Population Dynamics I:
Exponential Growth
• Assumptions
– All organisms have at least one parent
Population Dynamics I:
Exponential Growth
• Assumptions
– All organisms have at least one parent
– Population size increases through births and
immigration
Population Dynamics I:
Exponential Growth
• Assumptions
– All organisms have at least one parent
– Population size increases through births and
immigration
– Population size decreases through deaths and
emigration
Population Dynamics I:
Exponential Growth
• Assumptions
– All organisms have at least one parent
– Population size increases through births and
immigration
– Population size decreases through deaths and
emigration
– For simplicity’s sake, set immigration =
emigration = 0
Modeling Exponential Growth
Nt+1 = Nt + Births Deaths
Modeling Exponential Growth
Nt+1 = Nt + Births - Deaths
In words:
The population size N at at the next time sampled
(t+1) equals the population size now plus the births
in the population minus the deaths that occur in the
population during one time interval (1 unit of t)
Modeling Exponential Growth
Nt+1 = Nt + Births Deaths
How can we use this equation to estimate population size
N at any future time t from population size now (N0),
without resorting to recursion?
Modeling Exponential Growth
Nt+1 = Nt + Births - Deaths
First, subtract Nt from both sides of this equation:
Nt+1 - Nt = Nt - Nt + Births - Deaths
or
DN = Births - Deaths
Read: “the change in population size N = births - deaths”
Modeling Exponential Growth
DN = Births - Deaths
Assume that the population growth is continuous,
and let the time interval t to t+1 get very small...
dN/dt = Births - Deaths
Read: “the change in population numbers (dN) with
respect to changes in time (dt) equals births - deaths”
Modeling Exponential Growth
dN/dt = Births - Deaths
Assume that each individual produces the same number
of offspring in each time interval (we’ll violate this
assumption later), so total births B equals per capita
(each “head” or per person) births b times the number
of individuals in the population N: B = bN
Similarly, total deaths D equals per capita deaths d
times the number of individuals in the population N: D
= dN.
dN/dt = (b - d)N
Modeling Exponential Growth
dN/dt = (b - d)N
By convention, let r = (births-deaths). This parameter, r,
is commonly referred to as the Malthusian parameter,
the intrinsic rate of increase of a population, or the
instantaneous rate of increase. Then, our equation
becomes:
dN/dt = rN
Modeling Exponential Growth
dN/dt = rN
What was the question, again?
Modeling Exponential Growth
dN/dt = rN
How can we use this equation to estimate population size
N at any future time t from population size now (N0),
without resorting to recursion?
Modeling Exponential Growth
dN/dt = rN
How can we use this equation to estimate population size
N at any future time t from population size now (N0),
without resorting to recursion?
and then a miracle happened...
Modeling Exponential Growth
Nt = N0ert
How’d that happen? (calculus aficionados click here)
What’s this mean? (keep going)
Modeling Exponential Growth
Nt = N0ert
What’s this mean?
You can use this equation to predict population size at any
given time t in the future by multiplying the population size
now (at time t=0) times the exponential operator e raised to
the rt power (your calculator can do this for you).
how about some examples...?
Modeling Exponential Growth
Nt = N0ert
an example:
Can I do this myself…?
Modeling Exponential Growth
You can simulate exponential growth using
Microsoft Excel (or any other spreadsheet
software), or...
Modeling Exponential Growth
To get software for modeling population dynamics:
Download Populus free from the internet!
(it’s not Mac-compatible )
It’s also on the campus network, in the folder Populus
on the network drive //slab/biology
why should I care...?
Modeling Exponential Growth
• Many human populations are still growing
exponentially
Modeling Exponential Growth
• Many human populations are still growing
exponentially
• You can use this equation to estimate
population doubling time
Modeling Exponential Growth
• Many human populations are still growing
exponentially
• You can use this equation to estimate
population doubling time
• You can also use this equation to sensibly
choose a bank account
Modeling Exponential Growth
• Many human populations are still growing
exponentially
• You can use this equation to estimate
population doubling time
• You can also use this equation to sensibly
choose a bank account
• It will be on the first exam
Modeling Exponential Growth
Computing population doubling time
Recall: Nt = N0ert
For the population to double, Nt = 2N0
Modeling Exponential Growth
Computing population doubling time
Recall: Nt = N0ert
For the population to double, Nt = 2N0
So, 2N0 = N0ert , where t will be doubling time.
Modeling Exponential Growth
Computing population doubling time
Recall: Nt = N0ert
For the population to double, Nt = 2N0
So, 2N0 = N0ert , where t will be doubling time.
By division, 2 = ert , and taking logarithms of both
sides gives
ln(2) = rt, and by division again
ln(2)/r = time to doubling.
Modeling Exponential Growth
Computing population doubling time
Recall: Nt = N0ert
For the population to double, Nt = 2N0
So, 2N0 = N0ert , where t will be doubling time.
By division, 2 = ert , and taking logarithms of both sides
gives
ln(2) = rt, and by division again
ln(2)/r = time to doubling.
Note that doubling time is independent of
population size!
A brief assignment
• Determine the population sizes and the
approximate intrinsic rates of increase, r,
for your home town, your home state, and
your home country (note: many databases give r as
numbers per 1000, not per capita. What do you have to do
to correct for that?)
• Compute the estimated doubling time for
your home town, your home state, and your
home country.
Next:
violating assumptions and
approaching reality