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Gross growth efficiency as a function of food intake level in the “Pulpito” Octopus tehuelchus: A multimodel inference application
Aquaculture 284 (2008) 272–276
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Aquaculture j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a q u a - o n l i n e
Gross growth efficiency as a function of food intake level in the “Pulpito” Octopus tehuelchus: A multimodel inference application M.J. Klaich a,⁎, M.E. Ré b, S.N. Pedraza a,b a b
Universidad Nacional de la Patagonia San Juan Bosco, Boulevard Brown 3700, Sede Puerto Madryn, Chubut, Argentina Centro Nacional Patagónico, Boulevard Brown 2825, Puerto Madryn, Chubut, Argentina
a r t i c l e
i n f o
Article history: Received 5 February 2008 Received in revised form 3 May 2008 Accepted 30 July 2008 Keywords: Gross growth efficiency Food intake level Multimodel inference Octopus tehuelchus
a b s t r a c t Multimodel inference theory was used to investigate the relationship between gross growth efficiency (GGE) and food intake level (FI) in Octopus tehuelchus for immature octopus kept at 10 °C and 15 °C. Multimodel inference was carried out using the model averaging methodology over the entire set of candidate models. The main results indicated that in O. tehuelchus GGE is an increasing and asymptotic function of FI showing a maximum approaching the partial growth efficiency (PGE). The model-averaged estimates of PGE and maintenance level (MaL) did not differ from those estimated using the rectilinear relationship between growth and FI in previous work. For FI higher than MaL, both GGE and PGE were higher in immature octopus at 10 °C than octopus at 15 °C. It is noteworthy that as FI increases to a satiability state, the difference between GGE at 10 °C and 15 °C decreases as a potential function of FI. These results would indicate that O. tehuelchus maximizes the energy used in growth by keeping a constant MaL for any value of FI. Multimodel inference theory proved to be an efficient tool to build averaged functions taking each candidate model into account. On the other hand, the present work presents a method to understand the intensity feeding needed to reach an optimum growth efficiency, avoiding wasting of food, work energy and investment for octopus rearing or culture activities. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Cephalopods are among the most efficient animals using food for growing. This is reflected by their high instantaneous growth rates, feeding rates and food conversion indices. Such growth is promoted by efficient feeding and a protein-based metabolism that converts food into growth rather than storage (O'Dor and Webber, 1986). The most important features of octopus growth and its researching challenges have been reviewed by Semmens et al. (2004). The manner and efficiency with which an octopus invests the energy taken from its environment in growth, is among the most important topics of octopus growth research. Laboratory experiments have proved to be the best tool for studying octopus growth. Several studies have been carried out investigating the form of octopus growth curves as a time function (Van Heukelem, 1976; Joll, 1977; Forsythe, 1984; Hartwick et al., 1984; Forsythe and Toll, 1991; Domain et al., 2000), although few have studied the octopus growth as a function of food intake level (Van Heukelem, 1976; Joll, 1977; Klaich et al., 2006; Leporati et al., 2007). In octopus it is well known that growth curves have two phases, beginning with an exponential form, and shifting to a low power phase (Forsythe and Van Heukelem, 1987; Semmens et al., 2004). Recently, it was proven that a simple energy ⁎ Corresponding author. Boulevard Brown 2825, Puerto Madryn, Chubut, CP (U9120ACF), Argentina. Tel.: +54 2965 451024. E-mail address: [email protected] (M.J. Klaich). 0044-8486/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.aquaculture.2008.07.054
conservation constraint enforces the shift from the first to second phase. Although further investigations are still required regarding the effect of different rations of food as well as the effect of temperature on the point at which the curve shifts (Grist and Jackson, 2004). In addition, Moltschaniwskyj (2004) pointed out that understanding the form of the growth curve requires an understanding of the growth processes operating at several biological levels, such as organs, muscle fiber, proximal composition and protein synthesis. In octopus, growth is a rectilinear function of food intake level (Van Heukelem, 1976; Joll, 1977; Klaich et al., 2006; Leporati et al., 2007), implying that a) the weight gained for an octopus has no an upper limit, b) there is no additional energetic cost due to increasing food intake and c) the partial growth efficiency has no dependence on food intake. There are two basic parameters which can describe the efficiency of an octopus using food for growth: the gross growth efficiency (GGE), which is the fraction of ingested food used in growth, and the partial growth efficiency (PGE), which is the efficiency of the organism at converting food into growth after maintenance requirements have been satisfied (Kleiber, 1961). However, any value calculated for GGE will vary with circumstances (temperature, locomotor activity, etc.) and is also dependent on food intake level itself (specific dynamic action, maintenance level) (Wells and Clarke, 1996). Although basic, the relationship between GGE and food intake level (FI) have received little attention in the octopus literature. Van Heukelem (1976) observed in Octopus maya and Octopus cyanea that GGE increase with increasing FI and that GGE reaches a maximum
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value approaching PGE. These observations support Warren's (1971) hypothesis in which states that as FI increases, GGE must increase asymptotically from zero at the MaL value toward the PGE value. Although linear, exponential, logarithmic and parabolic models were fitted to O. maya and O. cyanea GGE vs. FI data, no satisfactory model was found in the sense that none showed x-intercept values close to the estimated MaL values from the growth vs. FI function (Van Heukelem, 1976). The author pointed out that sparse data could have been a possible problem in identifying the best model. Semmens et al. (2004) indicated that cephalopods appear to exhibit a great deal of individual variation in growth rate and final size, even within groups or siblings reared under identical conditions. Because of this, choosing the best model from a set of candidates for a specific data set becomes a difficult task. Burnham and Anderson (2002) indicate that given a set of models, specified independently of the sample data, formal inferences based on the entire set of models can be made. In addition, model-averaged parameter estimates can be easily computed. Recently, Katsanevakis et al. (2007) addressed the oxygen consumption of Pachygrapsus marmoratus using a multimodel inference (MMI) approach and recommended MMI as an effective method in finding a parsimonious approximating model. The “Pulpito”, Octopus tehuelchus (d'Orbigny 1834), has marked seasonal growth, increasing its food intake activity and growth rates as temperature increases and decreasing these rates when temperature decreases. Temperature means are 10 ± 1 °C for the colder months and 15 ± 1 °C for the warmer months. Their lifespan has a duration of 18 months and the dorsal mantle length and body weight of first maturation for males and females are 38 mm–18 g and 52 mm–40 g respectively (Ré, 1989). Klaich et al. (2006) have studied the effects of temperature and sex on growth, food intake and GGE in juveniles of O. tehuelchus under laboratory conditions. However, no attempt was made to model the relationship between GGE and FI. This work is aimed at providing a method to investigate that relationship in order to improve the culture or rearing of octopuses. Maximizing the growth efficiency of individuals helps to avoid the waste of supply (i.e. food), work and money. The prime objective of this work is to use model selection and multimodel inference to investigate the nature of the relationship between the GGE and FI in O. tehuelchus. This relationship is studied on the basis of the data support for hypotheses about the dependence of both the PGE and MaL on FI. Combinations of hypotheses about PGE and MaL will allow to identify the asymptotic or linear nature for the relationship between GGE and FI.
2.1. Model building The prime model was built using the GGE definition and the relationship between growth and FI. The GGE is defined as the fraction or the percentage of FI that is converted into body mass (growth) and can be expressed as: ΔBW FI
ð1Þ
where ΔBW is growth or the change in body weight in a fixed period of time. The units of ΔBW and FI can be expressed in grams, calories or any equivalent units which express either growth or an amount of food intake. For this study the wet weight of ingested food was used as well as the change in body weight. The rectilinear relationship between ΔBW and FI can be expressed as: ΔBW ¼ PGE⁎ðFI−MaLÞ
where MaL is the maintenance level, which is the amount of FI to maintain a constant body weight (i.e. ΔBW = 0); and PGE is the partial growth efficiency which follows the definition of Kleiber (1961). Thus, by substituting ΔBW of Eq. (2) in Eq. (1), we obtain: GGE ¼
PGE⁎ðFI−MaLÞ : FI
ð3Þ
Note that GGE assumes that PGE and MaL are independent of FI. The dependency of PGE and MaL on FI will be tested by constructing candidate models using PGE and MaL as either linear or exponential functions of FI. Table 1 resumes the functions and its associated hypotheses about the dependence of PGE and MaL on FI. Hypotheses associated with PGE0 and MaL0 implied in terms of PGE, that the cost of growing does not change with FI, while MaL0 assumes that maintenance level is independent of FI. The other hypotheses state that there is a dependence of PGE and MaL on FI, and this dependence could be linear or exponential. From PGE and MaL dependencies on FI stated in Table 1, we obtained 9 candidate models which were considered for model selection and multimodel inference approaching. Models were named using GGEij as general form, where i and j represent the sub-index related to the functions stated for PGE and MaL respectively. 2.2. Data sets used in the study The gross growth efficiency data set used in this study was calculated using both growth and food intake level data set in grams from Klaich et al. (2006). The growth and food intake level data sets were obtained by keeping immature individuals of O. tehuelchus of both sexes under laboratory conditions at 10 °C and 15 °C. Each octopus was fed with one live crab (Cyrtograpsus altimanus), which was chosen randomly from a daily ration between 3% and 17% of octopus's body weight. Growth as a change in body weight in grams (ΔBW) and FI in grams were recorded individually for a fixed time interval of 10 days. The authors concluded that temperature has an effect on GGE. Thus, two data sets: i) GGE and FI from immature octopus at 10 °C (n = 28) and ii) GGE and FI from immature octopus at 15 °C (n = 21) were used to evaluate their data support to each candidate model. Octopus body weight means were 11.8 g (s.e. = 3.33) at 10 °C and 11.06 (s.e. = 3.95) at 15 °C. 2.3. Model selection and multimodel inference The parameters for each model were estimated by means of the maximum likelihood method (Silvey, 1975). Akaike Information Criterion (AIC) values (Akaike, 1973) and AIC for small samples (AICc) (Hurvich and Tsai, 1989) were calculated for each model as follows:
2. Materials and methods
GGE ¼
273
ð2Þ
AICi ¼ −2 ln L ^ θ=y −2Ki AICCi ¼ AICi þ
ð4Þ
2Ki ðKi þ 1Þ : n−Ki −1
ð5Þ
Where the first term of Eq. (4) is the two times negative product of the log-likelihood of model i. In Eq. (5) Ki is the number of parameters Table 1 Functions and their associated hypotheses about the dependence in O. tehuelchus of partial growth efficiency (PGE) and maintenance level (MaL) on food intake (FI) Function
Associated hypotheses
PGE0 = k1 PGE1 = α1 ⁎ FI + α2 PGE2 = β1 ⁎ exp(β2 ⁎ FI) MaL0 = k2 MaL1 = δ1 ⁎ FI + δ2 MaL2 = γ1 ⁎ exp(γ2 ⁎ FI)
The dependence The dependence The dependence The dependence The dependence The dependence
of PGE on FI is null of PGE on FI is linear of PGE on FI is exponential of MaL on FI is null of MaL on FI is linear of MaL on FI is exponential
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Fig. 1. O. tehuelchus, gross growth efficiency (GGE) versus food intake level (FI) for immature octopuses reared at 10 °C (○) and 15 °C (●).
Fig. 2. O. tehuelchus, model-averaged curves for the gross growth efficiency (GGE) versus food intake level (FI) for immature octopuses reared at 10 °C (solid line) and 15 °C (dotted line). The dashed line represents the absolute difference between GGE estimated from 10 °C and 15 °C averaged curves.
for model i and n is the sample size for each data set. From AICc differences (Δi), Akaike weights (wi) (Akaike, 1978) were obtained as: wi ¼
expð−1=2Δi Þ ∑Rr¼1 expð−1=2Δr Þ
ð6Þ
where Δi = AICCi − AICCmin and R is the number of candidate models. For each data set, the R models were ranked by their wi values. The model with the biggest w was considered the model with the best data support. In order to compare the biggest w value with the others (R − 1) wi values, the ratios between the biggest w and the others (R − 1) w values were used as an additional criterion of model selection (Burnham and Anderson, 2002). Multimodel inference was carried out by computing weighted estimate curves for GGE, PGE and MaL as a function of FI. This concept leads to model-averaged estimates (MAE) as: ^ θi θ ¼ ∑Ri¼1 wi ^ where wi are the Akaike weights for model gi, θ i is the estimate value P from the gi model and θ^ denotes the MAE of θ (Burnham and Anderson, 2002). Model-averaged curves and MAE estimates were obtained using the set of candidate models for which the sum of wi was the nearest value to 0.95. Finally, using the model-averaged
Table 2 O. tehuelchus, values of small-sample bias-corrected form of Akaike information criterion (AICc), AICc differences (ΔAICc), Akaike weights (wi) and number of parameters (np) for each of the candidate models and temperatures used T (°C)
Model
AICc
ΔAICc
wi
np
10
GGE00a GGE10a GGE20a GGE01a GGE02a GGE11a GGE12a GGE21 GGE22 GGE00a GGE20a GGE10a GGE02a GGE01a GGE12a GGE21a GGE22 GGE11
2.9708 5.3037 5.3532 5.7099 5.7099 8.2919 8.2973 8.3413 8.3417 18.0991 20.9056 20.9922 21.1529 21.1873 23.7210 24.4056 24.4153 24.4658
0.0000 2.3329 2.3824 2.7391 2.7391 5.3210 5.3265 5.3705 5.3709 0.0000 2.8065 2.8931 3.0538 3.0882 5.6219 6.3065 6.3162 6.3667
0.4167 0.1298 0.1266 0.1059 0.1059 0.0291 0.0291 0.0284 0.0284 0.4765 0.1171 0.1122 0.1035 0.1017 0.0287 0.0204 0.0203 0.0197
2 3 3 3 3 4 4 4 4 2 3 3 3 3 4 4 4 4
15
a
Candidate models used for multimodel approaching.
