# Energy budget modeling of the environmental and individual determinism

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## Material and methods

The DEB theory

The present study will present a bioenergetic model based on the Dynamic Energy Budget theory. The DEB theory have been introduced by S.A.L.M. Kooijman in the 1980s (Kooijman and Metz 1984). The aim of this theory is to describe the use of mass and energy of an organism to stay alive and mature in a changing environment during its whole life.
In the standard DEB model, the individual doesn’t change of shape during growth (isomorph) and is modelled as composed by three components, (1) a reserve of energy (in J), (2) a structural volume (in cm3) and (3) the maturity/reproduction buffer (in J). The component named maturity is actually the quantity of energy invested in developing new structures. At the beginning of his life, the individual is only a pool of energy reserve with a quasi-null structural volume and a maturity and a reproduction buffer equal to zero. During the embryo phase, there is no feeding of the individual, the energy from the reserve is allocated to the structure and the maturity. A fraction named κ is allocated to the development of the structure and to its maintenance, with a priority to the maintenance. The rest of the energy allocated go to the increase of the maturity and to its maintenance, with, again, a priority to the maintenance. If maintenance costs are not payed, the individual die. When maturity reach the threshold of birth (in J), the individual became a juvenile. When the individual is juvenile, it begins to feed by himself. So the individual assimilate energy from food and reject it as faeces. And when the maturity is at its maximum value, named (in J), the individual achieves puberty and became an adult. In its adult phase, the individual is at its maximum of maturity, so the energy usually allocated to it is now allocated to the reproduction buffer. In the standard DEB model, in condition of food abundance, the modeled growth of the organism is the same as the Von Bertalanffy growth that is the most widely growth curve used (Bertalanffy 1938). With the Von Bertalanffy growth, the growth in length decrease linearly with the increase of the length of the organism if food and other conditions such as the temperature are constant. As the organism gets bigger, it grow less and less because the increase in maintenance (which is roughly proportional to structural volume) is higher than the increase in mobilization which means that less energy is available for growth. It stops growing when it achieves the ultimate length.
More precisely, the fluxes of energy in the organism are the following: initially the reserve of energy contain an amount of 0 J of energy, a structural volume quasi equal to 0 cm3 and an empty maturity and reproduction buffer (in J). During the embryo phase: the reserve of energy is drain by the mobilization rate of energy ̇ (in J.d-1). In the equation of ̇ (Figure 2), is always the energy reserve of the organism. [ ] (in J.cm 3) is the volume-specific cost of structure, that is the energy needed to build one unit of structural volume. ̇(in cm.d-1) is the energy conductance, 1⁄3 (in cm) is the structural length, κ is the fraction of mobilized reserve allocated to soma and [ ] (in J.cm-3) is the reserve density ( / ). And [ ̇] (in J.d-1.cm-3) is the power used to pay somatic maintenance costs divided by the structural volume. In the equation ̇ 2⁄3 2 [ ̇] -3 -1 of (Figure 2), (in cm ) is the structural surface, (in J.cm .d ) is the specific volume-linked somatic maintenance rate and { ̇} (in J.cm-2.d-1) is the specific surface area-linked somatic maintenance rate. The maturity increase by the power used for maturation ̇ (in J.d-1). In the equation of ̇ (Figure 2), ̇ (in J.d-1) is the power used to pay for maturity maintenance costs that equals the maturity maintenance costs ̇ (in d-1) times the maturity .
When the maturity reach the threshold of birth , the organism become a juvenile and start to feed, as a consequence the dynamic of the energy reserve that was equal to minus the mobilization rate of energy ̇, now become equal to the assimilation rate ̇ (in J.d-1) minus the mobilization rate of energy ̇. In the equation of ̇ (Figure 2), { ̇ } (in J.cm-2.d-1) is the surface-area-specific maximum assimilation rate, that indicates the maximum ingestion rate possible for the organism times the efficiency of assimilation food. ( ) is the functional response, it describe the ingested food and introduce the connection with the available food in the environment. It’s the available food divided by the available food plus the half saturation coefficient (the value of food density when ingestion is half its maximum). In consequence, the functional response is between 0 and 1. 2⁄3 is the structural surface that is responsible for the ingestion or assimilation, for example the gut or the searching area. The other dynamics remains the same.
When the maturity reach its maximum that is the threshold of puberty , the organism become an adult and start to allocate energy to the reproduction buffer. The dynamics of energy reserve and of structural volume remain the same, but maturity becomes constant and instead the increase in the reproduction buffer become equal to the efficiency of reproduction times the power used for maturation ̇ (Figure 2).
All parameters that are written with a dot on top indicate that they are rates, so of the dimension ‘per time’. Because the speed of biochemical reactions increases with temperature, these parameters have different values depending of the temperature of the environment. Based on the empirical observation that this effect of temperature is similar for the different biological processes such as feeding, growth or reproduction, the same correction, referred to as Arrhenius correction and is applied on each of those parameters. The Arrhenius correction is a ( − ) multiplication of the rate parameter by 1 with the Arrhenius temperature, 1 the reference temperature at which the rate parameter has been estimated and the environmental temperature. We assumed all rates were affected in the same way (Kooijman 2000).

