1. Quiz

2. Depensation and low density dynamics

3. References
Liermann M & Hilborn R (1997) Depensation in fish stocks: a hierarchic Bayesian meta-analysis. Canadian Journal of Fisheries and Aquatic Sciences 54:
Liermann M & Hilborn R (2001) Depensation: evidence, models and implications. Fish and Fisheries 2:33-58
Myers RA, Barrowman NJ, Hutchings JA & Rosenberg AA (1995) Population dynamics of exploited fish stocks at low population levels. Science 269:

4. Definitions
Compensatory: rate of increase declines at higher densities
Depensatory: rate of increase declines at lower densities

5. Why low density
This is where the conservation concern is!

6. What happens at low densities
Competition for resources should be at a minimum BUT
Reduced fertilization due to difficulty in finding mates
All births one gender
Random events with small numbers
Reduced survival due to predation
Genetic inbreeding depression
Lack of benefits from grouping behavior

7. The probability of finding a mate
Mate finding
The probability of finding a mate
Assume that the probability a female will encounter any individual male is p and there are N males in the population.
Expected average number of males encountered is λ = pN
The total number of encounters with males is Poisson distributed
The probability that no males will be encountered will be the 0 class of the Poisson distribution with mean = λ.

8. Poisson: probability of 0 class
Mate finding
Poisson: probability of 0 class

9. Mate finding
Proportion not mated and mated e.g. p = probability of finding a male
12 Depensation and extinction I.xlsx

10. (the population size at which 50% are mated)
Mate finding
Reparameterizing to N50
(the population size at which 50% are mated)

11. Mate finding
N50 10 20 50
100
200 p
0.0693
0.0347
0.0139
0.0069
0.0035
12 Depensation and extinction I.xlsx

12. Single-gender
All births one gender
If there are N total births in a population and the proportion of females is p = 0.5
What is the probability that zero births are female, or all births are female?
This is a binomial distribution, probability of 0 females out of N births OR N females out of N births

13. Single-gender cohorts
12 Depensation and extinction I.xlsx

14. Random events with small numbers
Randomness
Random events with small numbers
Population in equilibrium, births = deaths
Each time step for each individual, probability b of producing one offspring, probability d of dying, probability 1 – b – d of neither dying nor reproducing
Set b = d, simulate for 100 years with different starting population sizes
How often does the population go extinct?
This is an individual-based model, called a “random walk” model

15. R code: Random walk model.r
Randomness
R code: Random walk model.r

16. Randomness
5 individual population simulations (starting population size = 20, b = d = 0.2, 100 years)
12 Depensation and extinction I.xlsx
12 Random walk.r

17. Probability of extinction before year 100
Randomness
Probability of extinction before year 100
(100 simulations, 100 years, 100 individuals = 1 million calculations, 50 seconds)
12 Depensation and extinction I.xlsx
12 Random walk.r

18. Probability of extinction before year 100
Randomness
Probability of extinction before year 100
(1000 simulations, 100 years, 100 individuals = 10 million calculations, 15 minutes)
12 Depensation and extinction I.xlsx
12 Random walk.r

19. Probability of extinction before year 100
Randomness
Probability of extinction before year 100
(10000 simulations, 100 years, 100 individuals = 100 million calculations, 1 hr 23 min)
12 Depensation and extinction I.xlsx
12 Random walk.r

20. Predation at low densities
Predator might be very efficient
Predator population unaffected by prey abundance (i.e. this prey species is not the major part of the diet
Then predator functional response can lead to depensatory mortality (higher mortality per capita at low densities)

21. Predation

22. Functional response a = 300, b = 400 a = 300 b = 400 a = 300 b
Predation
a = 300, b = 400
Functional response
a = 300
Max prey eaten
b = 400
N when prey eaten
is at half of max
a = 300 b
12 Depensation and extinction I.xlsx

23. Depensation due to predation
Depensation: predation
Depensation due to predation
Add predation to stock-recruit relation (salmon)
R = Rstock-recruit – number eaten
Functional response leads to depensation (higher mortality per capita at low densities)
Rate of increase = recruits divided by spawners (salmon)
12 Depensation and extinction I.xlsx

24. Equilibrium and critical depensation
Depensation: predation
Equilibrium and critical depensation
Unfished equilibria
Critical depensation
12 Depensation and extinction I.xlsx

25. Depensation due to mating success
Depensation: mating
Depensation due to mating success
Replace spawners S in stock-recruit with pmated × S
Proportion mated
Number of spawners at which 50% successfully mate
A Beverton-Holt curve
12 Depensation and extinction I.xlsx

26. Low densities (summary)
Increased risk of extinction
All births one gender
Random events
Predation
Difficult to find mates
Other (inbreeding, lost group benefits, etc.)
The net effect is depensation: lower rate of increase at low densities

27. Ransom Myers 1952-2007 Myers analysis
Myers RA, Barrowman NJ, Hutchings JA & Rosenberg AA (1995) Population dynamics of exploited fish stocks at low population levels. Science 269:

28. Detecting depensation
Myers analysis
Detecting depensation
Model 1: δ = 1 (find MLE, likelihood L1)
Model 2: δ free (find MLE, likelihood L2)
Nested model
Likelihood ratio test: R = 2ln(L2/L1) is a chi-square distribution with degrees of freedom 1

29. Compare model 1 and model 2
Myers analysis
delta=1
delta free
alpha
0.87
0.07 K
3330.0
7116.9
delta 1
1.78
sigma
1.01
0.66
NLL
32.79
23.12
nparams 3 4
Likelihood ratio
19.35
Degrees of freedom
Chi-squared prob
1.1E-05
Compare model 1 and model 2
12 Depensation Myers.xlsx

30. Myers results Explored 128 data sets
Myers analysis
Myers results
Explored 128 data sets
Only 3 significant cases of depensation
Fewer than expected by chance
Of these data sets about 27 had high power

31. Problems with Myers method
Myers analysis
Problems with Myers method
Parameterization has no biological interpretation except δ > 1 implies depensation
Used p values to test for significant depensation, ignores biological significance
Confounding of environmental change (regime shifts) with depensation