1. Common conservation and management models
Biomass-dynamics models (logistic, Schaefer, Fox, Pella-Tomlinson)
Generation-to-generation models (Ricker, Beverton- Holt)
Delay-difference models (Deriso-Schnute)
Size- and stage-structured models
Age-structured models

2. Extensions to all Stochastic or deterministic
Adding environmental impacts
Extending to multiple species

3. Models with no age structure3

4. Non-age-structured models
Exponential growth
Logistic model (or “Schaefer” model)
Fox model
Pella-Tomlinson model
Schaefer MB (1954) Some aspects of the dynamics of the population important to the management of
the commercial marine fisheries. Inter-American Tropical Tuna Commission Bulletin 1:25-56
Fox WW (1970) An exponential surplus-yield model for optimizing exploited fish populations. Trans. American Fisheries Society 99:80-88
Pella JJ & Tomlinson PK (1969) A generalized stock production model. Inter-American Tropical Tuna Commission Bulletin 13: 4

5. Discrete exponential model
Numbers next year = numbers this year + births – deaths + immigrants – emigrants
2 Exponential model.xlsx: deterministic discrete 5

6. A differential equation version
Numbers
Solution:
Year
2 Exponential model.xslx: deterministic continuous 6

7. Model assumptions
Population will grow or decline exponentially for an indefinite period
Births and deaths are independent
There is no impact of age structure or sex ratio
Birth and death rates are constant in time
There is no environmental variability 7

8. Individual-based exponential model For every year For every individual
Pick random numbers X1~U[0,1], and X2~U[0,1] (uniform between 0 and 1)
If X1 < b then a new birth
If X2 < m then individual dies
2 Exponential indiv based.r 8

9. 2 Exponential indiv based.r

10. Individual-based model
binomial shortcut
Binomial distribution: how many successes out of N trials given probability p
Binomial has mean pN and variance p(1-p)N
If we have Nt individuals in year t, and probability b of a birth for each, this is a binomial distribution
Similarly for deaths, with probability m
2 Exponential indiv binomial.r 10

11. 2 Exponential indiv binomial.r

12. Types of stochasticity
Phenotypic: not all individuals are alike
Demographic: random births and deaths
Environmental: some years are better than others, El Nino, hurricanes, deep freeze etc.
Spatial: not all places are alike 12

13. Only demographic stochasticity was included in previous models
We often allow for a more general model of stochasticity:
i.e. wt is normally distributed with a mean zero and standard deviation s
numbers next year depend upon numbers this year, the parameters p, any forcing function u (such as harvesting) and random environmental conditions wt 13

14. Lognormal error: a little deeper
If wt is normally distributed with mean zero, then exp(wt) is lognormally distributed
When wt = 0, exp(wt) = 1, “average year”
When wt > 0, exp(wt) > 1, “good year”
When wt < 0, exp(wt) < 1, “bad year”
Since exp(wt) is not symmetric, the expected value is not 1
Therefore we use a correction factor: 14

15. Exponential model with lognormal error

16. Lognormal error correction
mean = 0.003
mean = 1.646
mean =
2 Lognormal error hist.r 16

17. Peak catch occurs when B = 0.5K
Logistic model
Peak catch occurs when B = 0.5K
Surplus production
Catch
Biomass at
time t+1
Carrying capacity
Intrinsic (maximum)
rate of increase

18. “Compensation”
At high densities there will be increasing competition for food, predator refuges or other critical requirements
Birth rates may decline, mortality rates may increase, or both
This is called compensation or density- dependence and is measured by the difference in population growth rate when resources are scarce vs. abundant 18

19. Logistic model Surplus production Biomass (fraction of K)
Surplus production and maximum sustainable yield (MSY)
Surplus production
Biomass (fraction of K)
2 Non-age models.xlsx: Fox vs logistic

20. Logistic model SP/biomass Biomass (fraction of carrying capacity)
Rate of population increase is surplus production divided by biomass
SP/biomass
Biomass (fraction of carrying capacity)

21. Peak catch occurs when B = 0.37K
Fox model
Peak catch occurs when B = 0.37K
Surplus production
Catch
Biomass at
time t+1
Intrinsic (maximum)
rate of increase
Carrying capacity 21

22. Logistic vs. Fox model Surplus production Biomass (fraction of K)
Note: Fox model will have lower SP for the same r value
Surplus production
Biomass (fraction of K)
2 Non-age models.xlsx: Fox vs logistic

23. Pella-Tomlinson model
Peak catch occurs anywhere from B = 0 to B = K, depending on n
Catch
Biomass at
time t+1
Surplus production
Determines biomass
that yields MSY
Carrying capacity
MSY: maximum
sustainable yield 23

24. Pella-Tomlinson model
These plots all have m = MSY = 500
Becomes the Fox model as n  1
Becomes the logistic model when n = 2
Surplus production
Biomass (fraction of K)
2 Non-age models.xlsx: Pella-Tomlinson

25. Advantages and disadvantages
Logistic, Fox, Pella-Tomlinson offer different hypotheses about the biomass level at which MSY would be obtained
The Pella-Tomlinson is more flexible but has three parameters instead of two
The most important and widely used is the logistic (know the assumptions and weaknesses) 25

26. Later in the course
Serial autocorrelation: What if environmental conditions are not independent in time, but tend to come in runs of good and bad years more often than would be expected by chance
Results in regime shifts
Does environment drive surplus production more than biomass? 26