General Ecology BIOL 3700 – Lecture Notes
Lecture 6 Parts II & III
Lecture 6 Part II. Exponential Growth
Outline
Part II. Logistic Growth
 Density independence vs. dependence, and carrying capacity
 The differential logistic equation for continuous growth and Maximum
Sustained Yield
 The difference equation for logistic continuous growth
Lecture organization & equations
Logistic
Continuous
Differential
dN/dt = rN (1–N/K)
Discrete
Difference
Nt = K/[1+ [(K – N0)/N0] e
-(rt)]
Difference
Nt+1 = Nt + R0 Nt (1 – Nt/K)
What were the assumptions of exponential growth?
• No I or E
• Resources are unlimited.
• Neither is realistic.
Density independence
• Factors that influence b and d that do not depend on the population size (i.e.
the severity of these processes does not fluctuate with pop. size).
Examples:
– Disturbances (floods, fires, hurricanes) can increase the death rate
independent from population size.
– Extreme temperatures can cause the birth rate to decline
independently of population size.
– Disease can increase the death rate both independently of population
size and dependent upon population size.
Density dependence
• In contrast, here… rates of reproduction and mortality are determined by
(depend on) population size. Factors whose severity depend on pop. size.
Examples:
– Crowding and competition for resources with other species or
individuals of the same species may increase the death rate and
decrease the birth rate as density increases.
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General Ecology BIOL 3700 – Lecture Notes
Lecture 6 Parts II & III
– Reproductive success may be slow as individuals struggle to find a
mate, then increase, then decrease as population size increases…
– Infection rates may increase with an increase in population size.*
*…but “herd immunity”
Assumption:
Ascidians example
• “Recruitment”
Definition: larvae leave the plankton and attach to solid substrate.
• Which is density dependent?
More:
• r does not vary in exponential growth… per capita births minus per capita
deaths.
• r does not vary with N in the exponential model. r is a constant.
• This is density independence. [Assume b & d are constant.]
• Graph:
•
•
2
If As N increases, r will decrease to zero under logistic growth.
This is density dependence.
Graph:
General Ecology BIOL 3700 – Lecture Notes
Lecture 6 Parts II & III
Intraspecific competition is often the density dependent “brake”
What is K?
What is K in terms of b and d? What do each of these mean?
How does r vary?
K is a stable point.
Lecture organization & equations
Logistic
Continuous
Differential
dN/dt = rN (1–N/K)
Difference
Nt = K/[1+ [(K – N0)/N0] e -(rt)]
Discrete
Difference
Nt+1 = Nt + R0 Nt (1 – Nt/K)
Logistic growth is density dependent growth
dN/dt = rN (1-N/K)
r will decrease as population density approaches K.
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General Ecology BIOL 3700 – Lecture Notes
Lecture 6 Parts II & III
How does (1-N/K) work?
What does the curve look like for continuous populations growing under density
dependence?
K
N
Time
K: a stable equilibrium point
•
•
•
What is the value of K?
What does the upper curve tell us?
At population size produces the maximum growth rate?
Maximum Sustained Yield (MSY)
• Fisheries managers tried to harvest enough fish to maintain the maximum
growth rate or maximum yield = K/2.
• Do you think this worked? Why or Why not?
Ed Ricketts, John Steinbeck, and Cannery Row
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General Ecology BIOL 3700 – Lecture Notes
Lecture 6 Parts II & III
Practice
• Suppose a population of ravens is growing according to the continuous logistic
equation. If K=200 and r = 0.1 individuals/(individual · year-1), what is the
maximum growth rate of this population over a year?
dN/dt = rN (1-N/K)
Lecture organization & equations
Logistic
Continuous
Differential
dN/dt = rN (1–N/K)
Discrete
Difference
Nt = K/[1+ [(K – N0)/N0] e
-(rt)]
Difference
Nt+1 = Nt + R0 Nt (1 – Nt/K)
What if you want to predict the pop. size at some point in time?
