458 Age-structured models Fish 458, Lecture 3

458 Why age-structured models? Advantages: Populations have age-structure! More realistic - many basic population processes (birth rate, death rate, growth, movement) are age-specific. Much of the data we collect are structured by age. Easily to build in actual processes and highly flexible.

458 Why age-structured models? Disadvantages: Increased complexity. The data needed to apply age-structured models are often not available. Some of the questions being addressed do not require information on age-structure. Age-lumped models often perform as well as age- structured models. Still not that realistic (no predation, competition, size and spatial structure)!

458 Some symbols

458 State variables, Forcing Functions and Parameters State variables: Numbers-at-age Fraction harvested Spawning biomass Forcing function: catch Parameters: Natural mortality, egg production-at-age, mass-at- age, vulnerability-at-age, survival-at-age, oldest age

458 The Basic Age-Structured Model Plus-group age

458 The Stock-Recruitment Relationship The function g determines the number of offspring (age 0) as a function of the egg production. Typical examples: Note that this model has no stochastic components, i.e. it is a deterministic model (sometimes called an “age-structured production model”).

458 Some Assumptions of this Model The fishing occurs at the start of the year. No immigration and emigration. Fecundity, natural mortality, mass and vulnerability don’t change over time. Vulnerability and mass don’t change with fishing pressure (i.e. no density-dependence in these parameters). Age x is chosen so that fecundity, natural mortality, mass and vulnerability are the same for all ages above age x.

458 Vulnerability, Selectivity and Availability Conventional definitions: Selectivity: The probability of catching an individual of a given age scaled to the maximum probability over all ages, given that all animals are available to be caught. Availability: The relative probability, as a function of age, of being in the area in which catching occurs. Vulnerability: The combination of selectivity and availability.

458 The Basic Model Again-I

458 The Basic Model Again-II (The steps in setting up a model) 1. Specify the initial (year y 1 ) age-structure. 2. Set y c =y Calculate the mortality (fishing and natural) during year y c. 4. Project ahead and hence compute the numbers-at- age for animals aged 1 and older at the start of year y c Compute the egg production at the start of year y c +1 and hence the number of 0-year-olds at the start of year y c Increase y c by 1 and go to step 3.

458 Building Age-Structured Models Be careful of timing. In the previous model: Spawning: start of the year Natural mortality: throughout the year Exploitation: start of the year Growth: instantaneous at the start of year These are not the only possible assumptions. Southern hemisphere krill – no growth in winter! The results may be sensitive to when population dynamic processes occur (especially if survival is low).

458 An Alternative Model (northern cod-like)

458 Assumptions of the alternative model The fishery occurs a fraction  after the start of the year. Vulnerability is age and time-dependent. Natural survival is independent of age. Only animals aged 2 and older are considered in the model. No stock-recruitment relationship, i.e. this is a stochastic model.

458 What about a population in equilibrium?? Equilibrium implies: Constant recruitment: Time-invariant exploitation rate: For the basic model therefore:

458 Calculating the plus-group

458 Building an age-structured model-I There are two fisheries with different vulnerabilities. One fishery operates from January-June and the other from July-December. Animals younger than 5 are discarded (dead) by fishery 1. Recruitment (age 0) is relate to egg production according to a stochastic Ricker stock-recruitment relationship. Survival is independent of age.

458 The Equations Note: This model implicitly ‘discards’ the catch of animals younger than 5 by not including then in the landed catch.

458 Readings Burgeman et al. (1994); Chapter 4 Haddon (2001); Chapter 2 Au and Smith (1997). Can. J. Fish. Aquat. Sci. 54: