458 Delay-difference models Fish 458, Lecture 14

458 Delay-difference models Provide an intermediate option between age- aggregated and full age-structured models. Are based on some key simplifying assumptions that allow age-structured dynamics to be simplified to a single equation involving total biomass or total numbers only. Do not require much information. Are useful when a “realistic” model is needed that doesn’t require substantial data / computer memory.

458 A delay difference model for the total number of animals in a population Note: s is natural survival if there is no exploitation or survival from all causes of mortality but then it is necessary to assume that the exploitable part of the population is the same as the mature part.

458 Some typical “recruitment” functions

458 Moving from numbers to biomass-I The previous delay-difference model ignored growth because it dealt only with numbers. We will now extend the framework to include: Growth Time-dependent mortality (due to, for example, harvesting).

458 Moving from numbers to biomass-II The simplest way to model changes in (mature) biomass is to assume that “recruitment” is in units of biomass and to take account of growth. g is the proportional change in mass from one year to the next, i.e. s t is the survival rate during year t from all causes. This model is the lagged recruitment, survival and growth (LRSG) model. Lets derive it from first principles.

458 Deriving the LRSG model - I is the biomass of animals aged L and older at the start of year t: The survival rate during year t is the product of survival from natural causes and from exploitation (note that vulnerability is assumed to be 1 for all animals older than age L and zero below this age): Don’t forget the standard age-structured dynamics equation:

458 Deriving the LRSG Model-II

458 Extending the biomass-based delay- difference model-I One of the fundamental assumptions of the LRSG model is its growth model. However, for g>1 this model implies that mass increases exponentially with age. In contrast, mass-at-age for most species exhibits asymptotic behavior.

458 Starting with an Asymptotic Growth Curve The von Bertalanffy growth curve is probably the most common used in fisheries. Assuming that mass-at-age (rather than length-at-age) follows a von Bertalanffy growth curve, mass- at-age is governed by the recursive equation: where  is the Brody growth coefficient.

458 Deriving the full Deriso-Schnute model

458 More on the Deriso-Schnute model Notes: “Survival” of biomass Growth Recruitment - mass-at-recruitment The original “Deriso” version corresponds to The virgin (no fishing) equilibrium is given by:

458 Extensions Allowing for “partial recruitment”. Harvesting can be continuous or occur instantaneously in the middle of the year. A variety of functions are available to predict recruitment. Delay-difference models based on size can also be derived.

458 Overview-I Provides an elegant link between the 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 Bertlanffy growth curve. Recruitment and maturity occur at the same time. Vulnerability is independent of age above the age- at-maturity. Natural mortality is independent of age above the age-at-maturity.

458 Overview-II Some of the assumptions are highly restrictive, particularly: Vulnerability independent of age; Age-at-recruitment equals age-at-maturity; Only one fishery. Not used much today because we have the computing resources to implement full age-structured models.

458 Comparison of Dynamic Schaefer, age- structured and Deriso-Schnute models (Cape hake) Age-structured model M assumed to be 0.4yr -1. Mass-at-age age-specific. Logistic vulnerability - inflection point at age 3. Maturity at age 4. Deriso-Schnute model M assumed to be 0.4yr -1. Estimate  from the mass-at-age data. Knife-edged vulnerability / maturity at age 3. Parameters: B 0, steepness (r and K), q and . Fitted to CPUE and survey data Stock assumed to be at B 0 in 1917.

458 Model Selection The three models are not nested but the data and likelihood functions are identical. We can compare these models using AIC (and AIC c )

458 Model Selection Schafer model: -LnL = Delay-difference model: -LnL = Age-structured model: -LnL =

458 Readings Hilborn and Walters (1992), Chapter 9 Deriso, R.B. (1980). Can. J. Fish. Aquat. Sci. 37: Schnute, J. (1985). Can. J. Fish. Aquat. Sci. 42: Schnute, J. (1987). Can. J. Fish. Aquat. Sci.. 44: Fournier, D.A., Doonan, I.J. (1987). Can. J. Fish. Aquat. Sci. 44: