Age-structured models Fish 458, Lecture 3
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.
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)!
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
The Basic Age-Structured Model Plus-group age
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”).
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.
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.
The Basic Model Again-II(The steps in setting up a model) • Specify the initial (year y1) age-structure. • Set yc=y1. • Calculate the mortality (fishing and natural) during year yc. • Project ahead and hence compute the numbers-at-age for animals aged 1 and older at the start of year yc+1. • Compute the egg production at the start of year yc+1 and hence the number of 0-year-olds at the start of year yc+1. • Increase yc by 1 and go to step 3.
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).
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.
What about a population in equilibrium?? • Equilibrium implies: • Constant recruitment: • Time-invariant exploitation rate: • For the basic model therefore:
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.
The Equations Note: This model implicitly ‘discards’ the catch of animals younger than 5 by not including then in the landed catch.
Readings • Burgeman et al. (1994); Chapter 4 • Haddon (2001); Chapter 2 • Au and Smith (1997). Can. J. Fish. Aquat. Sci. 54: 415-420.