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Stage- / Size-Structured Models

Stage- / Size-Structured Models. Fish 458, Lecture 16. Stage-Structured Models. Why not always use age-structured models: Ageing is difficult (or impossible) for several types of organisms.

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Stage- / Size-Structured Models

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

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

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

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

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

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

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