of the non-pathogenic factors that contributed to the disastrous disease epidemic that reduced Taiwan from the world’s leading exporter of marine shrimp to a net
importer. Given a set of environmental and water management practices, such as pond
tank size, water flow rate and others, a production system can support a given number of shrimp of given size. Beyond an upper biomass limit, the water quality
begins to degenerate, and as the biomass increases, so does the extent and frequency of disease and the reduction in growth rate.
This upper biomass limit is often called the ‘critical standing crop’ or CSC. For any given production system and set of management practices, the CSC is a
function of shrimp biomass in the system. For a single-stage production system, the stocking density can be set so that the CSC density will occur at harvest time. In
other words, the initial stocking density is set so that the CSC will be reached when the shrimp are ready to be harvested. In this way, the system is below the CSC level
and thereby operating below maximum capacity up until the day the shrimp are harvested.
In an ideal shrimp production system, the stocking density would be set so that the CSC level is reached on the first day and the pondtank would expand
continuously; the CSC level would be maintained since the continuous increase in shrimp biomass would be balanced by the continuous increase in water surface area
and supporting inputs. This obviously cannot be done economically. In practice, the initial stocking density is set at below the CSC level and, as the shrimp grow
and the biomass reaches the CSC level, the excess shrimp biomass is transferred to another pondtank so that the CSC level in any production pondtank is not
exceeded.
3. Multi-stage production system design
The fact that the shrimp growth rate is approximately inversely proportional to stocking density can be used to optimize the number of stages in a shrimp
production system. Fig. 1 shows four idealized stocking-density-dependent growth rate curves. Curve d represents the best growth rate curve, that is to say, the
stocking density is set such that at the end of the production period when the shrimp reach market size, the shrimp continue to maintain their maximum
biological growth rate attainable under the environmental and feeding regimes. Curves a, b and c show that shrimp growth rates are affected by shrimp stocking
densities a, b and c, where densities are a \ b \ c \ d. During the beginning of curves a, b and c, when the shrimp biomass is below the CSC level, curves a, b, c
and d are identical. Each growth curve departs from curve d as its CSC level is reached. With an initial stocking density of a, when the biomass reaches the level
represented by CSC
a
, the biomass in curve a begins to retard the shrimp growth. At CSC
b
and CSC
c
, the shrimp growth rates, as shown by curves b and c with initial stocking densities of b and c, begin to decrease. Curve d, with an initial stocking
density that allows the maximum physiological growth rate, represents the maxi-
mum growth rate that can be attained with the given biological and water management constraints. For a single-stage production system the appropriate
stocking density is d. According to Fig. 1, for a two-stage shrimp production system, we have three
choices: either a – d, b – d or c – d. An a – d system will begin with high stocking density a, and at time CSC
a
the density is reduced to d. For a three-stage production system, the combinations of a – b – d, a – c – d or b – c – d can be used.
There is only one possible design choice for a four-stage production and that is a – b – c – d.
The number of stages that can be considered in the design process is limited by the available data. As demonstrated by Fig. 1, the four curves, a, b, c and d limit
the potential choices to four stages. If more data is available to construct additional growth curves, then the potential number of stages can be correspondingly
increased.
In general, X = 2
n − 1
2 where: X is number of possible choices in the design of production systems in terms
of production stages; n is number of stocking density dependent growth curves. Table 1 gives a summary of all possible configurations using idealized data from
Fig. 1. Table 2 shows experimental shrimp growth data from Sturmer et al. 1991, Sandifer et al. 1988. These data give an indication of the effect on shrimp growth
rates of stocking densities at 1100, 550, 100, and 40 shrimpm
2
when other conditions were kept identical.
Fig. 1. Critical standing crop CSC theory.
247 J
.- K
. Wang
, J
. Leiman
Aquacultural
Engineering
22 2000
243 –
254
Table 1 Possible phase configurations and corresponding rearing period for CSC theory described in Fig. 3
Stage d days Stage a days
Stage a density Stage b days
Stage b density Stage c days
Stage c density Number of
Stage d density shrimpm
2
shrimpm
2
shrimpm
2
shrimpm
2
stages 0–21
1100 550
100 40
21–42 42–63
4a,b,c,d 63–154
550 100
40 –
3b,c,d –
63–154 42–63
0–42 1100
0–21 –
100 40
– 21–63
63–154 3a,c,d
42–154 1100
550 –
40 3a,b,d
0–21 21–42
– –
100 40
– 63–154
0–63 2c,d
– –
– –
550 –
40 0–42
– 42–154
2b,d 2a,d
– 0–21
– 40
– –
21–154 100
– –
40 –
– –
1d 0–154
–
248
J .-
K .