Parameters estimation

The model used is the standard DEB model presented above (Table 1).

### Models simulations

The aim of this study is to build a bioenergetic model for males Atlantic salmon until migration at sea. But the standard DEB model exposed above can’t be applied to this species in this state. Furthermore, the standard DEB model is for all individuals during their whole life, whereas in this study the search model is applied to juvenile males in freshwater. And, this model don’t take in account the precocious maturation nor the smoltification. For that, the model need to be modified in taking in account the maturation and the smoltification decisions. In this aim, a model using the standard DEB model has been created. This model is coded with the software R 3.5.2 (R Core Team 2014) and use the package deSolve. The model start from a single egg, like in the standard model, the individual hatch, and later emerge. And when it finished to consume its vitellus sac, it’s the birth in the standard DEB model. After that, during a window of time that spans the entire month of May (Adams and Thorpe 1989; Berglund 1992a; Lepais et al. 2017), if the maturity of the individual reach the threshold of , the organism become a maturing parr. In this case, the organism become an adult and start allocating energy for the reproduction buffer and will reproduce between November and December (Mangel 1994; Thorpe et al. 1998). If the individual don’t reproduce, as model in the following, the organism re use a part ( ) of the reproduction buffer to replenish its energy reserve and being able to re allocate it to pay maintenance costs for example. But this re use has its efficiency, noted .
If the organism does not reach this threshold, it remains in juvenile phase. During the August month (Metcalfe et al. 1986; Metcalfe et al. 1988; Thorpe et al. 1998), if this non-maturing parr reach the threshold size , it will smoltify the next spring (Skilbrei and Holmström 2011). In other case the non-maturing parr non future smolt will stay in juvenile phase and will feed less during winter than future smolts (Simpson et al. 1996; Thorpe et al. 1998; Morgan and Metcalfe 2001). After the reproduction period, if the mature parr reach the size of between the 31st December and 31st January (Metcalfe and Thorpe 1992), it will smoltify the next spring. With the smoltification, the organism leaves the freshwater to go to the ocean. In this case, the individual is not modelled anymore. If the mature parr don’t reach the threshold size, it will follow its life in freshwater with individuals that haven’t matured and that will not smoltify. The non futur smolt, which include mature and non-mature parrs that haven’t reach the threshold size at time, recover their usual rate of feeding the 1st March (Simpson et al. 1996; Thorpe and Metcalfe 1998; Morgan and Metcalfe 2001). If those individuals reach the threshold of maturity in May (Adams and Thorpe 1989; Berglund 1992a; Lepais et al. 2017), they will start (or continue) to allocate energy to the reproduction buffer. The simulation of the model end at 600 days.
The value of is 1750 J and the value of is equal to ( ∗ )3with equal to 120 mm. These values have been taken according to literature. Indeed, it is established that parrs have to reach the size of 12 cm during their first spring to be able to smoltify the next spring (Kristinsson et al. 1985; Nicieza et al. 1991; Berglund 1995). The value of maturity threshold has been chosen by taking the value of maturity reached when an individual have a fork length equal to 72 mm and a wet weight equal to 6 g according to Berglund (1992).
The resulting model is a system of ordinary differential equations. It was solved numerically with the function radau of the package deSolve. When the maturity (resp. physical length) reach the birth or precocious maturation (resp. smoltification) threshold within the corresponding seasonal time window, the ODEs reach a root and a corresponding event is triggered. Indeed, the decisions of first maturation, smoltification and second maturation have been modelled as state variables. Their initial values are defined as all equal to 0. Then during the simulation in time, if a root is reached, the value of the decision in question changes to 1.
