# The mathematics of ω-3

Modelling the impact of alternative omega-3 production on the global fish population

We modelled fish population dynamics. Our result: if we continue fishing in the current manner, by 2100, only a fraction of present day biomass levels will remain. Yet, there is hope. Indeed, realizing Team St Andrews' alternative production of omega-3 could be the measure necessary to save our seas. We investigate both the effect that alternative production can have on future fish biomass, as well as the practicalities of preserving life in this manner.

Our project can be split into four stages:

### 1: Fish biomass data – collection and manipulation

We performed meta-analysis to obtain information about the variation of total fish biomass in our oceans in the years between 1950 and 2006. We believe our time series to be one of the first of its kind.

### 2: Mathematical model

We hypothesised a differential equation model which we believe incorporates the key features responsible for fish population growth and decline.

### 3: Parameter tuning

We changed the parameters in our model until our model's predictions closely replicated the real world fish biomass data.

### 4: Model predictions

Content that our model succeeded in predicting past fish biomass values, we enabled it to forecast the future. We consider alternative futures with and without alternative omega-3 production schemes.

## Motivation

In order to anticipate the future of the global fish population, we hypothesized a mathematical (delay differential equation) model which incorporated, what we believed to be, the key features affecting population change. The success of our model and its ability to forecast the future relied on the careful definition of some parameter values. In particular, we performed parameter “tuning”: we took real world data and altered the values of the parameters in our equation, until our model’s predictions and our data resembled one another. Being able to precisely predict past biomass values ensured that we had some grounding for making future estimates.

Unfortunately the global fish biomass data, the cornerstone of the tuning process, was not something which was readily available. A “total fish biomass” time series did not, to our knowledge, exist. We had to distill it from existing lower-level data ourselves.

## RAM database

After further investigation, we found that there were some cases in which biomass data was available for specific species in specific regions (this data being produced mostly for the sake of commercial stock assessment). RAM Legacy Stock Assessment Database is a “compilation of stock assessment results for commercially exploited marine populations from around the world”. We believe that it is the most complete compilation of Stock Assessment Results to this date. Another advantage of the RAM Database, compared to other databases (NOAA, ICES, etc.), is that it combines data from different regional agencies, thus ensuring good global coverage. Ultimately, the RAM Database includes data from all sources known to us; therefore we decided to use it for our further work.

## Data manipulation

The data presented in RAM, in some cases, was not homogeneous. For example, the Spawning Stock Biomass (total mass of fish that have reached breeding age and the data figure we were interested in) was often presented in different measures. These measures ranged from mass in tonnes/kg, weight in pounds, to the biomass of the annually produced eggs and other unspecified measures. We had to omit the datasets which were not directly convertible to tonnes.

## Calculating total fish biomass

234 sets of data: refined and combined to give just one. This is the prized result of the data collection element of our modelling project. The only other attempt at a time series of total fish biomass was provided by Tremblay-Boyer et al. (2011). They used a very different approach to our own, however (they relied on the Ecopath ecological modelling software) and their time series consisted of only five data points.

Delay Differential Equations and Numerical Solution Approximation Methods - is it all really necessary?

## Why model?

In our project we sought to:

• Investigate the impact we, as humans, will have on the population of fish in our ocean if we continue to fish in our current manner. Today and in the past, we have fished at a rate proportional to the size of our population.
• Discover whether iGEM Team St Andrews can, with our alternative method of production of omega-3, influence the "future of fish", by preventing or, at least, slowing the suspected depletion of fish.

• In order to answer such questions about the future and theoretical, never before encountered, scenarios, one has to make assumptions about the nature of our world and how it 'works'. Very often, these assumptions can be expressed in a mathematical format. The mathematical format is often referred to as a "mathematical model" of the physical situation. Hence, as we sought to answer our own questions, we produced a mathematical model that predicted the population of world fish biomass at various times. Our model involved parameter values which could be changed to enable us to ask different questions of the same model.

## Why wet biomass?

Our model measured the total fish population at a specific time, in terms of the fish biomass present in our oceans at that time; and not in terms of total number of fish. It did this for various reasons:

• Most relevant data for fish population modelling, for example - recruitment rate, is expressed in terms of biomass. Therefore, we avoided unnecessary conversions and errors.
• More importantly, we modelled total fish populations with the aim of investigating their sustainability. To model fish numbers, when the definition of a sustainable number of fish varies so significantly from one species to another, would have been silly.

Having chosen to measure fish population in terms of (wet) fish biomass, it also became necessary to measure population in terms of adult fish biomass, instead of all fish biomass.

We sought to model fish biomass throughout time but to model all biomass would have required us to take into account the growth of fish. We would have had to model the population dynamics of multiple weight classes of fish, as well as the interaction between the weight classes. Instead, we chose to investigate adult (mature) fish biomass as we could assume, to a first approximation, that the biomass of an adult fish is constant throughout time (as suggested by Von Bertalanffy's fish growth model). We were, thus, able to produce a justifiable and relatively simple first model.

## The mathematics

Our model takes into account what we believe to be the most fundamental factors that alter adult fish biomass measurements between two years: the recruitment of junior fish into the adult population, the natural death of adult fish and the catching of adult fish.