curves for GGE at 10 °C and 15 °C, a third curve representing the absolute difference between GGE at 10 °C and 15 °C (adGGE) was calculated. 3. Results Fig. 1 shows GGE vs. FI for immature octopuses kept at 10 °C and 15 °C. Maximum values for 10 °C and 15 °C were 77% and 82% respectively. Data of GGE at 15 °C seem skewed to the right as compared to those GGE values recorded at 10 °C. This supports the fact that metabolic costs are higher for octopuses kept at 15 °C. Model selection indicated that no model with strong data support (i.e. wi N 0.9) was observed for either the 10 °C nor 15 °C data sets (Table 2). Model GGE00 showed the highest Akaike's weights (0.4167 and 0.4765) for 10 °C and 15 °C respectively. Although these results support the hypothesis that PGE and MaL are independent of FI, the required sum of wi (0.95) was obtained using 77% of the original set of candidate models for the 10 °C and 15 °C data sets (Table 2). Thus, MMI was carried out using 7 models from the original set of 9 candidate models (Table 2). Fig. 2 shows the model-averaged curves for GGE measured at 10 °C and 15 °C. GGE shows a positive dependence of FI and increases
Table 3 O. tehuelchus, the estimated model parameters of the 9 candidate models GGEij (as defined in Table 1) and temperatures used T (°C)
Model
Parameters
10
GGE00 GGE01 GGE02 GGE10 GGE11 GGE12 GGE20 GGE21 GGE22 GGE00 GGE01 GGE02 GGE10 GGE11 GGE12 GGE20 GGE21 GGE22
k1 = 0.7115; k2 = 0.4939 k1 = 0.7343; δ1 = 0.0310; δ2 = 0.4786 k1 = 0.7115; γ1 = 0.4939; γ2 = 0.0000 α1 = 0.0955; α2 = 0.4445; k1 = 0.3555 α1 = 0.1100; α2 = 0.5119; δ1 = 0.1317; δ2 = 0.3086 α1 = 0.1021; α2 = 0.3511; γ1 = 0.4689; γ2 = − 0.6969 β1 = 0.5067; β2 = 0.1244; k2k2 = 0.3892 β1 = 0.5067; β2 = 0.1244; δ1 = 0.0000; δ2 = 0.3892 β1 = 0.4859; β2 = 0.1265; γ1 = 0.4049; γ2 = −0.1066 k1 = 0.8764; k2 = 1.1633 k1 = 0.8382; δ1 = −0.0457; δ2 = 1.2164 k1 = 1.0106; γ1 = 1.0319; γ2 = 0.1026 α1 = −0.0649; α2 = 1.0937; k1 = 1.1802 α1 = −0.2159; α2 = 3.1819; δ1 = 0.6363; δ2 = 0.4402 α1 = 0.0109; α2 = 0.5321; γ1 = 5.1364; γ2 = −1.1742 β1 = 1.2898; β2 = −0.1101; k2 = 1.2203 β1 = 0.4276; β2 = −0.1101; δ1 = −2.0165; δ2 = 3.6810 β1 = 0.4008; β2 = 0.1019; γ1 = 3.3169; γ2 = −0.9381
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asymptotically as the FI level increases. GGE took positive values when FI was higher than 0.46 g for immature octopuses kept at 10 °C and 1.18 g for those kept at 15 °C. As FI increased, GGE of both temperatures tended to take a common value between 0.65 (at 10 °C) and 0.63 (at 15 °C). The model-averaged curve of adGGE indicated that as FI increases, the absolute difference decreases as a negative potential function of FI (Fig. 2). For FI higher than 3 g the adGGE reached a minimum of 0.03. Table 3 shows the parameter values obtained for all candidate models and data sets. Fig. 3 shows the model-averaged curves for PGE (Fig. 3a) and MaL (Fig. 3b) for octopuses kept at 10 °C and 15 °C. For octopuses kept at 10 °C PGE showed a positive and linear dependence on FI while for octopuses kept at 15 °C this dependence was negative and linear (Fig. 3a). For octopuses kept at 10 °C the dependence of MaL on FI was null while for octopuses kept at 15 °C MaL was slightly higher when FI was close to zero, though MaL remained constant at higher values of FI (Fig. 3b). The mean values for MaL were 0.46 g and 1.18 g for immature octopuses kept at 10 °C and 15 °C respectively. These MaL means expressed as a percentage of body weight were 3.89% and 10.66% for 10 °C and 15 °C respectively. 4. Discussion In O. tehuelchus, GGE is a positive-asymptotic function of FI. No model was observed with strong data support (wi′ N 0.9) allowing for based inference on a single model. In this case, model averaging
Fig. 3. O. tehuelchus, model-averaged curves for both the partial growth efficiency (PGE) (a) and maintenance level (MaL) (b) versus food intake level (FI) for immature octopuses kept at 10 °C (solid line) and 15 °C (dotted line).
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estimation allowed the consideration of the contribution of every single model for estimating MaL and PGE values. Wells and Clarke (1996) stated that in cephalopods the GGE is dependent of FI and Van Heukelem (1976) indicated that in O. maya and O. cyanea the GGE increases as FI increases. For the last case, the problem for formalizing a function to explain how GGE changes with FI was that no simple model was able to estimate a MaL near those obtained by means of the growth-FI function (Van Heukelem, 1976). In O. tehuelchus the response of GGE to FI follows the same pattern observed by Van Heukelem (1976) for O. maya and O. cyanea; and was formalized by means of a model-averaged curve instead of a single model. Moreover, the model-averaged estimates of MaL at 10 °C and 15 °C for immature octopuses agree with those obtained by Klaich et al. (2006) for O. tehuelchus using the growth vs. FI relationship. Few works involving cephalopods species have dealt with the relationship between GGE and FI. From laboratory experiments Hirtle et al. (1981) found that in Illex illecebrosus the relation between growth and FI could be positive-asymptotic. Assuming this as true, GGE should reach a maximum value before decreasing toward a lower asymptotic GGE. Koueta and Boucaud-Camou (2001) found in Sepia officinalis that after reaching a maximum the GGE decreases as FI increases. Note that similar to I. illecebrosus, S. officinalis show a positive-asymptotic function for growth vs. FI while for octopus species like O. maya and O. cyanea (Van Heukelem, 1976); O. tetricus (Joll, 1977), O. tehuelchus (Klaich et al., 2006) and O. pallidus (Leporati et al., 2007), this relationship is rectilinear. Possibly, the relationship between growth vs. FI (linear or asymptotic) and the cost of feeding influence the proportion of food, which is finally used in growth. As for Sepia officinalis, fishes show high GGE values at low levels of FI decreasing GGE values for larger amounts of FI. Brett and Groves (1979) stated that in fishes the increment of food intake level lead to elevated energetic cost for activity, which reaches 25% of food intake. In addition, FI is a positive-asymptotic function of swimming activity (Ware, 1975); and the energetic costs (i.e. specific dynamic action) processing the ingested food increase as well (Waterlow, 1980). The results obtained in the present work indicate that in O. tehuelchus, the asymptotic value of GGE can be reached when FI tends to large values, possibly near to a state of satiability. At this point the adGGE indicated a little difference between GGE measured at 10 °C and 15 °C. This leads to the conclusion that in the normal range of temperature, O. tehuelchus has the potential to reach a maximum GGE by ingesting large amounts of food. Katsanevakis et al. (2004) found in O. vulgaris that if the available food is restricted, a temperature of 20 °C favors growth in relation to a temperature of 28 °C. The author stated that it is largely because of the equity of the Specific Dynamic Action (SDA) magnitude between the two temperatures. In addition, Klaich et al. (2006) proved that in O. tehuelchus for an equal amount of FI immature octopuses under 10 °C showed a higher