Lecture 4 review

• Most stock-recruitment data sets show the pattern predicted by Beverton and Holt when they assumed cohorts die off at rates dN/dt=-MN, with varying over time as M=mo+m1N, i.e. linearly density-dependent M.

• Foraging arena theory predicts this pattern of density dependence to arise from risk-sensitive foraging behavior where fish try to balance need to grow with risks of foraging to achieve that growth.

• Estimation of stock-recruitment model parameters can be severely biased due to time-series and errors-in-variables effects.

• The Beverton-Holt “a” parameter (KRo/Eo) involves a risk ratio: ln(a)=f-(predation risk)/(food supply)

• The Beverton-Holt “b” parameter (K-1)/Eo is dependent on foraging arena habitat size b=(constant)/(habitat size)

Assignment 3: typical biologists: you asked about the effect of M on your stock-recruit assessment, but not about the effect of q.

Pacific Yellowfin tuna, Goodyear compensation ratio (K ) 3.21 assuming Unow=0.5, S=0.6

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0 10 20 30 40 50 60

N(t-2)

R(t

)

U2000=0.5S=0.6est K =3.21

Your result with q=0.01(current U=0.5):

Pacific Yellowfin tuna, Goodyear compensation ratio (K ) 1.47 assuming Unow=0.25, S=0.6

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101520253035404550

0 20 40 60 80 100 120

N(t-2)

R(t

)

U2000=0.25S=0.6est K =1.47

Effect of assuming q=0.005(current U=0.25):

Uncertainty about the observation parameter q (i.e. about current stock size and U) is a much bigger policy issue!

Lecture 5: growth, vulnerability, and fecundity

• Age structured population analysis and prediction requires that we estimate schedules of body size, vulnerability to capture, and fecundity

• Growth curves are very easy to model, but the parameters of such curves are pathologically difficult to estimate

• Expect complex vulnerability schedules (dome-shaped, changing over time)

• There is no need to measure absolute fecundity, but relative fecundity at age is important to assess

Analysis of growth

• We now mainly use the VonBertalanffy model:La=L[1-e-K(a-ao)]

L is the asymptotic (maximum) length K is the metabolic (!) parameter

ao is the apparent age at zero length

• Two basic approaches to estimation:– Collect lots of fish and age them (otoliths)– Tag lots of fish and look at how growth varies with

length at tagging: (La+Δ-La)=(L -La)(1-e-KΔ)

Problem 1: growth varies a lot among individuals

• Individual growth trajectories show similar K, different L

• This would not be a big statistical issue if all fish died at the same rates and were equally vulnerable to sampling

Age

Length

Why do individual growth trajectories vary so much?

• Variance in food intake (IBMs do not predict enough variation)

• Persistent differences in habitat quality (not for shoaling, migratory fish)

• Size-dependent dominance hierarchies

Individual growth trajectories, P(feed)=0.3CV length = 0.075

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Age

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gth

Problem 2: Faster growing individuals get caught first

(pikeminnow example)

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Age from tagging

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lati

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Problem 3: faster growing individuals have higher mortality rates

• This means that the individual growth curves are not sampled randomly

Mean length at age can even decrease

Length

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Age (years)

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igh

t (g

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Length (cm)

Hayes model Length

wt (kg)

Hayes Model weight

Brown troutexample

Size and maturation• Fish can either do this

(mature at fixed age, fishing selection causes slow growth)

• Or they can do this (mature at fixed size, fishing selection causes fast growth and early maturity)

Length Length

Age Age

Age

Length(Or something in between those extremes; Heino calls the observed pattern the “reaction norm”)

Here is how Heino (Evolution 2002) explains reaction norms:

There is typically a linear relationship between fecundity and body weight, with zero fecundity

occurring at some weight at maturity

Weight (W)

Eggs

This is caused by a shift in allocation of net energy (assimilated food consumption minus metabolism) from growth to gonads:

R

Repn

G

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0.00 0.25 0.50 0.75 1.00

W/Winfinity

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oca

tio

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f u

sab

le e

ner

gy

eQ

Total mass loss rate (metabolism+egg production is approximately a fixed

proportion of body weight• This means that weight growth can be

modeled asdW/dt=eQ – kW

• The vonBertalanffy model is derived by assuming further that eQ=HW2/3, i.e. that net energy intake varies as 2/3 power of W

• To convert from weight growth to length growth, we note that

• dL/dt=(dL/dW)(dW/dt)=(dL/dW)(eQ-kW)

Derivation of the VonBertalanffy (continued)

• dL/dt=(dL/dW)(dW/dt)• But W=aL3 (length weight relationship)

and dW/dt=HW2/3-kW• L=(W/a)1/3, so dL/dW=1/3a-1/3L-2/3

• Substituting this into the dL/dt equation gives dL/dt=H/(3a1/3)-(k/3)(W/a)1/3

• Note that dL/dt is thus a rate equation of the form dL/dt=constant-KL; the integral of this is called the “vonBertalanffy model”

• A critical point: the vonBertalanffy K parameter is thus equal to k/3, i.e. it is 1/3 of the slope of the relationship between annual total energy loss rate and W.

If you think that math was nasty, just wait until next week.

• If you got lost, review your basic calculus.• If that doesn’t help, get someone who did follow

the derivation to go over it with you.• There is a really good reason for trying hard to

understand this material: it ultimately leads to very simple bioenergetics models for estimating food consumption. Without such models, your only alternative will be the Dreaded Wisconsin Bioenergetics Model, which is infintely harder to understand (and gives the same answers after much, much more work on your part).