1. Lecture 4: Logistic growth equation
Comments on problem set
Sigmoidal growth curve
“Logistic Model” equation
Population dynamics
Management applications

2. Logistic growth: growth with limits
Because growth is typically slowed relative to exponential by density-dependent factors, the logistic model better mimics most pop’n growth
Calculate population change with logistic growth using: dN/dt = rNt(1-(N/K))
K: the carrying capacity of the local environment
Growth is diminished due to competition, which is more apparent as N approaches K
For continuous: Nt = K/ (1+[(K-N0)/N0]e-rt)

3. In lab, Paramecium show logistic growth

4. logistic growth equation
Q2. For r = 0.667, N = 300, K = 400. By how much will the population change in a time step (= what is the population-level growth rate)?

5. Q1. Let’s try using the logistic growth equation
For r = .667, N = 300, K = 400. By how much will the population change in a time step (= what is the population-level growth rate)?
dN/dt = * 300 * (1-(300/400))
= 200*(1-(3/4))
= 200 * ¼
= 50 = dN/dt
So, the population is expected to grow by 50 individuals in the next time step when population size is 300 (so will be 350).

6. Q3. How does population growth change across population size
Q3. How does population growth change across population size? At what point will you have greatest population-level growth? Why?
Calculate & plot values for r = 0.2, K = 100,000 and: N = 10, N = 100, N = K/2, N = 75,000, N = K, N = 150,000. Why is the population growth low at low N? Why is low (or negative) at high N?
Population-level growth rate, dN/dt
Population size, N
0 K/2 K 3K/2
Multiply out: dN/dt = rN – rN2/K
Take derivative, respect to N and set = 0: 0 = r - 2rN/K
Factor: 0 = r * (1-2N/K)
If r = 0, no growth at max; so 1-2N/K = 0 must be the max,
solve for N: 1 = 2N/K; K/2 = N = max

7. Q2. How population growth changes across N
Q2. How population growth changes across N? At what point will you have greatest population-level growth? Why?
r = 0.2, K = 100,000;
N = 10, 100, K/2, 75,000, K, 150,000
Why is the population growth low at low N? Why is low (or negative) at high N? N
dN/dt 10 2
100 20
50000
5000
75000
3750
100000
150000
-15000
K/ K K/2
Multiply out: dN/dt = rN – rN^2/K
Take derivative, respect to N and set = 0: 0 = r - 2rN/K
Factor: 0 = r * (1-2N/K)
If r = 0, no growth at max; so 1-2N/K = 0 must be the max,
solve for N: 1 = 2N/K; K/2 = N = max

8. Usually growth of natural populations is messier than model curves (though usually still generally fits w/ logistic model)
Populations fluctuate
Overshoot & Die offs (predicted by the logistic model)
Variation around K due to Temp
Fig 10.4, Cain et al. 2011, Ecology, Sinauer

9. K is assumed to be constant but birth and death rates vary over time.
High population size may exhaust resources. Q3. Where do you think K was for reindeer on St. Paul Island? Why?

10. Birth and death rates vary as abiotic conditions change, so carrying capacity fluctuates. What does this mean to model predictions? To management?
From Cain et al. 2011
Ecology Fig 10.5
Sinauer
𝑁 = 𝐾 − 𝜎 𝐾 2 2
Birth rates and death rates are not exactly constant over time. There is a range. This range in birth and death rates results in a range for carrying capacity that varies over time rather than a specific, fixed number.
or if cyclic variation in K:
𝐾 𝑡 = 𝐾 + amplitude of cycle*
[cos(2πt/cycle length)]

11. Population fluctuations can also be caused by predator–prey dynamics. E.g., Lynx and Hares

12. E.g. Wolves & Moose

13. Life History Remember, b-d = intrinsic growth rates (r).
Therefore need to understand generally life strategies regarding birth and death.
These are developed over evolutionary time scales.

14. Life History
Type Type 1

15. Management options for populations varies based on their life histories (reproductive & longevity strategies).
Survivorship curves are plots of the number of individuals from a hypothetical cohort that will survive to reach different ages.
Type I: Most individuals survive to old age (Whales, Dall sheep).
Type II: The chance of surviving remains constant throughout the lifetime (some birds, squirrels).
Type III: High death rates for young, those that reach adulthood survive well (species that produce a lot of offspring).

16. Q4. Type survivorship curve? What does this mean for management?
Crouse et al and Crowder et al estimated how population growth for loggerhead sea turtles might change given various management practices.
Early conservation efforts focused on egg and hatchling stages. However, there’s high mortality for early stages (eggs, nestlings, 1-yr olds).
Q4. Type survivorship curve? What does this mean for management?
Example: Loggerhead sea turtles are threatened by development on nesting sites and commercial fishing nets.
Early efforts focused on egg and hatchling stages.
What kind of survivorship curve do you think these turtles might have? (Type 3) High egg/nestling/1 year old mortality. Really important not to lose the animals that make it to adulthood, especially as it takes years for turtles to reach reproductive age.

17. Even if hatchling survival were increased to 100%, loggerhead populations would continue to decline.
Population growth rate was most responsive to decreasing mortality of older juveniles and adults. Prompted laws to add turtle-escape hatches to shrimp nets. These decreased net-caused mortality 44%.
Because of these studies, Turtle Excluder Devices (TEDs) were required to be installed in shrimp nets.
The number of turtles killed in nets declined by about 44% after TED regulations were implemented.
It will be decades before we know whether TED regulations help turtle populations to increase in size.

18. Management Applications

19. Tragedy of the Commons
Garrett Hardin’s classic theory of depletion of common pool resources: the tendency of a shared, limited resource to become depleted because people act from self interest
Common grazing area
What’s best for each farmer in the short term? long term?

20. Tragedy of the Commons
Tragedy of the commons is explained best using game theory
One player compromises – one has high yield, other low yield
Both players compromise – everyone has moderate yield
Neither compromise – high yield, then resource crash to v. low yield
What’s best for each farmer in long term?
What might keep them honest?

21. Overfishing and the collapse of the Northern Cod (Atlantic Cod)
Cod collapse often used as an example of MSY gone wrong, but also bad management of common pool resources (CPR)

22. Maximum Sustainable Yield (MSY)
Maximum sustainable yield: greatest harvest of a renewable resource that does not compromise the future availability of that resource. (pp 264-5)
Why is this concept useful?
How do you determine the level at which to harvest?
obtain the most of a resource; manage wildlife to avoid overpopulation,

23. Maximum Sustainable Yield (MSY)
Assumption: population growth is fastest at K/2
Theory: Use the logistic growth curve as the basis for a harvesting plan. To keep the population sustainable, try to maintain it at K/2.

24. Maximum Sustainable Yield
H = rate of harvest
The system is at equilibrium when the number of individuals removed is same as growth rate.
For almost all harvest rates, there can be two pop sizes yielding the same growth rates, far from vs. close to carrying capacity
a. not many individuals are there to reproduce; b. intense competition for resources

25. Maximum Sustainable Yield Problems
Predicting the carrying capacity and the maximum growth rate in natural populations is difficult. These vary across time due to natural fluctuations.
If calculated wrong, harvest often happens at the H3 level (see previous slide) rather than the H2 level.
Harvest usually occurs at all size and age ranges – but each of these can drastically affect current and future populations

26. Q5. Maximum sustainable yield revisited
What should you do to manage for fluctuation?
What does fluctuation mean for your estimate of Maximum sustainable yield?