Lecture 6
BSC 417
More models
• Logistic growth
• Overshoot and collapse
Population – a group of individuals of a single
species that occupies the same general area.
• Exponential growth model – the rate of expansion of
a population under ideal conditions
• Population-limiting factors – hunting, amount of
space suitable for breeding, restricted population
growth, food availability
• Logistic growth model – idealized population
growth slowed by limiting factors as the population
size increases
• Carrying capacity – the maximum population size
that an environment can support at a particular time
with no degradation to the habitat
Exponential growth of bacteria
Logistic growth and exponential growth
compared
Population Regulation:
A Multitude of Forces
Density-dependent growth
• The logistic equation (p 49)
The change in population size over time can be
written as:
dR(t)/dt = k(t) x R(t)
Where k(t) = unconstrained growth rate x (1-R(t)/cc)
Solution to the rate equation
• R(t) = cc/(1+Ae^-unconstrained growth rate x t)
• A = (cc-R0)/R0
• Steady state when?
Modeling density-dependent growth
• The Logistic Equation (cont.)
The next part,
Cc-R0
R0
can be thought of as the “braking term”, in that it causes
the growth rate to slow as population size increases –
making it dependent on density.
As the population size approaches CC…
actual growth rate slows down
stable equilibrium at R(t) = CC
Graphing Logistic Growth
CC
Logistic phase:
growing at a
decreasing rate
R(t)
● inflection point
Exponential phase:
growing at an
increasing rate
Time 
Modeling density-dependent growth
• Real Populations
Real populations do not always behave as smoothly
as our graph suggests.
Why not?
Examples of real populations:
Growth of a population of fur seals
Growth of Yeast Cells
Population of yeast cells grown under laboratory
conditions: R0 = 10, CC = 700, k = .54, Δt = 20 hours
US Population Prediction: Logistic
Logistic model prediction of the US
population for the period 1900 – 2050, with
initial data taken in 1900:
t0 = 1900; R0 = 76.2M; k = 0.017, CC = 661.9
Logistics Growth with Harvesting
Harvesting populations, removing members
from their environment, is a real-world
phenomenon.
Assumptions:
– Per unit time, each member of the population
has an equal chance of being harvested.
– In time period dt, expected number of
harvests is f*dt*P where f is a harvesting
intensity factor.
What does the logistic growth model suggest
about real populations in nature?
• A population’s growth rate will be small when the
population size is either small or large and highest
when the population is at an intermediate level
relative to the carrying capacity.
• Limiting factors make the birth rate decrease, the
death rate increase or both
• Eventually the population will stabilize at the carrying
capacity when the birth rate equals the death rate
• These are mathematical models and no population
fits either perfectly
Some factors that limit population growth
• As density of song sparrows
increase, the number of
eggs laid decreases because
of food shortages
• Plants grown under
crowded conditions tend to
be smaller and less likely to
survive
• Disease transmission or
accumulation of toxic waste
products can increase
mortality
Continued……
• A predator may capture
more of a particular kind of
prey as the prey becomes
abundant
• White-footed mice stop
reproducing at a colony size
of 30-40 even when food
and shelter are provided.
Stress?
• The graph shows aphids
which feed on the phloem
sap of plants increase in
population in the summer
and then die-off in the fall
and winter
Continued….
• Some populations remain
fairly stable in size close to
carrying capacity
• Most populations fluctuate
as seen at the left
• This graph shows song
sparrow populations, with
periodic catastrophic
reductions due to severe
winter weather
Boom and bust cycles
• Hare cycles may be caused
by increasing food
shortages during winter
caused by overgrazing
• They may be due to
predator-prey interactions
• Cycles could be affected by
a combination of food
resource limitation and
excessive predation
• Predators reproduce more
slowly than their prey so
they always lag behind prey
in population growth.
Exponential growth of the human
population
• Throughout human history
parents had many children
but only two on average
survived to adulthood
• Estimates that by 2025 the
world will have to double
food production, 2/3 of the
available fresh water on
earth will be in use, 60,000
plant species will be lost to
support the population
• Issues: overgrazing, rivers
running dry, decrease in
groundwater, energy?
Human carrying capacity estimates
• Ecological footprint with
multiple constraints such as
food, fuel, water, housing,
and waste disposal used.
• Calculates current demand
on resources by each
country in hectares of land
per person
• World ecological capacity is
1.7 ha per person alive in
1997
How to achieve population stability?
• Zero population growth –
when birth rates equal
death rates
• Two ways to reach ZPG.
High birth and death rates
or low birth and death
rates.
• Demographic transition is
moving from the first to the
second. Most developed
countries have made the
transition
• See the demographic
transition in Mexico at the
left.
Collapse of northern cod fishery
• Renewable resource
management – harvesting
crops without damaging the
resource
• Maximum sustainable yield
– harvest at a level that
produces a consistent yield
without forcing a
population into decline
• Can be just as tricky to
reduce population sizes
Overshoot and collapse
• Nonrenewable resource and a population
that depends on it
• Population dynamics linked to resource
consumption
• Resource base affects death rate
Key features
• Uses a coupled set of rate equations: one for
resource consumption, one for population
• In the beginning, when R(t)≈ R(t=0), birth rate
is maximum and exponential growth occurs
• R(t) always decreases at a rate proportional to
the size of P(t)
• Both reservoirs need to reach a steady state
for the overall system to be in steady state
– Achieved as t∞
Examples: famous “overshoot and
collapse” theories
• Peak oil
– Decline in extraction rate reflecting the end of
“easy oil”
• “Limits to growth”, “Carrying capacity”
• Malthusian demography
Peak oil
More examples:
• The boy who cried wolf
• Stress
• Others: