Introduction

While Team St Andrews' "Omega Squad" works to produce EPA Omega 3 Fatty Acids in the Laboratory; our "Mod Squad" motivates their endeavours, quantitatively. In particular, we seek to model the time evolution of human-available EPA and DHA Omega 3 Fatty Acids from the year 1950 and into the future. We will also investigate recent trends indicating increasing demand for these resources: due to population growth, as well as heightened awareness of the health benefits associated with a diet containing these Fatty Acids.

Our Model

Our Model relies on a number of assumptions:

Assumption 1: Humans obtain their necessary EPA and DHA Omega 3 Fatty Acids from fish, and fish alone. (Insert justification for this assumption)

Assumption 2: In addition to Assumption 1, only mature fish (insert clear definition of “mature”) can be considered a source of EPA and DHA. (Justification: current catch is composed mostly of mature fish (data to sup- port statement). This assumption also greatly simplifies our model: refer to Equation (4) later in this document)

Let $\omega(t)$ represent the DHA and EPA Omega 3 available for human consumption at time $t$ years (in tonnes). Further, let $B(t)$ represent total mature fish biomass in the world at time $t$ years (again, in tonnes) and $\omega_B(t)$ represent the average Omega 3 (DHA + EPA) content per tonne of mature fish biomass at $t$.

Then:

$$\omega(t)=B(t) \cdot \omega_B(t)$$ (1)

Assumption 3: The average Omega 3 content per tonne of mature fish biomass, $\omega_B(t)$, does not depend explicitly on time (an initial approximation that does not take into account the effects of, for example, climate change). Correspondingly, as a constant value, Omega 3 per tonne of biomass will be represented as $\omega_B$ throughout the rest of this document.

Differentiating (1) then yields:

$$\frac{d\omega(t)}{dt}=\frac{dB(t)}{dt} \cdot \omega_B$$ (2)

In addition,

$$\omega_B = \sum_{all f} p_f \cdot \omega_f$$ (3)

where $p_f$ is the probability that you select a mature fish at random and it is of species $f$ and $\omega_f$ is the average Omega 3 content per tonne of biomass of that species.

Further:

$$\frac{dB}{dt} = r \cdot w \cdot B(t-\tau) \cdot e^{- \delta_J \, \tau} - (F + \delta) \cdot B(t)$$ (4)

The rate of change of mature fish biomass is, thus, represented by a Delay-Differential Equation in which r is the average Birth Rate of fish (average number of eggs produced per tonne of fish per year), w is the average mass of an adult fish; $B(t-\tau)$ is the mature fish biomass at time $t-\tau$ and $\delta_J$ is the average Death Rate of junior fish (a "junior fish" being any fish that has not yet reached maturity). $\tau$ is the average time taken (in years) for a fish to reach maturity; F is the average Fishing Mortality Rate (number of caught fish for every one fish per year) and $\delta$ is the average Natural Mortality Rate for mature fish (number of fish, for every one fish, that die of natural causes each year).

Modelling the time evolution of available mature fish biomass greatly simplified the construction of (4). Most (insert appropriate reference) fish species can be assumed to grow according to von Bertalanffy's Growth Model:

$$L(t)=L_\infty \cdot (1-e^{- k \,(t-t_0)})$$ (5)

where $L(t)$ is the length of a fish of a particular species at time $t$; $L_\infty$ is the length the fish will tend towards with time; $k$ measures that rate at which the fish tends towards $L_\infty$ and $t_0$ is the theoretical time when the fish will have zero length.

When plotted against time, (insert plot) von Bertalanffy's growth curve levels off: it is therefore, to a good approximation, unnecessary to take into account fish growth when considering the rate of change of biomass with time. As a result of the fact that the parameters in von Bertalanffy's model vary significantly both between and within species, being able to neglect growth greatly reduces complexity when producing and solving (4). (??For example if we assume, for a particular species, the total rate of change of mature fish biomass per year to be ``not too significant'' between the times of $t-\tau$ and $t$ then, if large numbers of fish exist at time t and they grow according to von Bertalanffy's model, the increase in total mature fish biomass due to this growth is insignificant compared to the increase in biomass due to the addition of "new recruit biomass" at $t$. We assumed the number of fish at $t-\tau$ was quite similar to the number at $t$, and hence if the total biomass at $t$ is large, it will also be large at $t-\tau$. Correspondingly, the amount of new biomass produced at $t-\tau$ will be large. The amount of new biomass produced at $t-\tau$, combined with the growth it will undergo whilst tending to maturity, mean that the "new recruit biomass" is, indeed, significantly greater than the increase in existing biomass due to growth that occurs at time $t$.

(Insert further Assumptions relating to (4))

Data Collection

In order to run our model and make future predictions about the size and scale of necessary Omega 3 "factories", we require data - not least of all, relating to the various parameters contained within (3) and (4).

Required data for (4):

• Average birth rate of fish, $r$
• Definition: "mature fish"
• Average time taken for a fish to reach maturity, $\tau$
• Average junior death rate of fish, $\delta_J$
• Average Fishing Mortality Rate, $F$, for mature fish. (Number of caught fish for every one fish per year - likely to change over time (we need Biomass in our oceans per year and Catch per year - for an extended time scale))
• Average Natural Mortality Rate, $\delta$ for mature fish

• Of total biomass in our oceans, a breakdown of species abundance (so as to calculate $p_f$)
• Average Omega 3 content per tonne of biomass of the most abundant fish species in our oceans, $\omega_f$