#### Our mathematical model

$$\textrm{Biomass (this year)} - \textrm{Biomass (last year)} = \textrm{Recruits} - \textrm{Natural Deaths} - \textrm{Fish Caught}$$

$$\frac{dB}{dt}=r w e^{-\delta_J \tau}(1-\frac{B(t-\tau)}{k})B(t-\tau) -\delta_M B(t) - F(t) B(t)$$

### Equation explained

Parameter Explanation Units
$\frac{dB}{dt}$ Biomass(this year)-Biomass(last year), when the time scale over which you are calculating these yearly changes is large Tonnes per year
$B(t)$ Biomass at time t Tonnes
$r$ Number of junior fish produced by 1kg of mature adult fish per year Per kg
$w$ Average mass of a mature fish kg
$\tau$ Average time for a junior fish to reach maturity (gain ability to breed) Years
$\delta_J$ Juvenile natural mortality rate (fraction of junior fish that die to natural causes in a year) Per year
$k$ Carrying capacity of fish population (maximum size population can reach before competition for resourses causes population to decrease) Tonnes
$\delta_M$ Natural mortality rate (fraction of adult fish that die due to natural causes in a year) Per year
$F(t)$ Fishing mortality rate (fraction of adult fish that die due to being caught at time t) Per year
Term from model Physical meaning
$rB(t-\tau)$ Maximum number of junior fish that could reach maturity at time t (if no natural death present)
$e^{-\delta_J \tau}$ Fraction of junior fish that survive to reach maturity
$r w e^{-\delta_J \tau}B(t-\tau)$ Biomass contributed to stock of adult fish biomass at time t due to junior fish reaching maturity at that point
$\delta_M B(t)$ Adult fish biomass lost from stock at time t due to natural death
$F(t) B(t)$ Adult fish biomass lost from stock at time t due to fishing

## Defining our model

Content with the formulation of our model, we then sought to assign values to the parameters involved (‘parameter’ refers to, for example, “r”, “w” or “k”). Further, we looked for values which enabled our model to make predictions throughout time that resembled our biomass data. We could then use the tuned differential equation to make well grounded future biomass estimates.

## Data collection and initial values for parameters

We located values for the recruitment rate (r), the mass of an adult fish (w), the time for a fish to reach maturity ($\tau$) and the omega-3 content of a fish, for the 18 most abundant fish species by biomass (according to RAM Legacy Database. Taken together, these species comprise 83% of the fish biomass we could gain information about). Weighted averages provided estimates for these parameters in the general setting, where the parameters relate to all fish species. The range of uncertainty in a general setting parameter estimate was found by comparing the values for the 18 most abundant species and locating the greatest and smallest values.

We obtained values for the Fishing Mortality Rate (Catch/Biomass) throughout time using catch data from FAO (FAO, 2010) and our total fish biomass data obtained previously.

Our initial estimates and uncertainty ranges for the Natural Mortality Rates (adult and junior fish values), and for the Carrying Capacity (k) were somewhat arbitrary. These parameters cannot be readily measured in the physical world. We chose to use values for the Natural Mortality Rates that have been widely used by fish population ecological modellers in the past. As we anticipated that fish populations would not be so large that competition for resources would be significant, we set an initial estimate for k that was ten times greater than the biomass present at 1950.

## Tuning and refining our model

We sought to refine our parameters until the model’s predictions and our biomass data agreed well between 1950 and 2006, at least qualitatively. We varied the parameters $\delta_J$, $\delta_M$, $k$ and $\tau$ within their ranges of uncertainty and sought to reduce the error (the difference between our model’s prediction and actual biomass data value) at each year.

Unfortunately, even incrementing trial parameter values in small steps, the solution to our differential equation failed to reproduce the main features of the biomass data graph. It was clear that our model was failing to take into account some vital factor influencing total fish population dynamics. Due to the fact that the biomass data seemed to be broken into two halves - between 1950 and ~1980, biomass seemed to decrease almost linearly; after 1980 it started to level off - we proposed that the missing factor was death prior to 1978 of junior fish due to fishing and the subsequent reduction in this death due to changes in international legislation. (In 1978, an international agreement on mesh net sizes (Burd,1978) was reached and this had the effect of significantly reducing junior death, and doing so almost immediately). We, thus, amended our differential equation and our model took its final form:

$$\frac{dB}{dt}=r w e^{-(\delta_J +FJ(t)) \tau}(1-\frac{B(t-\tau)}{k})B(t-\tau) -\delta_M B(t) - F(t) B(t)$$

$$FJ(t) = \left\{ \begin{array}{lr} FJ & : t \leq 1978 \\ 0 & : t \geq 1979 \\ \end{array} \right.$$

We then varied the parameters $\delta_J$, $\delta_M$, $k$ and $\tau$ in order that our model predictions and biomass data post-1978 agreed well; we varied new parameter FJ until the model predictions and data pre-1978 were qualitatively similar.

