to demand is
the limiting one!!
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Assumptions of the Logistic Model
•
K is a constant
•
r is a constant
•
all individuals are identical
•
there are no time lags
K is a constant?
is the environment truly constant? (day
night, seasonality, disturbances)
its
NEVER constant
r is a constant?
well r is a function K, and since K changes so must r
all individuals identical?
NO!! newborns are not identical to adolescents and adolescents aren’t equal
to adults, etc.
no time lags?
maturation and reproduction take time
BUT,
we still use the logistic model b/c it works!! (populations increase rapidly when they are small and
grow slowly when they’re crowded
Rate of Increase (r) vs. Size
from viruses to large animals, intrinsic rate of increase, r, declines
predictably with increasing size
Two populations are still growing exponentially!!
humans and the AIDS virus (HIV)
Metapopulations:
the idea that populations of a species in an area are actually subdivided due in part
because separate patches of the preferred habitat are separated by less suitable habitat the populations
have some degree of gene flow (migration) between them (they are connected but not strongly)
Spreading of Risk (Den Boer):
metapopulations are more stable than populations which are not
subdivided each subpopulation is both a source and sink for the other subpopulations so risks are spread
across the subpopulations
they’re connected but the loss of one doesn’t endanger the others
Exponential growth
occurs when population isn’t crowded
Logistic growth
occurs when population is crowded
Logistic growth suggests a regulated population!
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 Summer '07
 MITCHELL
 Ecology, Population Ecology

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