## Delay Difference Models

Delay-difference models provide intermediate detail between (age-aggregated) production models and age-structured models. They are similar to production models, but allow a lag between birth and maturity. This lag accounts for one effect of age structure on reproduction, at least implicitly.

In simplest form, delay-difference models predict the number of adults in the next time period (Nt + based on current abundance, survival, and recruitment:

where ,t is the proportion of adults that survives year t, T N is a function describing the relationship between numbers of adults and recruits (see section titled Recruitment), and

T is the lag from birth to maturity. Survival rate it can reflect fishing, if one assumes that the exploited part of the population is the same as the mature part.

Equation [42] is based on numbers of fish, but the framework can also be based on biomass (B) to allow for effects of individual growth. Assuming von Bertalanffy growth, Deriso derived a delay-difference model in biomass:

B,+1 — i,B, + i, (B, - i, -1B, - 1)e - k + F b (B, - T+1)

where k is from the VBGF and TB is a function describing the relationship between biomass of adults and biomass of recruits. Equation [43] states that new biomass is the sum of surviving biomass, growth, and biomass of recruits.

Data requirements of delay-difference models are more extensive than those of production models. Besides time series of catches and relative abundance or biomass, additional information is needed on growth, natural mortality, and the spawner-recruit relationship. Many generalizations of delay-difference models have been proposed, for example, to allow for individual growth patterns other than the VBGF.

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