INTRODUCTION
Fisheries and aquaculture both involve the production of high quality fish
protein (Obaroh and AchionyeNzeh, 2011; Bozoglu
et al., 2006; Huda et al., 2002).
Fisheries are based on the extraction of living resources in water bodies (Bostock
et al., 2010; Akca et al., 2006).
Aquaculture is the farming of aquatic plants and animals (Iwama,
1991). Both disciplines provide most of the world’s aquatic edible
resources (Food and Agriculture Organization of the United Nations (FAO,
2010).
Aquaculture productivity is commonly measure as total biomass. However, in
many cases additional parameters are required to make thorough studies (Alagaraja,
1991). Simple algebraic models can be used to make important decisions in
farm (Roomian and Jamili, 2011). These methods are relative
easy to carry out and their implementation requires basic mathematical background.
In scientific research they can be used to compare similar experimental procedures
(AlatorreJacome et al., 2011).
In the case of fisheries, direct and indirect methods to assess productivity
have been extensively developed (Cochrane, 2002; Sparre
and Venema, 1998; Pauly, 1983). But as for inland,
smallscale aquaculture systems, the application of large and complex fisheries
analysis could be far to be done.
The purpose of the present study was to propose a synthetic methodology in order to assess smallscale fish productivity. Formulae for collective and individual fish growth determination are presented. A case study in aquaculture is analyzed for collective growth performance indexes. A smallfisheries study case is analyzed to explain the procedure for individual fish growth determination.
MATERIALS AND METHODS
Data measurement: For the following parameters, there were required
four types of response variables: Fish total length (mm), fish wet weight (g),
time (days) and number of fish measured. The correct techniques for accurate
measures can be found in the literature (Sparre and Venema,
1998; Brander, 1975). For explicative purposes, examples
on calculations were analyzed on results.
Total biomass production: According to Ricker (1971)
biomass is the amount of substance in a population expressed in material units,
such as living or wet weight, dry weight, ashfree weight, nitrogen contents,
etc. It is also termed as standing crop.
For the total biomass wet weight (W_{t}) we use:
where, W_{i} is the weight of the ith fish in the system.
Because the aquacultural systems are very heterogeneous about its size and
capacity, is useful apply the term relative density (prel) per volumetric unit
(kg m^{3}) to compare among them:
where, W_{t} is the total biomass on the system and V is its volume. In extensive systems if often used the area instead volume.
Akinwole and Faturoti (2007) use the next equations
as useful indicators for the system productivity. The Total Weight Gain (TWG)
function indicated the gain of biomass in a given time:
where, M_{f} is the final mass of the fish and M_{i} is the initial mass.
The Average Daily Growth Rate (ADGR) indicate the average weight gained each day:
where, TWG is the total weight gain (from Eq. 3) and D are
the culture day (Shnel et al., 2002).
According to Bwala and Omoregie (2009), the Specific
Growth Rate (SGR) is:
where, M_{f} is the final weight of the fish, M_{i} is the initial mass of the fish, l_{n} is the natural logarithm and D are the culture.
Metabolic Growth Rate (MGR) and the Feed Conversion Efficiency (FCE) can been
computed with the methodology exposed on the work of Frei
and Becker (2005):
where, M_{f} is the final mass of the fish, M_{i} is the initial mass of the fish and D are the interval time (in days). For FCE:
where, M_{f} is the final mass of the fish, M_{i} is the initial mass of the fish and F is the dry weight of the feed.
Individual fish growth: In hatchery or nursery system is also important
the length of the fish. Both variables (weight and length) are related by the
next equation (Sparre and Venema, 1998):
where, W_{(i)} is the weight for the ith fish, L(i) is the total length of the fish and the letters a and b are the growth parameters obtained by linearization.
In many cases is useful to predict the increment of the length and weight of the fish in a given time. It can be achieve using potential growth model. A very popular model among fish researchers is the von Bertalanffy’s growth function:
where, L_{t} is the total length of the fish on time t, L_{l} is the maximum total length at infinite time, k is the growth constant, t_{o} is the initial time to growth and t is time. In the case of weigh, the equation is the following:
where, W_{t} is the total weight of the fish on time t, W_{l}
is the maximum total weight at infinite time, k is the growth constant, t_{o}
is the initial time to growth and t is time (Pauly, 1983).
RESULTS AND DISCUSSION
Total biomass production: SotoZarazua et al.