## Browse the data

Ever wondered about the average mass of a fish? Well we've calculated a value for you. Browse our data files and "Mathematica" notebooks if you desire a more in depth understanding of what we did. An introduction is included, in case you get lost. In addition, please feel free to contact us if you seek additional assistance.

## A sensible result?

There was a clear resemblance between our model’s output and the total fish biomass data for 1950-2006. Yet how much trust could we place in our parameter values to predict future outcomes? We performed some relevant tests.

We firstly used our refined model to predict past catch values. We used Fishing Mortality Rate data between 1950 and 2006, as well as our model’s predictions of biomass for the same time period. The outcome: there was close agreement between true catch data and our model’s predictions of (F(t) x Biomass(t)) = Model’s Catch(t) values.

After we enabled our model to run to 2100 with our Fishing Mortality Rate function for the future (values obtained through correlation with population data - refer to section “Model predictions”), we altered the parameters in our model, one by one, to be one, two or three increments above and below accepted values. We then checked, qualitatively, whether the prediction for the evolution of fish biomass in the future was similar to the result we predicted with our accepted values. Qualitatively, in all cases, exponential decay was predicted for fish biomass between 2006 and 2100. If this had not been the case, we would have decreased increment size in our variation of parameter, parameter refinement stage.

## Tuned parameters

The set of parameters arising within our model: this table displays our initial estimates of their values and the uncertainty associated with these initial estimates. The table also displays the increments in which these parameter values were varied during the tuning process, as well as their final refined values.

Parameter Initial estimate Range for tuning Step size for tuning Final value
$rw$ 4.9 N/A N/A 4.9
$\delta_J$ N/A 0.7-2 0.05 0.7
$\tau$ 2.5 1-4 0.25 3.25
$k$ N/A 5*10^9-11*10^9 1*10^9 5*10^9
$\delta_M$ N/A 0.1-0.5 0.5 0.3
$FJ$ N/A 0.01-0.3 0.01 0.02
• Fish biomass depletion between 2006 and 2100 An alternative outcome

##### Two futures, one world: the choice is ours

Fish biomass depletion between 2006 and 2100: if we continue to fish in the manner we do today, the future is bleak both for fish and for us.

An alternative outcome. Suggestion: we replace traditional aquaculture with a farmed fish industry that does not require wild fish as an input. We can replace fish meal in feed with soybean meal and now, we can replace fish oil with Team St Andrews' Alternatively Produced Omega-3.

## A fishy dilemma

We enabled our model to run to 2100 under the assumption that the fishing mortality rate at years in the future (the fraction of biomass caught every year) would vary in a proportional way to human population over this same time period. The correlation between past fishing mortality rate data and past human population figures was strong (Pearson’s r = 0.897; P-value < 0.00001), thus justifying our approach. The result: fish biomass decays exponentially in the years following 2006 until, at 2100, only a very small fraction of the biomass present in 1950, prior to the birth of industrial fishing, remains.

## Changing the future

Can this tale of death and decay be reversed? Are there ways in which humans and fish can live in the same world; swim in the same oceans? In terms of resources, is it viable to implement these suggestions?

In seeking answers, we focussed on the potential impact of Team St Andrews’ Alternative Production of Omega 3. In particular, we proposed that we could influence catch and biomass figures in the future by replacing the need for wild fish in aquaculture. Currently, in order to produce 1 tonne of farmed fish, an average of 0.7 tonnes (Tacon, 2008) of wild fish is required (farmed fish are fed fish meal and fish oil in their feed). There is research that suggests the fish meal in the feed can be replaced entirely by other sources, including soybean meal. With the work of our lab team, it is now the case that farmed fish need not rely on wild fish for their fish oil, either.

We proceeded to investigate the effect on fish biomass if aquaculture output was presumed to remain at its 2006 level (we acknowledge this is a rather conservative estimate) and, from 2006 onwards, farmed fish were produced using feed from non-fish based products. Thus, we could reduce our projected yearly catch figures for 2006-2100 by 0.7 x (Aquaculture Output at 2006).

The effect was remarkable. The outcome from our model was entirely unrecognisable compared to the story of death and near-extinction previously predicted. Fish survived into the future and indeed flourished, as their population grew exponentially!

## The cost of success

In order to produce farmed fish at a level resembling 2006 output, using non-fish based products for feed, how much omega-3 is required? Is it plausible that iGEM Team St Andrews can save our oceans in this way?

We calculated required omega-3 by examining the number of wild fish required to produce the 2006 aquaculture output and then multiplying this figure by the average omega-3 content per tonne of fish biomass. We also proceeded to investigate how much omega three our “factory” would have to produce if we terminated traditional aquaculture in 2006 and used our own idea of aquaculture (zero wild fish input) to produce enough fish to maintain the current fish (available for human use) to population ratio. (The current fish to population ratio was calculated by averaging ((catch(t)+aquaculture(t)-(catch required to produce aquaculture)(t))/population(t) over the years between 2000 and 2010). Finally, we examined how much omega-3 Team St Andrews would have to produce in order to, by means of our alternative aquaculture, provide every person in our world with their recommended 0.5g (Kris-Etherton, 2007) of omega-3 per day.

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