• The integrated form of the differential logistic equation for continuous
reproduction = the difference equation.
• Nt = K/[1 + [(K – N0)/N0] e-rt]
Now you try… with ewoks
• If r = 2.5, K = 1000, and N0 = 100, what will the ewok population size be in 10
years?
• Nt = K/[1 + [(K – N0)/N0] e-(rt)]
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General Ecology BIOL 3700 – Lecture Notes
Lecture 6 Parts II & III
What effect does increasing or decreasing r have on time to reach K?
Stability properties
The phase-plane plot
• There is a simple way to view this stable equilibrium point of continuous
logistic growth…
• We plot Nt+1 versus Nt through time… phase plane plots
• This is called the function line in a stability analysis.
• Time is the third dimension coming out of the board… it is implicit.
Overview of the procedure
1. Create a phase-plane plot with Nt+1 on the Y-axis and Nt on the X-axis.
2. Calculate the function line using the difference equation.
3. Draw the equilibrium line where Nt+1 = Nt.
4. Use “cobwebbing” to determine the stability properties of a population… more
in a minute.
What phase-plane plots look like:
600
500
Function line
Nt+1
400
300
200
Equilibrium line
100
0
0
100
200
300
400
500
600
Nt
How do we plot the function line in phase plane?
Now, create a phase plane diagram by plotting Nt+1 by Nt.
Draw in the equilibrium line where Nt+1 = Nt. Then plot the data from the above
logistic growth curve. For example, the first point will be (10, 16), the second (16,
27), the third (27, 42), etc.
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General Ecology BIOL 3700 – Lecture Notes
Lecture 6 Parts II & III
Function lines all look the same when we use the same equation (e.g. logistic growth
equation), except some may be steeper.
What is cob webbing?
It’s how we determine the stability properties.
Now, we can use cobwebbing to see what kind of equilibrium point is represented in
the graph… stable or unstable.
If we start at some point, say Nt = 100, we first draw a straight vertical line up to the
function line, a straight horizontal line to the equilibrium line, and this is repeated
until you reach an equilibrium point.
600
500
Nt+1
400
300
200
100
0
0
100
200
300
400
500
600
Nt
What do we mean by “stability properties?”
•
What is an equilibrium point?
The Allee effect: a case of positive density dependence
Some species require a minimum number of individuals or they spiral to
extinction.
Some species need a certain group size to find a mate (whales) or to hunt
(wolves), care for young (elephants), reproduce (mayflies), and avoid predators
(flocking birds, e.g. Passenger Pigeon?) more efficiently in groups.
What are the stability properties of the Allee effect?
• How many equilibrium points with an Allee effect?
• Which are stable which are unstable?
General concept: equilibrium
• Equilibrium is a balance between opposing forces…births – deaths.
• Non-equilibrium: conditions are constantly changing. A stable balance cannot
be achieved.
• Equilibrium forces produce a stable balance.
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General Ecology BIOL 3700 – Lecture Notes
•
Lecture 6 Parts II & III
Non-equilibrium forces disrupt a stable balance.
************* The End of Lecture 6. Part II. *************
Lecture 6 Part III. Effect of Time Lags and Increasing r (Chaos)
Outline
• The logistic equation for discrete growth
• Time lags and increasing r
• Chaos theory
• r and extinction
Lecture organization & equations
Logistic
Continuous
Differential
dN/dt = rN (1–N/K)
Discrete
Difference
Nt = K/[1+ [(K – N0)/N0] e -(rt)]
Difference
Nt+1 = Nt + R0 Nt (1 – Nt/K)
A population with discrete reproduction
• A population of mayflies started with 2 individuals, with a carry capacity of
100 and an R0 = 0.2. The population reproduces once each year. What is the
population size at Nt+1?
•
Nt+1 = Nt + R0 Nt (1 – Nt/K):
Increasing r versus R0
• What happens as r increases in the continuous equation?
