This is a powerpoint on a biological topic of population and logisitics growth. This lecture was given to me in one of my classes and is my teacher's material.

1. Lecture Outline
• Density Independent and Density Dependent
growth and the carrying capacity
• The differential logistic equation for continuous
growth and Maximum Sustained Yield
• The difference equation for logistic continuous
growth.
• The logistic equation for discrete growth
• r and extinction

2. Organization & Equations
Logistic
Continuous Discrete
Differential Difference Difference
dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)

3. Density Independence,
page 140 in McConnell
• Factors that effect b and d
do not depend on the
population size.
Density Dependence
• Factors that effect b and
d do depend on
population size.

4. Recruitment = larvae leave the plankton
and attach to solid substrate.
• Which is density dependent?

5. As N increases, r will decrease to zero.
This is density dependence.
r
N K

6. Organization & Equations
Logistic
Continuous Discrete
Differential Difference Difference
dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)

7. Logistic growth is density dependent growth.
• dN/dt = rN (1-N/K)

8. How does (1-N/K) work?
• Population 1: K = 100 individuals and N = 5
• Population 2: K = 1500 and N = 1000

9. What does the curve look like for continuous
populations growing under density dependence?

10. • What is the value of K?
• What does the upper curve tell us?

11. Maximum Sustained Yield…
• 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?

12. Ed Ricketts, John Steinbeck,
and Cannery Row
“Ricketts was accepting because he just
listened and tended to turn whatever people
said into something that sounded brilliant.”
Ricketts was Steinbeck’s friend and
philosophical/intellectual father.
1930s = 500,000 tons of sardines/year
1952 = gone

13. 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)

14. Organization & Equations
Logistic
Continuous Discrete
Differential Difference Difference
dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)

15. 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]

16. An overview of the procedure.
• Create a phase-plane plot with Nt+1 on the Y-axis and Nt
on the X-axis
• Calculate the function line using the difference equation.
• Draw the equilibrium line where Nt+1 = Nt
• Use “cob webbing” to determine the stability properties of a
population…more in a minute.

17. What is a phase plane plot?
0
100
200
300
400
500
600
0 100 200 300 400 500 600
N t
Nt+1
Function
line
Equilibrium line

18. How do we
calculate data for
the function line?
Use the difference equation . Example: K = 500; r = 0.5; N0 = 10.
• Nt = K/[1 + [(K – N0)/N0]e-rt]
N1 = K/[1 + [(K – N0)/N0]e-rt] = 500/[1+[(500-10)/10] e-0.5*1 ]= 16
N2 = K/[1 + [(K – N0)/N0]e-rt] = 500/[1+[(500-10)/10] e-0.5*2 ] = 27
• N3 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 42
• N4 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 66
• N5 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 100
• N6 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 145
• N7 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 202
• N8 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 264
• N9 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 324
• N10 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 372
• N11 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 417
• N12 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 446
• N13 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 466
• N14 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 479
• N15 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 487
• N16 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 492
• N17 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 495
• N18 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 497
• N19 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 498
• N20 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 499
• N21 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 499
• N22 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 500
0
100
200
300
400
500
600
0 5 10 15 20 25
Time
Nt

19. How do we plot the function line in phase
plane?
0
100
200
300
400
500
600
0 100 200 300 400 500 600
Nt
Nt+1
Function
line
Equilibrium line

20. What is cobwebbing?
0
100
200
300
400
500
600
0 100 200 300 400 500 600
N t
Nt+1

21. Stability properties?
• What is an equilibrium point?

22. General Concept of Equilibrium
• It 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.
• Non-equilibrium forces disrupt a stable balance.

23. The Allee Effect:
A special case of 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).
– r is negative below a critical N (Allee density) and then
becomes positive as N increases past the critical N.

24. What are the stability properties of the
Allee effect?
• How many equilibrium points with an Allee effect?
• Which are stable which are unstable?

25. Organization & Equations
Logistic
Continuous Discrete
Differential Difference Difference
dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)

26. How to solve for Nt+1
when a population has discrete growth
• What happens as r increases in the
continuous equation?
• What happens as r increases in the discrete
equation?

27. A population with discrete reproduction
• A population of mayflies started with 2 individuals, with a
carry capacity of 100 and an r = 0.2. The population
reproduces once each year. What is the population size at
Nt+1.
• Nt+1 = Nt + R0 Nt (1 – Nt/K):
• Nt+1 = 2+(.2)(2) [1- (2/100)]
• Nt+1 = 2+.4 [1-.02]
• Nt+1 = 2+(.4)(.98)
– the population is increasing at 98% of the exponential rate
• Nt+1 = 2.392
• Now what?

28. Your turn.
• A population with discrete reproduction has a
growth rate of 3.0. They reproduce once per year,
K = 1500 and Nt = 20.
• What is the population size after two generations?
Nt+1 = Nt + R0 Nt (1 – Nt/K)

29. 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 N begins to apply a brake to R0 when N
overshoots K.

30. Kiabab deer herd in northern Arizona
??
Present
Wolves
exterminated

31. What is the relationship between r and the
stability of a discrete population?
Pg 130 in McConnell

32. Dampening, stable limit, complex
stable limit, chaos
Time
R0 < 2.0
2.0 < R0 < 2.5
2.5 < R0 < 2.57 R0 > 2.57

33. 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.

34. 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

35. Large r and Chaos
• Chaos = a non-repeating, drastic fluctuation in
population size with a simple deterministic
model…shocking.
Lord Robert May
1936 – present
Oxford University
Attractor = K
Bifurcation 4 8 Chaos

36. General Meaning
• Deterministic, meant that the future behavior was fully determined by
the initial conditions, with no random elements involved.
• “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 (2007)

37. Sibly et al. 2007. Ecology Letters 10:1-7
• 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.
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