Body Size and Growth

1
Body Size and Growth

• Goals
• Explain the rationale behind the use of size
  data in
• year-class studies
• mortality
• Discuss potential problems with the above
  methods
• Explain the modeling of fish growth
• Ford-Walford Plots
• Weight/Length - Age Relationships
• Describe the construction of
• age-length keys
• size distribution matrix
• Discuss Length-Based VPA

2

• Consider the case of an imaginarypopulation
  where
• fish recruit on March 14
• recruitment is constant
• grow at the same rate
• growth rate declines over time
• maximum age of 4 years
• common Z

A Single Haul (constant q)
3
What A Single Sample Really Looks Like

• Length-Frequency of Coral Trout
• p. 421 in H W
• distributions, not spikes
• why?
• Why fewer of smallest fish?
• More than one possible answer ...

4
Can you draw conclusions aboutpopulation from
length data?

• Often not
• most gear is selective
• body size
• movements or distribution
• gear avoidance

• Remedial Measures
• compare size distributions in different gears
• perform depletion experiments by size class
• survey and tagging
• detailed fishing logs
• sub-sample catches
• from hauls, vessels, areas
• visit the field

5
Models of Fish Growth

• Why?
• Production estimates
• field, aquaculture
• Different Approaches
• All summarize data
• from size-age data
• corrected for gear selectivity
• based on average growth
• individual variation
• Tabular
• table of average weights by time
• Mathematical
• von Bertalanffy, logistic, Gompertz
• Schnute

6
Step I Look at the data

• Snapper (Etilis sp.)
• note variation in size-at-age
• length easier than weight for size

7
Modeling fish growth
l?

• a) simple asymptotic
• von Bertalanffy
• b) accelerating and then asymptotic
• Logistic
• Gompterz
• different inflection points
• Also generalized Schnute model
• All assume l?

8
An example von Bertalanffy
lt l??1 - e -K(t - t0 )
l? maximum length K Brody Growth Coef. t0
adjusts for non-zero size at t 0
As w a( l )b , b ? 3
wt w??1 - e -K(t - t0 ) b
to express growth in weight
9
The Walford Line
lt1 l??1 - e -K e -K lt
10
Analysis of Length Frequency Data
Concept - Modes represent cohorts -
cautions as before

• A single sample
• assume overlapping distributions
• estimate modes
• harder for older ages
• estimate mean and variance.
• mix of statistical and subjective
• estimate growth and mortality
• problems?

11
Repeated Length-Frequency Data

• cross checking for spurious modes
• generated by poor sampling?
• tracing specific modes in time
• cohort growth rates

• works well when fish are
• short lived
• recruited seasonally

12
Mortality from Length Samples

• Mortality Rates Difficult to know
• iff. (if and only if)
• stock at equilibrium
• recruitment mortality
• recruitment independent of stock
• all fish above lcrit are fished
• von Bertalanffy growth
• all fish experience one Z

Z K l??- lmean lmean - lcrit
use in conjunction with other estimates
13
Conversion from Length to Age Frequencies

• Often have much length data
• cheap, easy
• less age data
• difficult, expensive
• Estimate age from length
• smearing at older ages
• fish grow at different rates
• Good check on assumption that age sample is
  representative
• Still need unbiased samples for population
  inferences
• Different Methods
• Deterministic Age-Length Keys
• Size Distribution Matrix

14
Deterministic Age-Length Keys

• Full calculation yields age-length key
• solve t for upper and lower interval bounds
• assign single age to each length interval
  (round to integer)

15
Size Distribution Matrices

• no growth curve necessary
• data required for size distribution of each
  age class
• growth implicit in matrix
• age classes with common size structure treated
  the same
• use matrix methods to map size distribution to
  age distribution

AGE 1 2 3 SIZE 1 0.8 0.3
0.1 2 0.2 0.6 0.2 3
0.0 0.1 0.7
16
Length-Based VPA

• assumes population is at equilibrium
• recruitment mortality
• studies the time required for a fish to grow
  through a length interval
• uses Popes approximation
• calculates
• instantaneous fishing mortality rate for size
  class
• mean number in size class

17
A Caution in a Bad Example

• Application of Length Based VPA
• to examine Pilchard (Sardinops sp.)
• West Australian Fishery, 1982-89
• estimated stock size of 100,000 tonnes
• BUT
• declining seabird populations
• increased search times to schools
• seasonal scarcity of fish
• analysis may have overestimated due to
  developing (non-equilibrium) nature of the
  fishery

• Message
• use more than one method if possible
• know assumptions and limitations
• be wary of length alone