# Atlantic Cod Growth Model

#### Modelling temperature effects on growth function of Atlantic cod (Gadus morhua)

Growth is one of the fundamental life properties that is powered by metabolism and influenced by a diversity of internal, environmental and ecological factors (Campbell et al., 2008). Investigation of growth involves such research fields as ocean physics, climate science, fisheries biology, marine ecosystem studies and ecophysiology, and includes theoretical background obtained from empirical growth data (laboratory experiments and aquatic studies), on the one hand, and growth observations of wild populations (fisheries/scientific surveys, tagging studies etc.), on the other hand (Weatherley, 1966; Audzijonyte et al., 2020).

At an organism level “growth” is defined as a measurable increase in size over a period of time, where size is represented as length, volume or weight (von Bertalanffy, 1938; Weatherley and Gill, 1987; Lorenzen, 2016; Clarke, 2017). Graphically growth performance can be described by a sigmoid growth curve approaching an upper asymptote with increasing time step (Beverton and Holt, 1957). Growth curves are used to identify individual growth strategies under various internal and external factors (Sokolova, 2022).

#### Temperature and size effects on growth

Environmental temperature is suggested as a key factor driving metabolic rates and thus influencing the rates at which an individual grows: organisms grow faster in a warmer environment while colder temperatures cause slower growth (Clarke, 2017). Despite the wide range of temperature environments that fish experience across latitudes, experimental studies reveal species-specific temperatures that are optimal for growth. Besides environmental temperatures most organisms’ physiological rates are affected by body size which can be expressed by a power function: Y = axb where, Y is a measure of performance, x is a measure of size and a, b are species-specific coefficients (Brown et al., 2004; Clarke, 2017). This is called a principle of scaling: organism performance changes at a different rate during early life (smaller sizes) than later (larger sizes) (Chamber and Trippel, 1997). For Atlantic cod it has been observed that smallest-sized individuals grow faster, have the highest optimal temperatures whilst the growth of larger fish is greater at lower temperatures (Jobling 1988; Pörtner et al., 2001; Björnsson and Steinarsson, 2002; Drinkwater, 2005). Both laboratory studies and statistical analyses show that optimal temperatures for growth decline with body size: 14°C for a 5-g individual in contrast to 9°C for a 5-kg one (Fig. 1).

Fig. 1. Temperature effects on growth rates of Atlantic cod fed to satiation. Numbers above the curves represent the weight of individuals in gramm (g). Source: Björnsson et al., 2007.

#### Mathematical representation of growth

Mathematically growth process is described as “a growth model“ which can be based on theoretical findings, empirical or observational data (Lorenzen, 2016). Here we first represent a growth model for Atlantic cod which was derived from experimental data (Björnsson and Steinarsson, 2002; Björnsson et al., 2007) by Butzin and Pörtner (2016). The model assumes that growth rates are mainly temperature- and size-dependent.

A bio-physical mechanistic background of the model roots in controlled laboratory experiments for Icelandic cod where fish growth was measured in fish tanks with various constant temperature settings (T = [2, 4, 7, 8, 10, 12, 13, 16] °C) for any given individual weight from 0.001 to 8.7 kg (Fig. 2a). Laboratory experiments are characterised by the regimes of constant temperatures, salinity and oxygen. The time frame spans hours to months. Experimental output includes growth rates in relation to temperature and body weight (Fig. 2b). Based on these data a mathematical equation was derived that includes a temperature dependent nonlinear reaction model with growth inhibition at 12°C (coefficient a) and temperature dependent allometric scaling exponent b (Fig. 2c). As the model equations were derived from experimental data, the base model setup mimics the results of controlled laboratory experiments by simulating temperature-dependent growth for any given individual weight class (Fig. 2d).

Fig. 2. Conceptual framework behind the base model experiments (Sokolova et al. JORS, submitted Jun. 28, 2022)

## Aquarium Experiment

Environmental temperature is suggested as a key factor driving metabolic rates and thus influencing the rates at which an individual grows: organisms grow faster in a warmer environment while colder temperatures cause slower growth (Clarke, 2017). Despite the wide range of temperature environments that fish experience across latitudes, experimental studies reveal species-specific temperatures that are optimal for growth. Atlantic cod growth rates increase with temperature, reaching a maximum between 8 and 10°C (Righton et al., 2010; Pörtner et al., 2001).

Initial weight in $$kg$$
Temperature
3°C

The model results cover the temperature range from 2 to 16 °C (T 2, 4, 7, 8, 10, 12, 13, and 16 °C), and the body weight range 0.001–8 kg (Björnsson and Steinarsson, 2002; Björnsson et al., 2007).
Core model assumptions:

• the rate at which an organism grows depends on the value of its own body mass, i.e. allometric growth;
• an organism responds to temdperature changes immediately;
• individuals grow under unlimited food supply and in a homogeneous thermal environment (based on data from Björnsson and Steinarsson, 2002; Björnsson et al., 2007).
• larval growth variations, vertical, or horizontal movements and ontogenetic habitat shifts are not considered;

## Real world Experiment

By including multi-dimensional time-series data we extended the growth model functionality. The extended version of the growth model can calculate growth of Atlantic cod over its life-cycle (Sokolova et al., 2021). The life-cycle of the southern cod populations spans in average 10 years, in the northern - 14 years. Using transient growth model setup in combination with observational or simulated temperature time series allows us to mimic growth in natural environment and compare our simulation results to the observed data (see next page).

Using transient growth model setup in combination with observational or simulated temperature time series allows us to mimic growth in natural environment and compare our simulation results to the observed data

Atlantic cod is a widely distributed commercially important species found throughout the shelf ecosystems of the North Atlantic (Hutchings, 2004; Rose, 2004). Among the main population and individual characteristics of Atlantic cod are wide environmental tolerance, behavioral flexibility and local adaptation (Righton et al., 2010). Species are adapted to live under a wide range of environmental conditions (temperatures, salinity, oxygen, depth etc) which vary between and within populations (Brander, 2019 in Atlantic cod: a bio-ecology).
Below are some examples of historical and projected growth curves of the Celtic Sea and North Sea cod populations in comparison to the data reported by the International Council of the Exploration of the Sea (ICES). The history of the source code is available through a publicly available version history on GitLab: https://gitlab.hzdr.de/awi_paleodyn/growth-model-atlantic-cod/.

Despite the differences in oceanic conditions and life-history characteristics of the Celtic Sea and the Barents Sea populations, their growth patterns are shaped by the decadal mean temperature regimes However the modeling approach, doesn’t explain interannual variability in body size of different age groups which is rather the task for complex multivariate statistical analyses requiring high quality long-term observations such as primary and secondary productivity levels, prey availability etc

 $$a r$$ 8.66 $$b r$$ 0.3055 $$\Theta a$$ 18145 $$\Theta b$$ 4258 $$\Theta h$$ 25234 $$T r$$ 283 $$T h$$ 286