Lecture 5 Review

1. Lecture 5 Review • 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

2. Lecture 6: bioenergetics models • Bioenergetics models either • (1) predict growth from predictions of food consumption and metabolism/reproduction, or • (2) back-calculate food consumption from growth and predictions ofmetabolism/reproduction • Backcalculation of food intake (needed for analysis of trophic interactions) can be done from growth curves • Bioenergetics models that account for seasonal variation in food, temperature effects are needed for interpretation of tagging data • Allocation of energy to reproduction is critical for understanding growth curves

3. The basic bioenergetics model • Growth=Consumption-Egestion-Respiration-Reproduction • Better form: Growth=e(Consumption)-Standard metabolism-Reproduction • Here, “e” is around 0.5-0.6, and represents the proportion of consumption that is assimilated (not egested) and is not used for digestion (digestive cost=0.2 x Consumption is called “specific dynamic action”) or active metabolism proportional to food intake

4. Body size is a key determinant of the components of energy intake and allocation eQ = HW2/3 R = mW0.8 R+Repn = kW

5. Modeling net energy intake eQ • The assimilation-SDA-active metabolism coefficient e does not vary much • Q depends strongly on body weight W, water temperature T, and effective food availability which depends on foraging time and food density • Q in the field is typically only about 1/3 of the maximum ration found under laboratory conditions, even in habitats with high food availability, i.e. foraging time decisions often override any effects of food availability

6. Modeling net energy intake eQ • We usually model body size and temperature effects by assuming eQ=HWd fc(T) i.e. a power of W times a temperature effect multiplier, with d around 0.65-0.75 • The temperature multipler is modeled as a Q10 effect times a high temperature inactivation effect:fc(T)=Q10(T-10)/10 e-k(T-Ti)/(1+e-k(T-Ti)) • Q10 is usually in the range 2-5, k is typically around 0.5, and inactivation temperature Ti depends on the species (cold vs warm water)

7. Temperature effects • The Q10 function f(T)=Q10(T-10)/10 is the same as the exponential f(T)=ekT that you often see used in the literature, when expressed as fc(T)=elnQ10(T-10)/10, i.e. k=ln(Q10)/10 • The product of Q10 x (Inactivation) behaves as:

8. Modeling metabolism+gonads • Some metabolism is included in “e”; we model the rest, and energy lost to reproductive products, as mWn fm(T) • Here, n is usually near 1.0 except in models that represent gonads explicitly, in which case it is around 0.8 • fm(T) is the Q10 function again: fm(T)=Q10m(T-10)/10 • Q10’s for metabolism (Q10m) are usually in the range 2.0-3.0 (lower in acclimated temperature range, higher at extreme high temperatures)

9. Putting it all together • The net intake and metabolism models give the overall rate equation dW/dt=HWdfc(T)-mWnfm(T) • Analytical expressions for W(t) (and/or length L(t) exist only for n=1 (generalized vonB; d=2/3 gives standard vonB • Otherwise we have to integrate the rate equation by numerical methods, e.g. Wt+Δt=Wt + (dW/dt)Δt (Euler) Wt+Δt=Wt + (Δt/2)[3(dW/dt)t-(dW/dt)t-Δt] (more accurate Adams Basforth)

10. A key point about dL/dt when d is near 0.67 and n is near 1.0 • In that case, dL/dt varies linearly with L, with intercept proportional to feeding rate and slope proportional to mass loss rate m/3 (slope is minus the vonB K) Warm water Growth rate dL/dt Higher intercept when feeding rate is higher, steeper slope when metabolism is higher Cool water Length (L)

11. Examples of seasonal change in the dL/dt line with temperature Humpback chub Trout

12. ΔL is the difference between two measurements, so has variance 2 times the variance of each measurement Also there is among-individual “process” error variation in growth rateso var(ΔL/Δt)=(2σ2obs+σ2proc)/ (Δt2) Note here that dividing by a short Δt squared causes hugh increase in variance of growth rate measurement Unfortunately, there are severe statistical problems with using the empirical ΔL/Δt data directly

13. Parameter estimation from tag data • Various methods have been proposed to deal with variation in tagging data (James, Francis, Wang, Laslett, etc., etc.); these methods all work fine with fake data, but fail badly with most real data sets due to: • Incorrect statistical assumptions • Effects of seasonality on growth • The latest approach (Walters and Essington) treats individual growth deviations (and ages at first capture) as unknown “process error” parameters to be estimated along with population mean bioenergetics parameters. The approach is very messy to implement.

14. Di>1 Length Di=1 Di<1 Age The Walters-Essington method • Assume each tagged fish was growing along an individual trajectory Li(a), such that Li(a)= Di i.e. each fish had relative growth Di

15. Estimation of relative growth rate Di for each tagged fish • For each possible discrete age ai that fish i could have been when tagged, calculate the conditional (on a) maximum likelihood estimate of Di using • Here, L1i and L2i are lengths at tagging and recapture, Δi is time to recapture, and vr is the variance ratio σ2obs/σ2D

16. Estimation of relative growth rate Di for each tagged fish • Then use this Di estimate to calculate the log likelihood of the observed L’s, given a, evaluated at most likely Di given a: • This is just a weighted sum of three squared deviations, one for each measure length L and one for the D

17. Estimation of most likely age ai and overall likelihood for the data • Then do the LnLi(a) calculation for each possible age ai, and take the maximum likelihood estimate of ai to be that value ai that gives the highest LnLi(a). • For each fish, you then have a log-likelihood LnLi that does not depend on assumed age, i.e. is evaluated at the most likely age of fish i at tagging. • Adding these LnLi values over fish gives an overall likelihood function for the data, that you can use with nonlinear search procedures to do parameter estimation (e.g. give it to Solver, ask for maximum likelihood estimates of H, m, Q10’s etc)

18. You can think of the maximization over possible ages “a” as sliding your data along an age axis until you get the best fit Di>1 Length Di=1 L2i Di<1 L1i Δi Age

19. Notes about this formulation • The variance ratio vr=σ2obs/σ2D represents relative variance in the data due to measurement error vs growth rates D. • An estimate of σ2obs can be obtained by looking at data from recaptures near the tag time, typical values are around 25-60mm2. • The σ2D choice can represent “prior” uncertainty about changes in mean D with age due to selective sampling; absent such effects, a good choice is around 0.12, but values of at least 0.42 should be assumed initially.

20. Estimating food intake from growth once you’ve got a fitted model • Remember that catabolism is represented as eQ=HWd fc(T) • So instantaneous food intake rate Q is easy to calculate, as Q=HWd fc(T)/e, and integrated over time to give various totals Flathead catfish example

21. In summary • You probably didn’t understand a word of all this. Fortunately, there is a spreadsheet to do it all for you. • You are not alone; the sheer complexity of this statistical approach is probably why it was not discovered until 2007. • It is worth spending a bunch of time trying to understand it; you are sure to be working with tagging and growth data, and there are big benefits from being able to estimate fish ages along with the bioenergetics parameters.