Fall 2009 IB Workshop Series
sponsored by IB academic advisors
IB Opportunities in
C-U
Tuesday, Sept. 15
4:00-5:00pm 135 Burrill
There are many local opportunities for volunteering/internships!
Representatives in fields from medicine to ecology will speak
with students!
Ch 11: Population Growth + Regulation
dN/dt = rN
dN/dt = rN(K-N)/K
For Thursday lecture,
Bring lecture outline;
Complete Problem Set
2-1 and 2-2 (see pg. 75).
Bring pg. 73 too!
Objectives
• Population Dynamics
• Growth in unlimited environment
•
Geometric growth
Nt+1 =  Nt
•
Exponential growth Nt+1 = Ntert
•
dN/dt = rN
•
Model assumptions
• Growth in limiting environment
•
Logistic growth dN/dt = rN (K - N)/ K
•
D-D birth and death rates
•
Model assumptions
1.***Draw a graph of population
growth constrasting:
1) Growth with unlimited resources
2) Growth with limited resources
Label axes.
Indicate carrying capacity (K).
3) Add equations representing both
types of growth:
A) exponential
B) logistic
Population growth predicted by the
exponential (J) vs. logistic (S) model.
Population growth can be mimicked
by simple mathematical models of
demography.
• Population growth (# ind/unit time) =
recruitment - losses
• Recruitment = Births + Immigrants
• Losses = Deaths + Emigrants
• Growth (g) = (B + I) - (D + E)
• Growth (g) = (B - D) (in practice)
Two models of population growth with
unlimited resources :
• Geometric growth:
• Individuals added at
one time of year
(seasonal reproduction)
• Uses difference equations
• Exponential growth:
• individuals added to population continuously
(overlapping generations)
• Uses differential equations
• Both assume no age-specific birth/death rates
Difference model for geometric growth
with finite amount of time
• ∆N/ ∆t = rate of ∆ = (bN - dN) = gN,
• where b = finite rate of birth or
per capita birth rate/unit of time
• g = b-d, gN = finite rate of growth
Projection model of geometric growth
(to predict future (or past) population
size)
• Nt+1 = Nt + gNt
•
=(1 + g)Nt Let  (lambda) =(1 + g),
then
• Nt+1 =  Nt
•  = Nt+1/Nt
•  = finite rate of increase, /unit time
Geometric growth over many time intervals:
• N1 =  N0
• N2 =  N1 = ·  · N0
• N3 =  N2 = ·  ·  · N0
• Nt =  t N0
• Populations grow by multiplication rather
than addition (like compounding interest)
• So if know  and N0, can find Nt.
Example of geometric growth (Nt = t N0)
•
•
•
•
•
Let  =1.12 (12% per unit time) N0 = 100
N1 = (1.12) 100
112
N2 = (1.12 x 1.12) 100
125
N3 = (1.12 x 1.12 x 1.12) 100
140
N4 = (1.12 x 1.12 x 1.12 x 1.12) 100
157
2. Problem 2.1 on Pg 75 - due Thursday!
A moth species breeds in late summer and leaves
only eggs to survive the winter. The adult dies
after laying eggs. One local population of the
moth increased from 5000 to 6000 in one year.
1. Does this species have overlapping
generations? Explain.
2. What is  for this population? Show formula with
numbers; don’t solve.
3. Predict the population size after 3 yrs. Show
formula with numbers; don’t solve.
4. What is one assumption you make in predicting
the future population size?
If know Nt N0 and ,
can calculate t (in future and
past)
•
•
•
•
•
Nt =  t N0
t = Nt/ N0
 = (Nt/ N0)1/t
t ln  = ln (Nt/ N0)
t = [ln (Nt/ N0)]/ ln 
3. Invasive insect:
Geometric population growth:
Calculate:
A. lambda
B. t when n = 1
Geometric growth (dots) and
exponential growth (lines) on regular and
log scales (when  >1 and r >0).