• What happens as R0 increases in the discrete equation (populations)?
Discrete growth has a built in time lag
• In the continuous growth model we assume that per capita growth changes
instantaneously when an individual is born or dies.
• Discrete populations do not adjust instantaneously. There is a time lag before
NDD begins to apply a brake to R0 when N overshoots K.
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General Ecology BIOL 3700 – Lecture Notes
Lecture 6 Parts II & III
Kaibab deer herd in northern Arizona
 What happened?
 …workers/managers exterminated wolves in 1906 (1906-1924)
• http://www.youtube.com/watch?feature=player_embedded&v=AiGh6YRRR_s
• http://www.youtube.com/watch?feature=player_embedded&v=3Ys89FOUIW
k#!
Wolves
exterminated
??
Present
What is the relationship between R0 and the stability of a discrete population?
• R0 < 2.0 the population dampens to K
• 2.0 < R0 < 2.5 it forms stable cycles
• 2.5 < R0 < 2.57 forms complex stable cycles
• R0 > 2.57 and it causes the population to cycle chaotically.
How do you know the stability characteristics?
• The outcome of cobwebbing is determined by the slope of the function line
where it crosses the equilibrium line.
Now, we have 4 types of population “stability”…
1. Smooth increase to K… constancy over time.
2. Dampened oscillations… constancy with minor variation around K.
3. Stable cycles… continual variation around K over time.
4. Chaos… this is not stable!
Chaos theory: Lorenz discovered chaos using simple climate models
• Chaos theory is used to understand a variety of phenomena…the break point
between smooth laminar flow, stable cycles, and chaos.
• Over the short term chaos is the same as random. Over the long term,
patterns can arise and cycle around strange attractors.
• Butterfly wings and strange attractor for climate models.
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General Ecology BIOL 3700 – Lecture Notes
Lecture 6 Parts II & III
Chaos and the butterfly effect: initial conditions can matter a lot
• The different wings represent the effects of small changes in the initial
conditions of the model, which increase exponentially cause a chain of events
leading to large-scale changes in phenomena (e.g. population growth &
stability properties). However, sometimes differences in initial conditions do
not have a great effect on the trajectory of the system.
Large r and chaos
• Chaos = a non-repeating, drastic fluctuation in
population size with a simple deterministic model…
shocking.
General meaning of chaos…
• Deterministic meant that the future behavior was
fully determined by the initial conditions, with no random
Lord Robert May
(1936 – present)
elements involved.
Oxford University
• “The phenomenon of deterministic chaos… very
simple and purely deterministic laws or equations can give rise to dynamic
behavior that not merely looks like random noise, but is so sensitive to initial
conditions that long-term prediction is effectively impossible. This ended the
Newtonian dream that if the system is simple (very few variables) and orderly
(the rules and parameters are exactly known) then the future is predictable.”
Robert May (2007), Theoretical Ecology
r, R0, and Rmax summary
• r = instantaneous rate of change
• R0 = discrete growth factor
• Rmax = maximum growth rate
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General Ecology BIOL 3700 – Lecture Notes
r and extinction
What is the relationship between r or R0 and extinction?
Species with intermediate r ranges have been
“naturally selected”
Closed circles = mammals
Open cirlce = insects
Plus signs = birds
Open squares = bony fish
A total of 634 populations
The return rates are an approximation of r
• Many return rates are ≈1.
• Between 0.5 & 2 is considered stable.
• <0.5 and disturbance rates may exceed
return rates resulting in extinction. >2.0
can result in chaotic behavior and thus,
extinction.
Sibly et al. 2007,
Ecology Letters 10:1-7
How does NDD cause a population to crash?
The End of Logistic Growth
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Lecture 6 Parts II & III
General Ecology BIOL 3700 – Lecture Notes
•
Remember to continue to bring your calculators.
************* The End of Lecture 6. Part III. *************
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Lecture 6 Parts II & III