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

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

Models with no age structure 3

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 13:419-496 4

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

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

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

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

2 Exponential indiv based.r

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

2 Exponential indiv binomial.r

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

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

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

Exponential model with lognormal error

Lognormal error correction mean = 0.003 mean = 1.646 mean = 1.0005 2 Lognormal error hist.r 16

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

“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

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

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)

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

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

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

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

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

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