Team:St Andrews/Modelling

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The mathematics of ω-3

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

We sought to model fish population depletion. We succeeded. The result: unless we work together on a global scale and make drastic changes to our fishing habits, only a fraction of the total fish population that existed in 1950 will be present in our oceans by 2100. The work of our team in the laboratory - the creation of Omega-3 using E.coli - could be exactly the measure necessary to save our oceans.

Our approach: we took one of a multitude of different possible approaches to population modelling and our project can be broken down into approximately four different stages:

$   f(x) = \left\{     \begin{array}{lr}       1 & : x \in \mathbb{Q}\\       0 & : x \notin \mathbb{Q}     \end{array}   \right. $



We performed meta-analysis to obtain information about the variation of total fish biomass in our oceans over time. We scaled our result to Villy Christensen’s (University of British Columbia) prediction of total fish biomass for 1950. We, thus, created a time series of total fish biomass in our oceans between 1950 and 2006. We believe our time series to be one of the first of its kind and certainly one of the first to be generated, largely, from real world data.

We hypothesised a differential equation model which we believe incorporates the key features responsible for fish population growth and decline. Our final model takes into account recruitment of fish into the adult fish population, death of adult fish due to fishing and death of adult fish due to natural causes.

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

Content that our model could predict fish biomass in the past and present, we enabled our model to forecast future fish biomass. We discuss how alternative sources of omega three could influence this outcome.

Motivation

In order to model the future of the global fish population, we chose a differential equation modelling approach. Such an approach, however,does rely on precise parameter definition – and, as a result, we spent considerable time refining these parameters. In particular, this “tuning” was done by taking a set of observed data (in our case, fish biomass throughout the last 60 years) and changing these parameters until our model’s predictions resembled the data as closely as possible. 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 create it ourselves.

RAM database

After further investigation, we found that there were many 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. Effectively, the RAM Database includes data from all known to us sources; therefore we decided to use it for our further work.

  • RAM Database coverage

    Ricard D, Minto C, Jensen OP, Baum JK (in press) . 2011. Examining the knowledge base and status of commercially exploited marine species with the RAM Legacy Stock Assessment Database. Fish and Fisheries doi: 10.1111/j.1467-2979.2011.00435.x.

Data manipulation

The data presented in RAM, in some cases, was not entirely homogeneous. For example, the Spawning Stock Biomass (total weight of those fish that have reached the breeding age) – the data figure we were interested in - was often presented in different measures. These measures ranged from weight in tonnes/kg/pounds to the biomass of the annually produced eggs and non-specified 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.

    Why adult fish?

    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

    Parameter tuning

    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, qualitatively, 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 (place reference here) 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 as ten times the biomass present at 1950.

    Our mathematical model

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



    Browse the data

    Initial parameters

    Ever wondered what is the average weight of a fish? This and many more surreal things inside. An introduction is included in case you get lost or want more information.

    Tuned parameters

    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
    $\delta_M$ 0.2 0.1-0.5 0.5 0.3
    $FJ$ N/A 0.01-0.3 0.01 0.02

    These three graphs show the amount of papers published each year containing certain search queries, as well as the number of times these papers were cited. All graphs show positive tendencies: the competition is becoming more wide-spread and more iGEM-related papers are being published and recognized. The search queries were chosen to show which part of iGEM is usually cited: the iGEM competition, the Registry of Standard Parts, or both. The data shows that only around half of papers will cite both elements. It should be noted that some outlying data points were ignored as they are obvious mis-searches.

    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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    University of St Andrews, 2012.

    Contact us: igem2012@st-andrews.ac.uk, Twitter, Facebook

    This iGEM team has been funded by the MSD Scottish Life Sciences Fund. The opinions expressed by this iGEM team are those of the team members and do not necessarily represent those of Merck Sharp & Dohme Limited, nor its Affiliates.