For the simulations of the DEB model of the Atlantic salmon, real environmental data was used to compare the simulations with real life-history data. For that, the temperature used in the simulations are taken from Jeannot (2019). They are Scorff river water temperature from the Environmental Research Observatory (ERO) on Diadromous Fish in Coastal Rivers (DiaPFC). Temperatures are sampled at 5 sites since 1995 for one and since 2005 and later for others. The Scorff river is a small coastal river in southern Brittany (France). In the simulations, the temperatures of the year 2005 to 2007 have been used from the Moulin des Princes site. The mean temperature per day has been fitted on a function with splinefun from the package stats. For the functional response, there is not a lot of information on the food availability for the Atlantic salmon in the Scorff River. Because of that, in this study there are several approximations of the functional response dynamics in time. Several models were used (Table 2), to see which one will be the more representative of the patterns known in the literature and to compare the simulations of these different models with real data. The first one called M1 use the same constant temperature and constant functional response for all individuals of the population. The second one named M2, use a same constant value of functional response and a same temporally-variable temperature corresponding to the Scorff river temperature data for all individuals. The third model M3 use the same temperature as in M2 but the functional response change in time with a peak in April-May and a low f in winter (Simpson et al. 1996; Thorpe et al. 1998; Morgan and Metcalfe 2001) (Figure 5). As with the temperature, a spline function is used for the functional response to implement it later in the ODEs system. The M4 and M5 models were made to compare two behavior hypothesis. The M4 model follow a behavior of dominant individuals versus dominated individuals in the acquisition of food (Metcalfe 1998).
The fourth model M4 use the same temperature as in M2 and M3 but for the functional response, half of the individuals are in a low group and the others in a high group. This means that the low group, have a lower functional response than the individuals of the high group. To apply that, the functional response curve of M3 is multiplied by a proportion ( ) randomly chosen between 30% and 70% for the low group and between 70% and 100% the high group according to a uniform law. Thanks to that, there is two groups of individuals that have inter and intra group differences of functional response. The fifth model M5 use always the same temperature as in models M2, M3 and M4 but the functional response is different for each individual at each time step. Indeed, out of the iteration by individuals and by time step, a data frame is created with 21 time steps (one by month) (columns) for each individual (rows) and each of those values are multiplied by a random proportion between 30% and 100% according to a uniform law. After what there is a fitting of a function on each rows (for each individual) with splinefun (package stats) that are put in a list of functions, used lately in the function radau.
To represent the results of the simulations and compare them with available data such as the weight and the fork length of Atlantic salmon, the weight and the length have been calculated from the components of the DEB model. Wet weight (in g) has contributions from the structure ( ), the reserve ( ) and the reproduction buffer ( ). Wet weight in this study is defined as follows with the whole-body energy density of dry biomass:
0.2 + + = ∗ 1000 0.2
The proportions of adopted strategies in the simulated population are compared with the ones observed by the Environmental Research Observatory (ERO) on Diadromous Fish in Coastal Rivers (DiaPFC) between 2005 and 2007 in the Scorff River. These data are based on a mark-recapture program that starts in 2005. This program is based on individual following of a part of the local population in which the individual recognition is done with specific mark that attribute a unique code at each marked individual. Marked individuals are followed during their whole life during several recapture occasions where they are registered, sized and weighted and identify as parr, mature parr or smolt. In this study, only the information on the proportion of life histories (mature parr and later smolt during the first year for example) in the cohort is used.

