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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 structured models
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).
stage structured models examples of stage structured populations
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.
modeling stage structure ii differences from age based models
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.
modeling stage structure iii example loggerhead turtles
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

stage structured models advantages and disadvantages
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.
stage structured models advantages and disadvantages8
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.
moving to size structured models
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:
slide10

expanded

Natural survival

Harvest survival

Growth

The matrix X is often constrained to prevent “negative growth” (e.g. lobsters, abalone)

fitting size structured models
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).
estimating the size transition matrix
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:
estimating the size transition matrix13
Estimating the Size-Transition Matrix

Size-increment information for Tasmanian rock lobster

(note the large fraction of zero increments)

fitting size structured models14
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

fitting size structured models15
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.
size structured models advantages and disadvantages
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.
size structured models advantages and disadvantages17
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.
readings
Readings
  • Burgman et al. (1993); Chapter 4.
  • Quinn and Deriso (1999); Chapter 9.