1. 458 Stage- / Size-Structured Models Fish 458, Lecture 16

2. 458 Stage-Structured Models Why not always use age-structured models: Ageing is difficult (or impossible) for several types of organisms. Division of the population into “stages” may be more natural than into “ages” (the most common type of “stage” is size-class). Animals in a stage share “demographically similar” characteristics. The data available to fit the model may be “stage- based” (e.g. the fraction of new-borns, juveniles, mature animals).

3. 458 Stage-Structured Models (Examples of Stage-structured Populations) Trees: seed, in the understory, in the canopy Trees may spend decades in any of these stages and the age of a tree may have little to do with moving from one stage to another. Insects: eggs, larvae, pupae, adults. Seabirds: new borns, fledglings, juveniles, adults giving birth this year, adults resting between giving birth.

4. 458 Modeling Stage-Structure-I (back to age-based models)

5. 458 Modeling Stage-Structure-II (differences from age-based models) The age-based model can be written: We now generalize this by: defining each row in N as the abundance of a stage (rather than an age-class); allowing recruitment to occur to any stage (though usually recruitment only occurs to the first stage); and allowing animals to move between any stages.

6. 458 Modeling Stage-Structure-III (example: Loggerhead turtles) 1 – first years; 2 – small juveniles; 3 – large juveniles; 4-subadults; 5-novice breeders; 6 – first-year remigrants; 7-mature breeders

7. 458 Stage-Structured Models (Advantages and Disadvantages) Advantages: Highly flexible: Some fisheries models keep track of age and whether an animal is mature and whether it is recruited to the fishery (i.e. each age is associated with four stages). Realistic: It is reasonably easy to build in assumptions regarding behavior that cannot be captured using standard age-structured models.

8. 458 Stage-Structured Models (Advantages and Disadvantages) Disadvantages: The flexibility makes designing the model more difficult (how to select the “stages”?) A stage-structured model may have many more (rather than fewer) parameters than the equivalent age-structured model.

9. 458 Moving to Size-Structured Models For these models, each “stage” is a size-class (usually all of equal width). The general equation for these models is:

10. 458 Natural survivalHarvest survival Growth expanded The matrix X is often constrained to prevent “negative growth” (e.g. lobsters, abalone)

11. 458 Fitting Size-Structured Models The typical parameters of a size-structured model are: The numbers-at-size for the first year (analogously with age-structured models, one can assume that the population was in equilibrium at that time). The recruitments. The parameters that define vulnerability at size. The parameters of the size-transition matrix (the growth parameters).

12. 458 Estimating the Size-Transition Matrix This can be the most data-demanding step of applying a size-structured model. Typically, the size-transition matrix is estimated by postulating a growth curve (including its uncertainty) and fitting it to tagging data. A typical choice is the normal distribution :

13. 458 Estimating the Size-Transition Matrix Size-increment information for Tasmanian rock lobster (note the large fraction of zero increments)

14. 458 Fitting Size-Structured Models Size-structured models are almost always fitted to information on population (or catch) size-structure in addition to some index of abundance Example: rock lobster off Tasmania, Australia

15. 458 Fitting Size-Structured Models The likelihood function for the length- frequency data (often the fraction in each size-class) is usually assumed to be multinomial.

16. 458 Size-structured models (Advantages and Disadvantages) Advantages: Requires no ability to age animals (crabs, abalone, rock lobsters). Uses the data actually available (size- compositions). Vulnerability / maturity are often functions of size and not age.

17. 458 Size-structured models (Advantages and Disadvantages) Disadvantages: Potentially very many parameters that are difficult to estimate (e.g. the entries in the size-transition matrix). Still needs an estimate of M (in years -1 ). Computationally much more intensive that age- structured models. Many of the animals to which these models are applied exhibit small-scale spatial differences in growth.

18. 458 Readings Burgman et al. (1993); Chapter 4. Quinn and Deriso (1999); Chapter 9.