growth rate than those at 15 °C. The authors concluded that for an equal amount of ingested food the GGE is higher for immature octopuses kept at lower temperatures. This effect on GGE is particularly true only when octopuses show FI values near to MaL of individuals kept at higher temperatures. At this point, the portion used in growth is smaller for octopuses kept at higher temperatures than those kept at lower ones. In the present work the highest differences between GGE of both temperatures were observed at this point. However, when FI can either reach larger values or it is possible to feed the animals until satiability, the portion of the total FI that is represented by MaL decreases. This condition increases the proportion of food used in growth after covering the maintenance costs leading GGE to reach its maximum at the PGE value. This is possible only if MaL remains either constant or with little change over the whole range of FI for a given temperature. Regarding the result for O. tehuelchus, MaL showed little change over the entire range of FI. For individuals of O. vulgaris, Mangold and Boletzky (1973) fed ad libitum and held at 10 °C, 15 °C and 20 °C, recording mean values for GGE of 56%, 55% and 48%
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respectively. The similarity between GGE values could be explained because animals were fed ad libitum allowing GGE to reach its maximum for each temperature. Although the tendency of GGE to have higher values for lower temperatures remains. Moreover, Aguado Giménez and García García (2002), by means of a multiple regression analysis, observed in O. vulgaris that GGE increased from 13 °C to an optimal temperature value at 16.5 °C. After this point, GGE decreased to zero at 23 °C. This tendency of GGE could be interpreted as a thermal effect, in which MaL increases faster than FI. Recently, André et al. (2006) found in O. pallidus that fluctuations in FI did not appear to affect GGE. The authors pointed out that this negative crosscorrelation suggests that the physiological mechanism underlying food conversion may have an immediate antagonist effect on feeding. This was not observed in O. tehuelchus although additional studies are required for testing such hypotheses. However, in this study, some individuals of O. tehuelchus show no food intake for a single day over each time interval of 10 days. This could have a negative effect on the estimation of GGE. The ability of O. tehuelchus to maintain a stable MaL value for any value of FI could be interpreted as a metabolic strategy, in which GGE and growth rate are maximized during periods of both high temperature and feeding activity. Because other octopus species cited above exhibited a rectilinear function for growth vs. FI, the same rule may apply for GGE vs. FI observed in O. tehuelchus. In terms of rearing or culture activities of octopus, feeding ad libitum (i.e. satiability) seems to be the best option in order to reach optimum values of GGE and growth rate. However, the satiability state is closely related to the value of MaL for each species and temperature. In addition to the methods proposed by Wells and Clarke (1996), the MaL could be estimated by means of a model-averaged curve for GGE vs. FI applied in O. tehuelchus as demonstrated in present work. Finally, in this work, the relationship between GGE and FI was studied for O. tehuelchus. The multimodel inference approach proved to be an efficient tool when no single model was capable of explaining the relationship between GGE vs. FI. Multimodel inference allowed every single model contribution to be taken into account from the entire set of candidates. Experimental work in combination with multimodel inference appears to be a useful alternative tool for taking into account the high variability observed in cephalopod data; and testing hypotheses about both growth and feeding processes in other aquatic animals. Acknowledgments The authors thank Sarah O'Neal for her English proofreading and PhD Stelios Katsanevakis for his valuable comments on the earlier version of this manuscript. They are also grateful to two anonymous reviewers and the section editor for their helpful comments. References Aguado Giménez, F., García García, B., 2002. Growth and food intake models in Octopus vulgaris Cuvier (1797): influence of body weight, temperature, sex and diet. Aquac. Int. 10, 361–377.
Akaike, H., 1973. Information theory and an extension of the maximum likelihood principle. In: Petrov, B.N., Csaki, F. (Eds.), Second International Symposium on Information Theory. Akademiai Kiado, Budapest, pp. 267–281. Akaike, H., 1978. A Bayesian analysis of the minimum AIC procedure. Ann. Inst. Stat. Math. 30, 9–14. André, J., Pecl, G.T., Semmens, J.M., Grist, E.P.M., 2006. Early life-history processes in benthic octopus: relationships between temperature, feeding, food conversion, and growth in juvenile Octopus pallidus. J. Exp. Mar. Biol. Ecol. 354, 81–92. Brett, J.R., Groves, T.D.D., 1979. Physiological energetics. In: Hoar, W.S., Randall, D.J., Brett, J.R. (Eds.), Fish Physiology. Vol. III. Academic Press, New York, pp. 279–352. Burnham, K.P., Anderson, D.R., 2002. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd edn. Springer, New York. 488 pp. Domain, F., Jouffre, D., Caveriviére, A., 2000. Growth of Octopus vulgaris of tagging in Senegalese water. J. Mar. Biol. Assoc. UK 80, 699–706. Forsythe, J.W., 1984. Octopus joubini (Mollusca: Cephalopoda): a detailed study of growth through the full life cycle in a closed see water system. J. Zool. 202, 393–417. Forsythe, J.W., Van Heukelem, W.F., 1987. Growth. In: Boyle, P.R. (Ed.), Cephalopod Life Cycles. Vol. II Comparative Reviews. Academic Press, London, U.K., pp. 135–156. Forsythe, J.W., Toll, R.B., 1991. Clarification of the Western Atlantic Ocean pygmy octopus complex: the identity and life history of Octopus joubini (Cephalopoda: Octopodinae). Bull. Mar. Sci. 49, 88–97. Grist, E.P.M., Jackson, G.D., 2004. Energy balance as a determinant of two-phase growth in cephalopods. Mar. Freshw. Res. 55, 395–401. Hartwick, E.B., Ambrose, R.F., Robinson, S.M.C., 1984. Dynamics of shallow-water populations of Octopus dofleini. Mar. Biol. 82, 65–72. Hirtle, R.W.M., DeMont, M.E., O'Dor, R.K., 1981. Feeding, growth, and metabolic rates in captive short finned squid, Illex illecebrosus, in relation to the natural population. J. Shellfish Res. 1 (2), 187–192. Hurvich, C.M., Tsai, C.L., 1989. Regression and time series model selection in small samples. Biometrika 76, 297–307. Joll, L.M., 1977. Growth and food intake of Octopus tetricus (Mollusca: Cephalopoda) in aquaria. Aust. J. Mar. Freshw. Res. 28, 45–56. Katsanevakis, S., Protopapas, N., Miliou, H., Verriopoulos, G., 2004. Effect of temperature on specific dynamic action in the common octopus, Octopus vulgaris (Cephalopoda). Mar. Biol. 146, 733–738. Katsanevakis, S., Xanthopoulos, J., Protopapas, N., Verriopoulos, G., 2007. Oxygen consumption of the semi-terrestrial crab Pachygrapsus marmoratus in relation to body mass and temperature: an information theory approach. Mar. Biol. 151, 343–352. Klaich, M.J., Ré, M.E., Pedraza, S.N., 2006. Effect of temperature, sexual maturity and sex on growth, food intake and