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#### Sensitivity analysis

To estimate the consequences of variation in the estimated parameters on the models predictions, two sensitivity analyses have been done on the M0 model on one individual. The aim of a sensitivity analysis is to study the uncertainty of a model depending on the uncertainty of his entries. The methods used for the two analysis are the One-At-a-Time (OAT) and the Morris method. 17 parameters have been tested, their nominal values were those specified below (Table 3 plus and ). The bounds of the intervals of variability of the parameter values (terminals) used in this sensitivity analysis are equal to 75% of the nominal value for the lower one and equal to 125% of the nominal value for the upper one. The Morris method have regular grids between two variables as experimental design. This permit to show interactions of two parameter variability in the analysis in contrary to the OAT method.

Results

Parameters estimation

Ten parameters were estimated by the estimation procedure (Table 3). Comparison with Salmo trutta parameters (Thrane and Kooijman 2015), Chinook salmon (Pecquerie and Kooijman 2016) and the Chum salmon (Kooijman 2019) is made. The energy conductance ̇(0.02707 cm.d-1) is almost 10 times lower than the trout parameter value (0.1648 cm.d-1) but is closer to the Chinook (0.07753 cm.d-1) and the Chum (0.045117 cm.d-1) salmons parameter values and close from the generalized animal value (0.02 cm.d-1). But the value of Atlantic salmon is the double of the ̇parameter values of chinook and chum. The κ value is very low compare to the generalized animal; indeed the value estimated for the Atlantic salmon is equal to 0.4146 whereas the generalized animal κ value is equal to 0.8. But the Chinook salmon also have a low κ (κ = 0.4612). The value of the [ ̇] estimated (101.9 J.cm-3.d-1) is close to the values of brown trout (115.3 J.cm-3.d-1) and chum salmon (117.362 J.cm-3.d-1) but not of the chinook one (20.56 J.cm-3.d-1) and the generalized animal (18 J.cm-3.d-1). The values of [ ]of all the 4 species are closed (Atlantic salmon: 5221 J.cm-3, trout: 5229 J.cm-3, Chinook salmon: 5242 J.cm-3 and chum salmon: 5234.16 J.cm-3). The values of are very different (Atlantic salmon: 69.33 J, chum salmon: 28.37 J) with the trout that have the lowest one (5.601 J) and the chinook which have the bigger one (1389 J). The Arrhenius temperature (9800 K) is between the values of the three other salmonid species (chum salmon 8000 K, Chinook salmon 10000 K and brown trout 15690 K). The shape coefficient (0.08254) that is species specific is quite low compare to the three other ones (trout: 0.1571, chum salmon: 0.18483, Chinook salmon: 0.1958). The energy density of dry biomass (22.9328 kJ.g-1) is very close to the other three ones (trout 22.2492 kJ.g-1, Chinook salmon 22.0405 kJ.g-1 and chum salmon: 22.5405 kJ.g-1)

1. Introduction
2. Material and methods
The DEB theory
Parameters estimation
Models simulations
Sensitivity analysis
3. Results
Parameters estimation
Standard DEB model simulation
Sensibility analysis
OAT
Morris
Models simulations
4. Discussion
Parameters estimation
Standard DEB model simulation
Sensibility analysis
OAT
Morris
Models simulations
5. Conclusion
References
Résumé
Abstract

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