What is a Population
A collection of potentially interacting
organisms of one species within a
defined geographic area.
Figure 14.1
Estimates of Population Size
Recall the Lincoln index from recitation:
M/N = m/n
M is the total number marked;
m is the number marked in the sample;
n is the sample size.
Elementary Postulates
1. Every living organism has arisen from at
least one parent of the same kind.
2. In a finite space there is an upper limit
to the number of finite beings that can
occupy or utilize that space.
Populations grow by multiplication.
A population increases in proportion to its size,
in a manner analogous to a savings account
earning interest on principal:
at a 10% annual rate of increase:
a population of 100 adds 10 individuals in 1 year
a population of 1000 adds 100 individuals in 1
year
allowed to grow unchecked, a population
growing at a constant rate would rapidly
climb toward infinity
Geometric population growth
Usually think of animals increasing by
distinct generations.
N(1) = N(0) + B – D + I – E
N(1) = N(0)R
N(2) = N(0)RR
N(t) = N(0)Rt
Critical parameter is
 = net replacement rate (Ro).
N(t+1) = N(t) 
where:
N(t + 1) = number of individuals after 1 time unit
N(t) = initial population size
 = ratio of population at any time to that 1 time unit
earlier, such that λ = N(t + 1)/N(t)
To calculate the growth of a population over
many time intervals, we multiply the original
population size by the geometric growth rate
for the appropriate number of intervals t:
N(t) = N(0)  t
For a population growing at a geometric rate
of 50% per year ( = 1.50), an initial
population of N(0) = 100 would grow to N(10)
= N(0)  10 = 5,767 in 10 years.
Given No = 5000; N1 = 6000;
What is N2 ?
N2 = (6/5)2 x 5000 = 7200
Exponential Growth
Generations overlap, usually not discrete
generations.
For convenience, most of our models are
continuous.
Think about a complex model approximated by
may term in a potentially infinite series. Then
consider how many of these terms are needed for
the simplest acceptable model.
dN/dt = a + bN + cN2 + dN3 + ....
From parenthood postulate, N = 0 ==> dN/dt = 0,
therefore a = 0.
Simplest model ===> dN/dt = bN, (or rN, where r is
the intrinsic rate of increase.)
Solve equation: N = No e rt
Alternative form: dN/dt = bN - dN = (b-d)N
Rarely do b and d remain constant,
but if well below what environment can
support, then OK assumption.
Each species has optimum environment
with r = max
Figure 14.3
Figure 14.6
Human lice; r=.111/day
How fast will a population that starts at
100 lice increase?
(i.e., what is rate of increase of 100 lice?)
dN/dt = rN = .111 x 100 = 11.1 lice/day
Human population in 1993 = 5,600,000,000
b = 26/1000, d = 9/1000
How fast was the population growing?
dN/dt = rN = (.017)(5,600,000,000) =
95,200,000
(i.e., in excess of 1/3 US population per year)
Humans currently have b and d of 26 and 9
per 1000. How many years to double the
population?
N = No e rt = Nox2
2 = ert
ln2/.017 = 40.77 yrs
1700-1800 Human population from
600,000,000 ==> 900,000,000.
Calculate r.
r = ln (N/No) / t
= ln(9/6)/100
= .0040547
Logistic Growth
There has to be a limit. Postulate 2.
Therefore add a second parameter to equation.
dN/dt = rN + cN2
call c = -r/K
dN/dt = rN ((K-N)/K)
Nt = K/[1+((K-No)/No)e-rt]
Figure 14.18
Figure 14.16
Optimal yield problem.
dN/dt = rN - rN2/K
d2N/dt2 = r - 2rN/K
set = 0  N = K/2
If want maximum yield, should exercise
continual cropping around N = K/2
Figure 14.17
Data ??
Further Refinements of the Theory
Third term to equation?
More realism? Symmetry;
No reason why the curve has to be a
symmetric curve with maximal growth
at N = K/2.
What if the population is too small?
Is r still high under these conditions?
Need to find each other to mate
Need to keep up genetic diversity
Need for various social systems to work
Examples of small population problems
Whales,
Heath hens,
Bachmann's warbler
dN/dt = rN[(K-N)/K][(N-m)/N]
Instantaneous response is not realistic.
Need to introduce time lags into the system
dN/dt = rNt[(K-Nt-T)/K]
Three time lag types
Monotonic increase of decrease: 0 < rT < e-1
Oscillations damped: e-1 < rT <  /2
Limit cycle: rT > /2
Finite difference equations and Chaos
Nt+1 = aNt(1-Nt)
Models populations with discrete, nonoverlapping
generations, like many temperate zone insects.
if 1<a<3, population settles to a steady state.
if 3<a<3.57.., population settles into a stable cycle.
if 3.57..<a<4, population apparently random or chaotic.
if 4< a, N runs away to minus infinity.
This weird range of behaviors is generic to
most difference equations that describe a
population with a propensity to increase at
low values and to decrease at high values.
Similar behavior arises if there are many
discrete but overlapping generations.