gross growth efficiency in the “pulpito” Octopus tehuelchus (d'Orbigny, 1834). J. Shellfish Res. 25 (3), 979–986. Kleiber, M., 1961. The Fire of Life, an Introduction to Animal Energetics, John Wiley & Sons, Inc., New York, USA. 453 pp. Koueta, N., Boucaud-Camou, E., 2001. Basic growth relations in experimental rearing of early juvenile cuttlefish Sepia officinalis L. (Mollusca: Cephalopoda). J. Exp. Mar. Biol. Ecol. 265, 75–87. Leporati, S.C., Pecl, G.T., Semmens, J.M., 2007. Cephalopod hatchling growth: the effects of initial size and seasonal temperatures. Mar. Biol. 151, 1375–1383. Mangold, K., Boletzky, S.v., 1973. New data on reproductive biology and growth of Octopus vulgaris. Mar. Biol. 19, 7–12. Moltschaniwskyj, N.A., 2004. Understanding the process of growth in cephalopods. Mar. Freshw. Res. 55, 379–386. O'Dor, R.K., Webber, D.M., 1986. The constraints on cephalopods: why squid aren't fish. Can. J. Zool. 64, 1591–1605. Ré, M.E., 1989. Estudios ecológicos sobre el crecimiento y la alimentación de Octopus tehuelchus d'Orbigny en Puerto Lobos, golfo San Matías. Tesis Doctoral, Facultad de Ciencias Naturales y Museo de la Universidad Nacional de La Plata. Semmens, J.M., Pecl, G.T., Villanueva, R., Jouffre, D., Sobrino, I., Wood, J.B., Rigby, P.R., 2004. Understanding octopus growth: patterns, variability and physiology. Mar. Freshw. Res. 55, 367–377. Silvey, S.D., 1975. Statistical Inference. Chapman and Hall, London. Van Heukelem, W.F., 1976. Growth, bioenergetics and life-span of Octopus cyanea and Octopus maya. PhD Thesis, University of Hawaii. Ware, D.M., 1975. Growth, metabolism and optimal swimming speed of pelagic fish. J. Fish. Res. Board Can. 32, 33. Warren, C.E., 1971. Biology and Water Pollution Control. Philadelphia., 434 pp. Waterlow, J.C., 1980. Protein turnover in the whole animal. Invest. Cell Pathol. 3, 107. Wells, M.J., Clarke, A., 1996. The costs of living and reproducing for an individual cephalopod. Philos. Trans. R. Soc. Lond. B. 351, 1083–1104.
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Aquaculture j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a q u a - o n l i n e
Gross growth efficiency as a function of food intake level in the “Pulpito” Octopus tehuelchus: A multimodel inference application M.J. Klaich a,⁎, M.E. Ré b, S.N. Pedraza a,b a b
Universidad Nacional de la Patagonia San Juan Bosco, Boulevard Brown 3700, Sede Puerto Madryn, Chubut, Argentina Centro Nacional Patagónico, Boulevard Brown 2825, Puerto Madryn, Chubut, Argentina
a r t i c l e
i n f o
Article history: Received 5 February 2008 Received in revised form 3 May 2008 Accepted 30 July 2008 Keywords: Gross growth efficiency Food intake level Multimodel inference Octopus tehuelchus
a b s t r a c t Multimodel inference theory was used to investigate the relationship between gross growth efficiency (GGE) and food intake level (FI) in Octopus tehuelchus for immature octopus kept at 10 °C and 15 °C. Multimodel inference was carried out using the model averaging methodology over the entire set of candidate models. The main results indicated that in O. tehuelchus GGE is an increasing and asymptotic function of FI showing a maximum approaching the partial growth efficiency (PGE). The model-averaged estimates of PGE and maintenance level (MaL) did not differ from those estimated using the rectilinear relationship between growth and FI in previous work. For FI higher than MaL, both GGE and PGE were higher in immature octopus at 10 °C than octopus at 15 °C. It is noteworthy that as FI increases to a satiability state, the difference between GGE at 10 °C and 15 °C decreases as a potential function of FI. These results would indicate that O. tehuelchus maximizes the energy used in growth by keeping a constant MaL for any value of FI. Multimodel inference theory proved to be an efficient tool to build averaged functions taking each candidate model into account. On the other hand, the present work presents a method to understand the intensity feeding needed to reach an optimum growth efficiency, avoiding wasting of food, work energy and investment for octopus rearing or culture activities. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Cephalopods are among the most efficient animals using food for growing. This is reflected by their high instantaneous growth rates, feeding rates and food conversion indices. Such growth is promoted by efficient feeding and a protein-based metabolism that converts food into growth rather than storage (O'Dor and Webber, 1986). The most important features of octopus growth and its researching challenges have been reviewed by Semmens et al. (2004). The manner and efficiency with which an octopus invests the energy taken from its environment in growth, is among the most important topics of octopus growth research. Laboratory experiments have proved to be the best tool for studying octopus growth. Several studies have been carried out investigating the form of octopus growth curves as a time function (Van Heukelem, 1976; Joll, 1977; Forsythe, 1984; Hartwick et al., 1984; Forsythe and Toll, 1991; Domain et al., 2000), although few have studied the octopus growth as a function of food intake level (Van Heukelem, 1976; Joll, 1977; Klaich et al., 2006; Leporati et al., 2007). In octopus it is well known that growth curves have two phases, beginning with an exponential form, and shifting to a low power phase (Forsythe and Van Heukelem, 1987; Semmens et al., 2004). Recently, it was proven that a simple energy ⁎ Corresponding author. Boulevard Brown 2825, Puerto Madryn, Chubut, CP (U9120ACF), Argentina. Tel.: +54 2965 451024. E-mail address: [email protected] (M.J. Klaich). 0044-8486/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.aquaculture.2008.07.054
conservation constraint enforces the shift from the first to second phase. Although further investigations are still required regarding the effect of different rations of food as well as the effect of temperature on the point at which the curve shifts (Grist and Jackson, 2004). In addition, Moltschaniwskyj (2004) pointed out that understanding the form of the growth curve requires an understanding of the growth processes operating at several biological levels, such as organs, muscle fiber, proximal composition and protein synthesis. In octopus, growth is a rectilinear function of food intake level (Van Heukelem, 1976; Joll, 1977; Klaich et al., 2006; Leporati et al., 2007), implying that a) the weight gained for an octopus has no an upper limit, b) there is no additional energetic cost due to increasing food intake and c) the partial growth efficiency has no dependence on food intake. There are two basic parameters which can describe the efficiency of an octopus using food for growth: the gross growth efficiency (GGE), which is the fraction of ingested food used in growth, and the partial growth efficiency (PGE), which is the efficiency of the organism at converting food into growth after maintenance requirements have been satisfied (Kleiber, 1961). However, any value calculated for GGE will vary with circumstances (temperature, locomotor activity, etc.) and is also dependent on food intake level itself (specific dynamic action, maintenance level) (Wells and Clarke, 1996). Although basic, the relationship between GGE and food intake level (FI) have received little attention in the octopus literature. Van Heukelem (1976) observed in Octopus maya and Octopus cyanea that GGE increase with increasing FI and that GGE reaches a maximum
