Continuous logistic model Source: Mangel M (2006) The theoretical ecologist's toolbox, Cambridge University Press, Cambridge This equation is quite different to the discrete model! 1 Logistic.xlsx – Deterministic continuous

How to decide between discrete and continuous models Are the processes discrete or continuous? In individual-based models discrete processes are almost always appropriate For most computer software discrete models are easier Differential equations provide analytic solutions for simple problems Tradition

The modeling toolbox for conservation of single populations 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 (total biomass, total numbers)

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

A differential equation version Solution: 2 Exponential model.xslx: deterministic and continuous

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

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

Binomial shortcut Binomial distribution: probability p and N trials, with mean pN and variance p(1-p)N Births: probability b, N t trials, expected mean bN t, variance b(1-b)N t Deaths: probability m, N t trials, expected mean mN t, variance m(1-m)N t 2 Exponential indiv binomial.r

Normal approximation With large number of trials (individuals) the binomial approaches the normal distribution Randomly sample deviates from a normal distribution with mean bN t and standard deviation sqrt(b(1-b)N t ) mean + X×SD where X is a random normal deviate In Excel norminv(rand(),0,1) produces random normal deviates with mean 0 and SD 1 Small numbers (<30) this approximation does not work well, then use binomial draws (R is good at this) Repeat for deaths with b replaced by m 2 Exponential model.xlsx: Normal approx to binomial

Why the normal approximation? We do not have to calculate the probability of each individual living or giving birth This is very helpful with populations in the thousands or millions

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

Only demographic stochasticity was included in previous models We often allow for a more general model of stochasticity: i.e. w t 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 w t

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

Adding lognormal error to the exponential model

Normal and lognormal error

Non-age-structured models Exponential growth Logistic (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:

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

Rate of increase declines as density approaches K

Rate of increase approaches 0 near K

“Compensation” At high densities there will be a shortage of food, refuge from predators or some critical requirement Birth rates may decline, or mortality rates will increase This is called compensation and can be quantified as the difference between the rate of increase when resources are abundant and a rate of increase of 0.

“Depensation” Rates of increase may decline at low densities This is known as depensation (or the Allee effect) It makes local or total extinction much more likely Will be discussed later in course

Number of spawners Offspring produced Expected curve Depensation Visualizing rates of change with compensation and depensation Liermann M & Hilborn R (1997) Depensation in fish stocks: a hierarchic Bayesian meta-analysis. CJFAS 54: Myers RA et al. (1995) Population dynamics of exploited fish stocks at low population levels. Science 269:

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

Logistic vs. Fox model Note: Fox model will have lower SP for the same r value 2 Non-age models.xlsx: Fox vs logistic

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

Pella-Tomlinson model These plots all have m = MSY = 500 Becomes the Fox model as n  1 Becomes the logistic model when n = 2 2 Non-age models.xlsx: Pella-Tomlinson

Advantages and disadvantages Logistic, Fox, Pella-Tomlinson offer different hypotheses about at what biomass level MSY would be obtained The Pella-Tomlinson is more flexible but has more parameters

Things to do later in course What happens 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 This is called serial autocorrelation Results in regime shifts Does environment drive surplus production more than biomass?