Raymond Lindeman (1941) was one of the first scientists ... K. Radway Allen in the 1940s and '50s, ... Allen (1950) came up with an intuitive way of illustrating.
Fish production: integrating growth, mortality, and population density K. Limburg lecture notes, Fisheries Science
Reading to accompany this lecture: Chapman, D.W. 1978. Production, pp. 202-217 In Methods for the Assessment of Fish Production in Fresh Waters (T.Bagenal, editor). IBP Handbook #3. Blackwell Scientific Publisher, Oxford, UK.
Outline: 1. Biological production – a critical ecological parameter 2. How to compute production from a simple biomass model 3. Production:biomass ratios 4. Growth: mortality ratios
1. Biological production – a critical ecological parameter Biological production is the processing of energy and matter to produce living tissue. Raymond Lindeman (1941) was one of the first scientists rigorously to define trophic levels: Primary producers Secondary producers = primary consumers Tertiary producers = secondary consumers …and so on… Q: difference between gross and net production?
Gross production = all metabolic activity + growth = Rtotal + G Net production = only tissue elaboration (G)
In fisheries, “production” simply refers to the net production (in terms of weight or biomass) of fish populations over time, taking account of weight gains through growth and biomass losses through death. This method assumes that metabolic costs are incorporated into growth.
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2. Production from a simple model of biomass accumulation.
This is analogous to the modeling of mortality in a cohort, as we saw before: dN/dt = -ZN .
The famous fisheries scientist (and aquatic entomologist) William Ricker noted that if you consider the average growth of a cohort, you could model biomass, B, with the simple exponential model (just like modelling the change in population size, N):
These equations are virtually identical in form, except that there is a (-) sign in front of the Z. Needless to say, the models behave similarly, one curving up and one curving down: Mortality in a cohort: effect of Z
Ricker's biomass model: effect of G
where G is the instantaneous growth rate in biomass.
1200
700
G = 0.1
600
G = 0.05
Number in cohort, N
1908 - 2001
dB/dt = GB,
Biomass B (kg)
800
500 400 300 200 100 0 0
5
10
15
20
25
1000
Z = -0.1
800
Z = -0.05
600 400 200 0 0
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tim e (for exam ple, year)
Ricker also stated that if a population was at steady state (births = deaths), then this equation should work for the entire population as well (because each cohort behaves like all the others), at least for sufficiently short intervals of time. Just about the simplest expression for production was formulated by Ricker and also by another Canadian fisheries scientist, K. Radway Allen in the 1940s and ‘50s, respectively.
P G t B
10
15
20
25
tim e (years)
Ricker and Allen’s equation:
P G t B where P = production of new tissue (kg) G = inst. growth rate (1/time) t = a time period (units of time)
B = average biomass (kg)
.
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Usually we deal with units of mass… …but we could, if we wished, convert this to energy units or units of various elements like carbon (C) or nitrogen (N), if we know the conversions (which we would get from direct analytic measurements). This would enable us to make comparisons with other components of the ecosystem, for example if we wanted to compare the magnitudes of phytoplankton production with fish production to look at the efficiency of energy transfer down the food chain.
N
Allen (1950) came up with an intuitive way of illustrating production, which links up • population size, • mortality, • and the change in biomass (weight or W). It is called the Allen curve.
The Allen curve Nt = 1
time t Nt = 2
Some fish die, the others grow until the next census at time t+1… W1
W
W2
W
If we look at a small change in the mean weight of fish in a cohort, W, over a small time interval t, and multiply that by the number of fish alive in that time interval (Nt), the production is approximately P Nt W .
Time t+1: the numbers of survivors, times their average weight, minus the numbers at time t times their average weight, is production.
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To integrate under the curve by hand, you can draw the curve on gridded paper and count the number of grid cells, or you can weigh the paper.
(e.g., grams)
Just as when we considered mortality, you can think about chopping up time into infinitesimally small slices, and production is then considered as a continuous process. We would then find the total production between {t = 1} and {t = 2} by integrating under the curve.
(e.g., months)
A very approximate solution:
To integrate under a curve by weighing it, you need to, once again, define a reference area (a known area, Aref) and weigh it (Wref), then cut out the curve and weigh it (Wcurve) to get the area (Acurve).
Aref Wref
Aref A curve , Acurve Wcurve Wref Wcurve
P N W N
N (W2 W1 )
Also see Figure 8.2 in paper by Chapman.
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Date
Mean weight, W (g)
1-May 1-Jun 1-Jul 1-Aug 1-Sep 1-Oct 1-Nov 1-Dec 1-Jan 1-Feb 1-Mar 1-Apr 1-May
1.5 2 2.5 3.5 4.5 6.5 6.9 6.9 6.8 6.6 6.6 6.9 7.4
Instantaneous Growth, G
Stock Stock Mean numbers, Biomass, Biomass, N B (kg) B_bar (kg)
Production, P (kg)
= W * N = (Bt+Bt-1)/2
= G * B_bar
= ln(W t/W t-1)
The Allen curve is an example of what we’d call an empirical method of estimating something – in this case, production. No underlying model is really needed; all you need are point-to-point estimates of the changes in N and W over a time period of interest.