Values of , r, and Ro indicate whether
population is: ***
Ro < 1
Ro =1
Ro >1
Differential equation model of
exponential growth:
dN / dt
=
rate of
change
in
=
population
size
r
contribution
of each
individual X
to population
growth
N
number
of
individuals
in the
population
4. Calculate daily increase
in population.
dN / dt = r N
• r = difference between birth (b) and
death (d)
• Instantaneous rate of birth and death
• r = (b - d) so r is analogous to g, but
instantaneous rates
• rates averaged over individuals (i.e. per
capita rates)
• r =intrinsic rate of increase
5. Calculate r.
E.g.: exponential population growth
r = .247
Exponential growth: Nt = N0ert
r>0
r=0
r<0
• Continuously accelerating curve of increase
• Slope varies directly with population size.
• (N) (gets steeper as size increases).
6. Calculate
A. r
B. population size in 6 mo.
Environmental conditions and species
influence r, the intrinsic rate of increase.
Population growth rate depends on the
value of r; r is environmental- and
species-specific.
Value of r is unique to each set of
environmental conditions that influenced birth
and death rates…
•…but have some general expectations of pattern:
•
•
•
High rmax for organisms in disturbed habitats
Low rmax for organisms in more stable habitats
Rates of population growth are
directly related to body size.
• Population growth:
• increases inversely with mean
generation time.
• Mean generation time:
• Increases directly with body size.
Assumptions of the model
• 1. Population changes as proportion of current
population size (∆ per capita)
•
∆ x # individuals -->∆ in population;
• 2. Constant rate of ∆; constant birth and death
rates
• 3. No resource limits
• 4. All individuals are the same (no age or size
structure)
Objectives
• Growth in unlimited environment
•
Geometric growth
Nt+1 =  Nt
•
Exponential growth Nt+1 = Ntert
•
dN/dt = rN
•
Model assumptions
• Growth in limiting environment
•
Logistic growth dN/dt = rN (K - N)/ K
•
D-D birth and death rates
•
Model assumptions
Populations have the potential to increase
rapidly…
until balanced by extrinsic factors.
Population growth rate =
Intrinsic
growth
rate at
N close
to 0
X
Population
Reduction in
size
X growth rate
due to crowding
Population growth predicted by the
***
model.
K = ***
On which island(s) is there evidence of
limited resources affecting population
growth ?
Assumptions of the exponential model
• 1. No resource limits
• 2. Population changes as proportion of current
population size (∆ per capita)
•
∆ x # individuals -->∆ in population;
• 3. Constant rate of ∆; constant birth and death
rates
• 4. All individuals are the same (no age or size
structure)
1,2,3 are violated ***
Population growth rates become lower
as population size increases.
• Assumption of constant birth and death rates is violated.
• Birth and/or death rates must change as pop. size changes.
Population equilibrium is reached when b = d
Those rates can change with density.
Density-dependent factors: + or -?
.
Habitat quality affects reproductive variables
affected (How is K affected?)
Population size is regulated by densitydependent factors affecting birth and/or death
rates.
1)
Density-dependence in plants first
decreases growth.
2. Size hierarchy develops.
skewed
3) Density-dependence
thirdly decreases
survival. Intraspecific
competition causes
“self-thinning”.
r (intrinsic rate of increase)
decreases as a function of N
.• Population growth is negatively densitydependent.
.
rm
slope = rm/K
r
r0
N
K
Positive density-dependence
Allee effect: greater pop size increases
chance of recruitment.
2) (+) density-dependence secondly increases
some components of reproduction;
decreases others…
Logistic equation
• Describes a population that experiences
negative density-dependence.
• Population size stabilizes at K = carrying capacity
• dN/dt = rmN(K-N)/K
• where rm = maximum rate of increase w/o
resource limitation
= ‘intrinsic rate of increase’
K = carrying capacity
•
(K-N)K = environmental break (resistance)
= proportion of unused resources
Logistic (sigmoid) growth occurs when the
population reaches a resource limit.
• Inflection point at K/2 separates accelerating
and decelerating phases of population
growth; point of most rapid
growth
Logistic curve incorporates influences of
decreasing per capita growth rate and
increasing population size.
Specific
Assumptions of logistic model:
• Population growth is proportional to the
remaining resources (linear response)
• All individuals can be represented by an
average (no change in age structure)
• Continuous resource renewal (constant E)
• Instantaneous responses to crowding
.
***
• K and r are specific to particular organisms
in a particular environment.
7. Complete 2-2 (see pg. 75)
due Thursday.