Wang ,
J .
Leiman Aquacultural
Engineering
22 2000
243 –
254
Table 2 Experiment data used to develop the Von Bertallanfy growth curves
Stocking density: 550 shrimpm
2
; Stocking density: 100 shrimpm
2
; Stocking density: 1100 shrimpm
2;
Stocking density: 40 shrimpm
2
; Pe- Penaeus 6annamei; Sturmer et al.,
naeus 6annamei; Sandifer et al., Penaeus 6annamei; Sandifer et al.,
Penaeus 6annamei; Sturmer et al., 1988
1988 1991
1991 Day
Shrimp mean Day
Shrimp mean Day
Shrimp mean Shrimp mean
Day weight g
weight g weight g
weight g 0.003
0.003 0.003
0.003 0.5
0.018 35
1.3 7
0.018 24
7 2.7
0.0545 63
4.22 11
0.0545 53
11 91
8.97 6.0
80 14
0.145 14
0.109 120
14.99 18
0.127 18
0.164 151
20.65 21
0.273 0.345
21 0.418
176 25.13
25 0.545
25 0.436
203 26.67
28 0.688
28 32
1.0 32
0.618 0.727
35 35
1.108 39
1.234 0.889
39 42
1.326 42
1.653
Table 3 Von Bertallanfy growth parameters solved by GAMS
Stocking density W
inf
g k day
Best-fit line plot of theoretical equations vs. real R
2
data shrimpm
2
1100 y = 0.0046+0.98x
0.022 0.993
4.1 y = 0.0028+0.95x
0.988 550
0.014 15.0
y = 0.22+0.93x 0.993
28.6 100
0.011 y = −0.110+0.97x
40 0.993
0.009 44.9
Fig. 2. Shrimp mean weight versus rearing period at various stocking densities.
The Von Bertallanfy equation is one of the best to describe penaeid shrimp growth rates Tien et al., 1993:
W
t
= W
inf
[1 − e
− kt-t
]
n
3 where: W
t
is weight at time t g; W
inf
is asymptotic weight g; t is shrimp age days; t
is theoretical age at which W = 0 g days; k is instantaneous growth rate gdays; n = 2.8025 for Penaeus monodon Mercy et al., 1993.
The data in Table 2 were compiled using G
ENERAL
A
LGEBRAIC
M
ODELING
S
YSTEM
GAMS Release 2.25 Brooke et al., 1992 to solve for k and W
inf
in Eq.
3. The calculated constants are shown in Table 3 Leiman, 1995. It can be seen that as stocking densities decrease from 1100 to 40 shrimpm
2
, the k values decrease from 0.022 to 0.009day, while the W
inf
values increase from 4.1 to 44.9 g. Using Eq. 3 and the constants from Table 3, we can construct Fig. 2.
While Fig. 1 is an idealized representation, Fig. 2 is based upon limited real data. We can see that Fig. 2 gives an excellent facsimile of Fig. 1.
The total water surface area of the pondstanks for each production stage can now be calculated with the following equations and Eq. 1.
Ii = Oi Si 100 4
Ai = Ii SDi 5
where: Ii is input or stocking population of shrimp in phase i; Oi is output or transferharvest population of shrimp in phase i; Si is shrimp alive by the
end of rearing period in phase i 100 refers to the shrimp alive at stocking time of phase i ; Ai is water surface area of individual pondtank in phase i
m
2
; SDi is stocking density of shrimp in phase i shrimpm
2
. Ni = [t
th
i − t
s
i + t
d
i ]t
pd
6 where: Ni is number of pondstanks in phase i integer c ; t
th
i is transfer harvest time in phase i weeks; t
s
i is stocking time in phase i weeks; t
d
i is down period between cycle in phase i weeks; t
pd
is period between product deliveries weeks = lfrequency of delivery.
4. Optimization