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value approaching PGE. These observations support Warren's (1971) hypothesis in which states that as FI increases, GGE must increase asymptotically from zero at the MaL value toward the PGE value. Although linear, exponential, logarithmic and parabolic models were fitted to O. maya and O. cyanea GGE vs. FI data, no satisfactory model was found in the sense that none showed x-intercept values close to the estimated MaL values from the growth vs. FI function (Van Heukelem, 1976). The author pointed out that sparse data could have been a possible problem in identifying the best model. Semmens et al. (2004) indicated that cephalopods appear to exhibit a great deal of individual variation in growth rate and final size, even within groups or siblings reared under identical conditions. Because of this, choosing the best model from a set of candidates for a specific data set becomes a difficult task. Burnham and Anderson (2002) indicate that given a set of models, specified independently of the sample data, formal inferences based on the entire set of models can be made. In addition, model-averaged parameter estimates can be easily computed. Recently, Katsanevakis et al. (2007) addressed the oxygen consumption of Pachygrapsus marmoratus using a multimodel inference (MMI) approach and recommended MMI as an effective method in finding a parsimonious approximating model. The “Pulpito”, Octopus tehuelchus (d'Orbigny 1834), has marked seasonal growth, increasing its food intake activity and growth rates as temperature increases and decreasing these rates when temperature decreases. Temperature means are 10 ± 1 °C for the colder months and 15 ± 1 °C for the warmer months. Their lifespan has a duration of 18 months and the dorsal mantle length and body weight of first maturation for males and females are 38 mm–18 g and 52 mm–40 g respectively (Ré, 1989). Klaich et al. (2006) have studied the effects of temperature and sex on growth, food intake and GGE in juveniles of O. tehuelchus under laboratory conditions. However, no attempt was made to model the relationship between GGE and FI. This work is aimed at providing a method to investigate that relationship in order to improve the culture or rearing of octopuses. Maximizing the growth efficiency of individuals helps to avoid the waste of supply (i.e. food), work and money. The prime objective of this work is to use model selection and multimodel inference to investigate the nature of the relationship between the GGE and FI in O. tehuelchus. This relationship is studied on the basis of the data support for hypotheses about the dependence of both the PGE and MaL on FI. Combinations of hypotheses about PGE and MaL will allow to identify the asymptotic or linear nature for the relationship between GGE and FI.
2.1. Model building The prime model was built using the GGE definition and the relationship between growth and FI. The GGE is defined as the fraction or the percentage of FI that is converted into body mass (growth) and can be expressed as: ΔBW FI
ð1Þ
where ΔBW is growth or the change in body weight in a fixed period of time. The units of ΔBW and FI can be expressed in grams, calories or any equivalent units which express either growth or an amount of food intake. For this study the wet weight of ingested food was used as well as the change in body weight. The rectilinear relationship between ΔBW and FI can be expressed as: ΔBW ¼ PGE⁎ðFI−MaLÞ
where MaL is the maintenance level, which is the amount of FI to maintain a constant body weight (i.e. ΔBW = 0); and PGE is the partial growth efficiency which follows the definition of Kleiber (1961). Thus, by substituting ΔBW of Eq. (2) in Eq. (1), we obtain: GGE ¼
PGE⁎ðFI−MaLÞ : FI
ð3Þ
Note that GGE assumes that PGE and MaL are independent of FI. The dependency of PGE and MaL on FI will be tested by constructing candidate models using PGE and MaL as either linear or exponential functions of FI. Table 1 resumes the functions and its associated hypotheses about the dependence of PGE and MaL on FI. Hypotheses associated with PGE0 and MaL0 implied in terms of PGE, that the cost of growing does not change with FI, while MaL0 assumes that maintenance level is independent of FI. The other hypotheses state that there is a dependence of PGE and MaL on FI, and this dependence could be linear or exponential. From PGE and MaL dependencies on FI stated in Table 1, we obtained 9 candidate models which were considered for model selection and multimodel inference approaching. Models were named using GGEij as general form, where i and j represent the sub-index related to the functions stated for PGE and MaL respectively. 2.2. Data sets used in the study The gross growth efficiency data set used in this study was calculated using both growth and food intake level data set in grams from Klaich et al. (2006). The growth and food intake level data sets were obtained by keeping immature individuals of O. tehuelchus of both sexes under laboratory conditions at 10 °C and 15 °C. Each octopus was fed with one live crab (Cyrtograpsus altimanus), which was chosen randomly from a daily ration between 3% and 17% of octopus's body weight. Growth as a change in body weight in grams (ΔBW) and FI in grams were recorded individually for a fixed time interval of 10 days. The authors concluded that temperature has an effect on GGE. Thus, two data sets: i) GGE and FI from immature octopus at 10 °C (n = 28) and ii) GGE and FI from immature octopus at 15 °C (n = 21) were used to evaluate their data support to each candidate model. Octopus body weight means were 11.8 g (s.e. = 3.33) at 10 °C and 11.06 (s.e. = 3.95) at 15 °C. 2.3. Model selection and multimodel inference The parameters for each model were estimated by means of the maximum likelihood method (Silvey, 1975). Akaike Information Criterion (AIC) values (Akaike, 1973) and AIC for small samples (AICc) (Hurvich and Tsai, 1989) were calculated for each model as follows:
2. Materials and methods
GGE ¼
273
ð2Þ
AICi ¼ −2 ln L ^ θ=y −2Ki AICCi ¼ AICi þ
ð4Þ
2Ki ðKi þ 1Þ : n−Ki −1
ð5Þ
Where the first term of Eq. (4) is the two times negative product of the log-likelihood of model i. In Eq. (5) Ki is the number of parameters Table 1 Functions and their associated hypotheses about the dependence in O. tehuelchus of partial growth efficiency (PGE) and maintenance level (MaL) on food intake (FI) Function
Associated hypotheses
PGE0 = k1 PGE1 = α1 ⁎ FI + α2 PGE2 = β1 ⁎ exp(β2 ⁎ FI) MaL0 = k2 MaL1 = δ1 ⁎ FI + δ2 MaL2 = γ1 ⁎ exp(γ2 ⁎ FI)
The dependence The dependence The dependence The dependence The dependence The dependence
of PGE on FI is null of PGE on FI is linear of PGE on FI is exponential of MaL on FI is null of MaL on FI is linear of MaL on FI is exponential
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Fig. 1. O. tehuelchus, gross growth efficiency (GGE) versus food intake level (FI) for immature octopuses reared at 10 °C (○) and 15 °C (●).