For example, Chapman (1978, Table 8.1) gives the following hypothetical data for a single year-class over a 12 month period:
0.29 0.22 0.34 0.25 0.37 0.06 0 -0.01 -0.03 0 0.04 0.07
8,000 4,500 3,500 3,000 2,500 2,000 1,900 1,700 1,500 1,400 1,300 1,100 1,000
12 9 8.8 10.5 11.3 13 13.1 11.7 10.2 9.2 8.6 7.6 7.4
10.5 8.9 9.6 10.9 12.1 13.1 12.4 11 9.7 8.9 8.1 7.5
Annual production = P
3 2 3.2 2.7 4.5 0.8 0 -0.2 -0.3 0 0.4 0.5
= 16.6
Here, the instantaneous growth parameter G is calculated from one month to the next, and since the time increment is one month, t = 1, so G = ln(Wt+1/Wt). Similarly, the monthly biomass B is computed as W (in grams) * N 1000 (to get kg), and B is the average biomass between two adjacent months ( [Bt + Bt-1] 2).
If we make an Allen curve by plotting N against mean weights over the time period, it looks like this: Note that negative production occurs in winter, because the mean weights of the fish decrease.
Allen curve example May 1
8,000
Estimated N
6,000 June 1 4,000
July 1 negative production
2,000
1.5
2.5
3.5
4.5
5.5
mean W (grams)
6.5
7.5
8.5
If you integrated under this curve, as shown above, you should arrive at the same estimate of annual P as obtained in the spreadsheet.
The Allen curve is useful to compile data already collected, but to forecast biomass and production over some time horizon, we usually need to model it. We do this by recalling our simple models of biomass growth (change in weight) and population decline (change in N):
W ln( t ) W0 Wt W0 e Gt , G t ln( N t N 0 e Zt , Z
Nt ) N0
t
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The quantity (G – Z) represents the net rate of change in the population biomass and we can use this to calculate average B, and finally, P:
B0 (e ( G Z ) t 1) (G Z )t
B0 (1 e ( Z G ) t ) B ( Z G ) t and P GB
G Z,
P 3 K N 0 W (
G Z,
1 2 1 ) Z K Z 2 K Z 3K
G > Z - blue 200 Mean Biomass
B
Allen (1971) does provide another way to compute production that uses another growth model, namely the von Bertalanffy model of growth in weight, together with the same mortality model. This looks hairy, but all you need are a few parameters:
Here,
150 100
N0 = initial population size
50 0 -0.1
0.1
0.3
0.5
0.7
0.9
Z > G - pink
Note: assumes the simplest form of exponential growth in weight and exponential “decay” for death. More complex models may not have exact solutions.
1.1
(page 208 in reading)
K = von Bertalanffy coefficient W = von Bertalanffy asymptotic weight (g) Z = instantaneous mortality coefficient.
3. Production:Biomass ratios. Ratios are often useful “quick-and-dirty” indexes. The ratio of production to biomass, or P/B ratio, gives you an index of growth or decline in biomass. This is used not only in fisheries, but also in ecosystem ecology, where we may be interested in the production and standing crop of an entire ecosystem. Chapman (reading) discusses P/B ratios calculated for some fish: Salmonids: 1.5 – 2 Cyprinids: 0.6 – 1.1
At least for some fish species, there’s an allometric relationship between mean weight in a population and their P/B ratio. In this paper, the slope of this log-log relationship is -0.35
Vistula River predatory fish: 0.3 – 0.7 {freshwater invertebrates: 2.5 – 5}
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4. Growth:mortality ratios.
(in this figure, M is the same as Z, b/c it’s assumed there’s only natural mortality for larvae…)
When considering the models of exponential growth in weight and exponential decay for mortality together, the difference (G - Z) governs whether production will be positive or negative. In studies of fish early life history, some researchers, led by Ed Houde at the University of Maryland, like to use the ratio of G over Z (G/Z) as an index of potential recruitment. (Alternatively, Z/G) time
References: Allen, K.R. 1950. The computation of production in fish populations. N.Z. Sci. Rev. 8: 89. Allen, K.R. 1971. Relation between production and biomass. J. Fish. Res. Bd. Canada 28: 1573-1581. Chapman, D.W. 1978. Production, pp. 202-217 In Methods for the Assessment of Fish Production in Fresh Waters (T.Bagenal, editor). IBP Handbook #3. Blackwell Scientific Publisher, Oxford, UK. Ricker, W.E. 1946. Production and utilization of fish populations. Ecological Monographs 16: 373-391.
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