Delay-difference models

Readings Ecological Detective, p. 244–246 Hilborn and Walters Chapter 9

References Deriso RB (1980) Harvesting strategies and parameter estimation for an age-structured model. CJFAS 37: Schnute J (1985) A general theory for analysis of catch and effort data. CJFAS 42: Schnute J (1987) A general fishery model for a size- structured fish population. CJFAS 44: Fournier DA & Doonan IJ (1987) A length-based stock assessment method utilizing a generalized delay- difference model. CJFAS 44:

What is a delay-difference model? A model with two components Written as difference equations With a time-delay

Simple numbers model (difference but not delay) Constant recruitment Proportional recruitment Density-dependent recruitment Survival Recruitment function

Logistic delay-model (numbers) Additional survival, S=1 unless there is harvest New births

Delay difference.r

Years Abundance Logistic model Delay-logistic model There is a lag in the initial increase There is a lag in density-dependence too, overshoots K When rL > 1 there will be stable limit cycles At higher r values there will be damped oscillations Delay-logistic 22 Delay difference.r r = 0.05, L = 10 r = 0.2, L = 5 r = 0.3, L = 5

Plus a delay of L years in recruitment Survival Constant recruitment Recruitment function Proportional recruitment Density-dependent recruitment

Going from numbers to biomass The big difference is that individuals grow as they get older, so that even if there is no recruitment, the biomass next year will not decline by 1–S Delay-difference models include the LRSG model and the Deriso-Schnute model

The LRSG model Lagged recruitment, survival and growth Survival in year t Constant recruitment Recruitment function Proportional recruitment Density-dependent recruitment Somatic growth Biomass of individuals above the age at recruitment (L years old)

Biological meaning of g g is the proportional weight gain from one year to the next, i.e. g = 1.1 implies a 10% gain in body weight per year In other words the implied relationship is:

Deriving the LRSG model part I Total survival in year t Catches Natural survival Harvest rate Biomass above age L in year t Weight at age a Numbers at age a in year t Surviving numbers (not including recruitment)

Key assumptions The “biomass” is not total biomass, but “recruited” biomass (i.e. only the fishable part of the population above age L) The S t term combines natural survival, s, and survival from harvest 1–u t Vulnerability to harvest is assumed to be 1 for animals older than age L and 0 for animals below this age

Deriving the LRSG model part II Numbers equation Convert numbers to weight Key assumption: annual weight change Sum over all ages above L Note: B is biomass above age L, R is new biomass Substitute assumption into equation Summation change

Deriving the LRSG model part III Since both S t and g are multiplicative constants, combine them into a single new constant Substitute in and rearrange Note that S’ now represents combined survival and growth

Basic modeling approaches: two distinct modeling camps Camp 1: Total numbers or total biomass, describing the population with a single state variable (logistic, Fox, etc.) Camp 2: Age-structured (size-structured, stage- structured) models that track complex structure and time lags in recruitment

The Deriso-Schnute model Provides an elegant link between the two camps Showed that a simple delay-difference set of equations could be used to exactly duplicate a full age-structured model (with some restrictions) First derivation: Rick Deriso (1980) University of Washington biomathematics student Later generalizations by Jon Schnute Deriso RB (1980) Harvesting strategies and parameter estimation for an age-structured model. CJFAS 37: Richard (Rick) Deriso Jon Schnute

Extending the LRSG model The fundamental assumption of the LRSG model is that growth can be modeled as follows: For g > 1 this model implies that mass increases exponentially with age But mass-at-age for most species actually exhibits asymptotic behavior

Common growth model for fish Combine (A) and (B) (A) von Bertalanffy length(B) Length-weight relation (β ≈ 3)

von Bertalanffy weight model V-B length model plus length-weight relation V-B weight model

von Bertalanffy weight model

The slope is constant! Since the slope is constant, the ratio between successive weights at age is equal The “Brody” growth coefficient

Deriving the Deriso-Schnute model I From the previous slide, weight for the next age depends on the weight of the two previous ages and the Brody growth coefficient ρ

To get biomass, multiply numbers by weight (still following a single cohort) Deriving the Deriso-Schnute model II

Next we replace numbers at t+1 and a+1 by the numbers in the previous age and time steps, multiplying by survival (combined from natural mortality and fishing mortality) in the corresponding years Deriving the Deriso-Schnute model III

Now move from a single cohort to summing across all cohorts to get total population biomass Deriving the Deriso-Schnute model IV

Digression I

Now substitute what we learnt from Digression I into the left hand side. Note that the B t also replaces the summation on the right, using the definition of B t as summing biomass from a = L to infinity. Deriving the Deriso-Schnute model V

Digression II

Substitute Digression II into the rightmost term Deriving the Deriso-Schnute model VI

The full Deriso-Schnute model Rearrange and multiply out for the final expression

More on the Deriso-Schnute model GrowthRecruitmentSurvival of biomass Unfished equilibrium biomass

Extensions Allowing for partial recruitment Allowing harvesting to be continuous or to occur instantaneously in the middle of the year Using a variety of functions to predict recruitment Delay-difference models based on size can also be derived (Fournier & Doonan 1987) Fournier DA & Doonan IJ (1987) A length-based stock assessment method utilizing a generalized delay-difference model. CJFAS 44:

Overview I The delay-difference model provides an elegant link between age-aggregated (e.g. logistic) models and more complicated (age-, size- and stage-structured) models Rests on some key simplifying assumptions – Mass-at-age follows a von Bertalanffy growth curve – Recruitment and maturity occur at the same age – Vulnerability is the same for ages greater than age-at- maturity – Natural mortality is the same for ages greater than age-at- maturity

Overview II Some of the assumptions are highly restrictive, particularly: – Vulnerability independent of age – Age-at-recruitment equals age-at-maturity – Only one fishery

Schnute extensions Schnute (1985, 1987) showed that the Deriso model and the logistic were special cases of the full age-structured model If you can specify ρ, age at maturity, and natural mortality, then you only estimate the two stock-recruitment parameters, and the model is not more complex than the two- parameter logistic model Schnute J (1985) A general theory for analysis of catch and effort data. CJFAS 42: Schnute J (1987) A general fishery model for a size-structured fish population. CJFAS 44: Jon Schnute

Current use Not used much today because we have the computing resources to implement full age- structured models Spatially structured models for MPA analysis Keeping track of age structure at hundreds of sites too computationally intense Easier to find “tricks” to solve for equilibrium conditions