Fig. 2. O. tehuelchus, model-averaged curves for the gross growth efficiency (GGE) versus food intake level (FI) for immature octopuses reared at 10 °C (solid line) and 15 °C (dotted line). The dashed line represents the absolute difference between GGE estimated from 10 °C and 15 °C averaged curves.
for model i and n is the sample size for each data set. From AICc differences (Δi), Akaike weights (wi) (Akaike, 1978) were obtained as: wi ¼
expð−1=2Δi Þ ∑Rr¼1 expð−1=2Δr Þ
ð6Þ
where Δi = AICCi − AICCmin and R is the number of candidate models. For each data set, the R models were ranked by their wi values. The model with the biggest w was considered the model with the best data support. In order to compare the biggest w value with the others (R − 1) wi values, the ratios between the biggest w and the others (R − 1) w values were used as an additional criterion of model selection (Burnham and Anderson, 2002). Multimodel inference was carried out by computing weighted estimate curves for GGE, PGE and MaL as a function of FI. This concept leads to model-averaged estimates (MAE) as: ^ θi θ ¼ ∑Ri¼1 wi ^ where wi are the Akaike weights for model gi, θ i is the estimate value P from the gi model and θ^ denotes the MAE of θ (Burnham and Anderson, 2002). Model-averaged curves and MAE estimates were obtained using the set of candidate models for which the sum of wi was the nearest value to 0.95. Finally, using the model-averaged
Table 2 O. tehuelchus, values of small-sample bias-corrected form of Akaike information criterion (AICc), AICc differences (ΔAICc), Akaike weights (wi) and number of parameters (np) for each of the candidate models and temperatures used T (°C)
Model
AICc
ΔAICc
wi
np
10
GGE00a GGE10a GGE20a GGE01a GGE02a GGE11a GGE12a GGE21 GGE22 GGE00a GGE20a GGE10a GGE02a GGE01a GGE12a GGE21a GGE22 GGE11
2.9708 5.3037 5.3532 5.7099 5.7099 8.2919 8.2973 8.3413 8.3417 18.0991 20.9056 20.9922 21.1529 21.1873 23.7210 24.4056 24.4153 24.4658
0.0000 2.3329 2.3824 2.7391 2.7391 5.3210 5.3265 5.3705 5.3709 0.0000 2.8065 2.8931 3.0538 3.0882 5.6219 6.3065 6.3162 6.3667
0.4167 0.1298 0.1266 0.1059 0.1059 0.0291 0.0291 0.0284 0.0284 0.4765 0.1171 0.1122 0.1035 0.1017 0.0287 0.0204 0.0203 0.0197
2 3 3 3 3 4 4 4 4 2 3 3 3 3 4 4 4 4
15
a
Candidate models used for multimodel approaching.
curves for GGE at 10 °C and 15 °C, a third curve representing the absolute difference between GGE at 10 °C and 15 °C (adGGE) was calculated. 3. Results Fig. 1 shows GGE vs. FI for immature octopuses kept at 10 °C and 15 °C. Maximum values for 10 °C and 15 °C were 77% and 82% respectively. Data of GGE at 15 °C seem skewed to the right as compared to those GGE values recorded at 10 °C. This supports the fact that metabolic costs are higher for octopuses kept at 15 °C. Model selection indicated that no model with strong data support (i.e. wi N 0.9) was observed for either the 10 °C nor 15 °C data sets (Table 2). Model GGE00 showed the highest Akaike's weights (0.4167 and 0.4765) for 10 °C and 15 °C respectively. Although these results support the hypothesis that PGE and MaL are independent of FI, the required sum of wi (0.95) was obtained using 77% of the original set of candidate models for the 10 °C and 15 °C data sets (Table 2). Thus, MMI was carried out using 7 models from the original set of 9 candidate models (Table 2). Fig. 2 shows the model-averaged curves for GGE measured at 10 °C and 15 °C. GGE shows a positive dependence of FI and increases
Table 3 O. tehuelchus, the estimated model parameters of the 9 candidate models GGEij (as defined in Table 1) and temperatures used T (°C)
Model
Parameters
10
GGE00 GGE01 GGE02 GGE10 GGE11 GGE12 GGE20 GGE21 GGE22 GGE00 GGE01 GGE02 GGE10 GGE11 GGE12 GGE20 GGE21 GGE22
k1 = 0.7115; k2 = 0.4939 k1 = 0.7343; δ1 = 0.0310; δ2 = 0.4786 k1 = 0.7115; γ1 = 0.4939; γ2 = 0.0000 α1 = 0.0955; α2 = 0.4445; k1 = 0.3555 α1 = 0.1100; α2 = 0.5119; δ1 = 0.1317; δ2 = 0.3086 α1 = 0.1021; α2 = 0.3511; γ1 = 0.4689; γ2 = − 0.6969 β1 = 0.5067; β2 = 0.1244; k2k2 = 0.3892 β1 = 0.5067; β2 = 0.1244; δ1 = 0.0000; δ2 = 0.3892 β1 = 0.4859; β2 = 0.1265; γ1 = 0.4049; γ2 = −0.1066 k1 = 0.8764; k2 = 1.1633 k1 = 0.8382; δ1 = −0.0457; δ2 = 1.2164 k1 = 1.0106; γ1 = 1.0319; γ2 = 0.1026 α1 = −0.0649; α2 = 1.0937; k1 = 1.1802 α1 = −0.2159; α2 = 3.1819; δ1 = 0.6363; δ2 = 0.4402 α1 = 0.0109; α2 = 0.5321; γ1 = 5.1364; γ2 = −1.1742 β1 = 1.2898; β2 = −0.1101; k2 = 1.2203 β1 = 0.4276; β2 = −0.1101; δ1 = −2.0165; δ2 = 3.6810 β1 = 0.4008; β2 = 0.1019; γ1 = 3.3169; γ2 = −0.9381
15
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asymptotically as the FI level increases. GGE took positive values when FI was higher than 0.46 g for immature octopuses kept at 10 °C and 1.18 g for those kept at 15 °C. As FI increased, GGE of both temperatures tended to take a common value between 0.65 (at 10 °C) and 0.63 (at 15 °C). The model-averaged curve of adGGE indicated that as FI increases, the absolute difference decreases as a negative potential function of FI (Fig. 2). For FI higher than 3 g the adGGE reached a minimum of 0.03. Table 3 shows the parameter values obtained for all candidate models and data sets. Fig. 3 shows the model-averaged curves for PGE (Fig. 3a) and MaL (Fig. 3b) for octopuses kept at 10 °C and 15 °C. For octopuses kept at 10 °C PGE showed a positive and linear dependence on FI while for octopuses kept at 15 °C this dependence was negative and linear (Fig. 3a). For octopuses kept at 10 °C the dependence of MaL on FI was null while for octopuses kept at 15 °C MaL was slightly higher when FI was close to zero, though MaL remained constant at higher values of FI (Fig. 3b). The mean values for MaL were 0.46 g and 1.18 g for immature octopuses kept at 10 °C and 15 °C respectively. These MaL means expressed as a percentage of body weight were 3.89% and 10.66% for 10 °C and 15 °C respectively. 4. Discussion In O. tehuelchus, GGE is a positive-asymptotic function of FI. No model was observed with strong data support (wi′ N 0.9) allowing for based inference on a single model. In this case, model averaging
Fig. 3. O. tehuelchus, model-averaged curves for both the partial growth efficiency (PGE) (a) and maintenance level (MaL) (b) versus food intake level (FI) for immature octopuses kept at 10 °C (solid line) and 15 °C (dotted line).