(2010) cultivated 1,200 tilapia fish fingerlings on circular tanks with
a capacity of 20 m^{3}. The average initial weight was 20 g and after
180 days the weight of all the fishes was measured. Applying the formula 1,
the total biomass production on one tank was 580.33 kg. For relative biomass
production (Eq. 2):
The initial total weight gain assumed 1,200 fish and 20 g per fish was 24 kg. For Eq. 3:
And the average daily growth rate per tank, according to Eq. 4:
The Specific Growth Rate ( SGR) was:
Metabolic Growth Rate (MGR) and the feed conversion efficiency (FCE) can been
computed with the methodology exposed on the work of Frei
and Becker (2005):
In addition, there was reported 940.13 kg of feed consumed during the experiment, so the feed conversion efficiency was:
Individual fish growth: In 2006, AlatorreJacome measured the following data for length and weight on M. salmoides located in a small lake in center México (Table 1).
Fitting the data for an exponential model (r^{2} = 0.97) the specific lengthweight relation was:
To obtain the parameter for Von Bertalanffy equation, K and L_{o} there
are several methods. In this case we used the following. In literature, In literature,
the propose value L_{∞} = 358.4. With the data of age and length (Table
2), in a third column there was calculated:
Plotting the values of the first and the third column, there was obtained by linear regression (R^{2 }= 0.9971) the parameters a = 0.33, which is the value x at y = 0 and b = 0.3255, the slope of the line. The value K = b and T_{o} was:
And the Von Bertalanffy’s weight equation for this population is:
Table 1: 
Length and weight measured on largemouth bass (M. salmoides)
on Camecuaro Lake, 2006 

Table 2: 
Linearization of time (age) and length values to parameter
determination on von Bertalanffy’s equation 

Calculating the fish weight at L_{∞}, we obtain L_{∞} = 652.07, so the Von Bertalanffy’s weight equation is:
DISCUSSION
There are many different values for productivity index on literature, which
explained the global performance for one system. In this case, the value of
29 kg m^{3} is obtained. Timmons et al. (2002)
recommended less than 40 kg m^{3} for systems with blower. However,
Rakocy et al. (2006) reported densities of 60
kg m^{3} for aquaponic systems. In extensive systems, Sarker
et al. (2005) reported lower values (479 kg ha^{1}) even
with strains of genetically improved farm tilapia. The principal causes of productivity
are due to managing practices, temperature (Ghosh et
al., 2008; Sarkar et al., 2007) and water
quality factors in culture water (Hossain et al.,
2007).
In ADGR, the index can be used to make more accurate feed management schedules.
A variation can be made with the data, dividing ADGR by number of fishes. Then
the average day growth rate per fish can be obtained. In this case, 2.57 g day^{1}
fish^{1} is reported. Rezk et al. (2002)
reported ADGR from 1.87 g day^{1} fish^{1} in O. aureus
after 35 days of culture. This value is lower than the observed on SotoZarazua
et al. (2010) but the main difference is than Rezk cultured fingerlings,
who have a different metabolism than adults. In other hand, Liti
et al. (2005) reported ADGR from 0.06 to 1.5 g day g day^{1}
fish^{1} found on tilapia fed on two formulated diets with locally
available feed in Kenia.
From SGR, Akinwole and Faturoti reported SGR from 2.656 to 2.86 measured on
C. gariepinus cultivated in recirculating aquaculture system. Velazquez
and Martinez (2005) reported 0.97 and 0.86 for C. auratus. Hlophe
et al. (2011) reported SGR values lower than 1.7 in T. rendalli
fed with kikuyu grass.
On the other hand, Cho and Bureau (2001) suggested the
elimination of the parameter due to the nonrealistic approaching of its calculation.
In MGR, the index is used in nutritional studies. Richter
et al. (2002) reported several index (From 1.76 to 5.04 g kg^{1}
day^{1}) in order to assess a more convenient maintenance diet formulations
in red tilapia. At last, the feed conversion efficiency of 0.59 means that almost
60% of the mass provided for the tilapia was assimilated as tissue. This is
very convenient, due the requirements of the fish to its energy for respiration
and the balance of nonassimilated food.
In the example presented for individual fish weight, it can be observed that
in Eq. 16 the parameter b˜3.05. When the parameter b = 3,
the growth is called isometric and if b<3 is called allometric negative and
if b>3 is allometric positive (Pauly, 1983). So we
can see that the fish measured have a good increment of weight. For the equations
of von Bertalanffy’s equations, the parameter k is very important, because
is the growth constant and is species specific (Lv and Pitchford,
2007). For this case, in the study of GuzmanArroyo made 35 years earlier
in the same place, the value was k = 0.56, so we can assume that the conditions
were more favorable to a faster growth for M. salmoides.
CONCLUSION
Selected fish biometric indexes were presented in this work. The use of the parameters mentioned can bring more information for the intrinsic factors in the fish culture that influenced growth. They also allow the comparison between different populations in space and time. This paper can be used as a quick guide to measure fish productivity in small systems. The following parameters can be used by scientist or producers to compare different systems each other.
ACKNOWLEDGMENT
The Fondo de Investigación de la Facultad de Ingenieria (FIFI, 2011) of Queretaro State University sponsored this work, the financial support is greatly appreciated.