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estimation allowed the consideration of the contribution of every single model for estimating MaL and PGE values. Wells and Clarke (1996) stated that in cephalopods the GGE is dependent of FI and Van Heukelem (1976) indicated that in O. maya and O. cyanea the GGE increases as FI increases. For the last case, the problem for formalizing a function to explain how GGE changes with FI was that no simple model was able to estimate a MaL near those obtained by means of the growth-FI function (Van Heukelem, 1976). In O. tehuelchus the response of GGE to FI follows the same pattern observed by Van Heukelem (1976) for O. maya and O. cyanea; and was formalized by means of a model-averaged curve instead of a single model. Moreover, the model-averaged estimates of MaL at 10 °C and 15 °C for immature octopuses agree with those obtained by Klaich et al. (2006) for O. tehuelchus using the growth vs. FI relationship. Few works involving cephalopods species have dealt with the relationship between GGE and FI. From laboratory experiments Hirtle et al. (1981) found that in Illex illecebrosus the relation between growth and FI could be positive-asymptotic. Assuming this as true, GGE should reach a maximum value before decreasing toward a lower asymptotic GGE. Koueta and Boucaud-Camou (2001) found in Sepia officinalis that after reaching a maximum the GGE decreases as FI increases. Note that similar to I. illecebrosus, S. officinalis show a positive-asymptotic function for growth vs. FI while for octopus species like O. maya and O. cyanea (Van Heukelem, 1976); O. tetricus (Joll, 1977), O. tehuelchus (Klaich et al., 2006) and O. pallidus (Leporati et al., 2007), this relationship is rectilinear. Possibly, the relationship between growth vs. FI (linear or asymptotic) and the cost of feeding influence the proportion of food, which is finally used in growth. As for Sepia officinalis, fishes show high GGE values at low levels of FI decreasing GGE values for larger amounts of FI. Brett and Groves (1979) stated that in fishes the increment of food intake level lead to elevated energetic cost for activity, which reaches 25% of food intake. In addition, FI is a positive-asymptotic function of swimming activity (Ware, 1975); and the energetic costs (i.e. specific dynamic action) processing the ingested food increase as well (Waterlow, 1980). The results obtained in the present work indicate that in O. tehuelchus, the asymptotic value of GGE can be reached when FI tends to large values, possibly near to a state of satiability. At this point the adGGE indicated a little difference between GGE measured at 10 °C and 15 °C. This leads to the conclusion that in the normal range of temperature, O. tehuelchus has the potential to reach a maximum GGE by ingesting large amounts of food. Katsanevakis et al. (2004) found in O. vulgaris that if the available food is restricted, a temperature of 20 °C favors growth in relation to a temperature of 28 °C. The author stated that it is largely because of the equity of the Specific Dynamic Action (SDA) magnitude between the two temperatures. In addition, Klaich et al. (2006) proved that in O. tehuelchus for an equal amount of FI immature octopuses under 10 °C showed a higher growth rate than those at 15 °C. The authors concluded that for an equal amount of ingested food the GGE is higher for immature octopuses kept at lower temperatures. This effect on GGE is particularly true only when octopuses show FI values near to MaL of individuals kept at higher temperatures. At this point, the portion used in growth is smaller for octopuses kept at higher temperatures than those kept at lower ones. In the present work the highest differences between GGE of both temperatures were observed at this point. However, when FI can either reach larger values or it is possible to feed the animals until satiability, the portion of the total FI that is represented by MaL decreases. This condition increases the proportion of food used in growth after covering the maintenance costs leading GGE to reach its maximum at the PGE value. This is possible only if MaL remains either constant or with little change over the whole range of FI for a given temperature. Regarding the result for O. tehuelchus, MaL showed little change over the entire range of FI. For individuals of O. vulgaris, Mangold and Boletzky (1973) fed ad libitum and held at 10 °C, 15 °C and 20 °C, recording mean values for GGE of 56%, 55% and 48%
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respectively. The similarity between GGE values could be explained because animals were fed ad libitum allowing GGE to reach its maximum for each temperature. Although the tendency of GGE to have higher values for lower temperatures remains. Moreover, Aguado Giménez and García García (2002), by means of a multiple regression analysis, observed in O. vulgaris that GGE increased from 13 °C to an optimal temperature value at 16.5 °C. After this point, GGE decreased to zero at 23 °C. This tendency of GGE could be interpreted as a thermal effect, in which MaL increases faster than FI. Recently, André et al. (2006) found in O. pallidus that fluctuations in FI did not appear to affect GGE. The authors pointed out that this negative crosscorrelation suggests that the physiological mechanism underlying food conversion may have an immediate antagonist effect on feeding. This was not observed in O. tehuelchus although additional studies are required for testing such hypotheses. However, in this study, some individuals of O. tehuelchus show no food intake for a single day over each time interval of 10 days. This could have a negative effect on the estimation of GGE. The ability of O. tehuelchus to maintain a stable MaL value for any value of FI could be interpreted as a metabolic strategy, in which GGE and growth rate are maximized during periods of both high temperature and feeding activity. Because other octopus species cited above exhibited a rectilinear function for growth vs. FI, the same rule may apply for GGE vs. FI observed in O. tehuelchus. In terms of rearing or culture activities of octopus, feeding ad libitum (i.e. satiability) seems to be the best option in order to reach optimum values of GGE and growth rate. However, the satiability state is closely related to the value of MaL for each species and temperature. In addition to the methods proposed by Wells and Clarke (1996), the MaL could be estimated by means of a model-averaged curve for GGE vs. FI applied in O. tehuelchus as demonstrated in present work. Finally, in this work, the relationship between GGE and FI was studied for O. tehuelchus. The multimodel inference approach proved to be an efficient tool when no single model was capable of explaining the relationship between GGE vs. FI. Multimodel inference allowed every single model contribution to be taken into account from the entire set of candidates. Experimental work in combination with multimodel inference appears to be a useful alternative tool for taking into account the high variability observed in cephalopod data; and testing hypotheses about both growth and feeding processes in other aquatic animals. Acknowledgments The authors thank Sarah O'Neal for her English proofreading and PhD Stelios Katsanevakis for his valuable comments on the earlier version of this manuscript. They are also grateful to two anonymous reviewers and the section editor for their helpful comments. References Aguado Giménez, F., García García, B., 2002. Growth and food intake models in Octopus vulgaris Cuvier (1797): influence of body weight, temperature, sex and diet. Aquac. Int. 10, 361–377.
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