Team:St Andrews/Modelling
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- | <h1> | + | <h1>The mathematics of ω-3</h1> |
- | <p class="lead"> | + | <p class="lead">Modelling the impact of alternative omega-3 production on the global fish population</p> |
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<ul class="nav nav-pills"> | <ul class="nav nav-pills"> | ||
- | <li><a href="#introduction"> | + | <li><a href="#introduction">Our model 101</a></li> |
- | <li><a href="#model">Model</a></li> | + | <li><a href="#data">Fish biomass data</a></li> |
+ | <li><a href="#model">Mathematical model</a></li> | ||
+ | <li><a href="#tuning">Parameter tuning</a></li> | ||
+ | <li><a href="#predictions">Model predictions</a></li> | ||
+ | <li><a href="#references">References</a></li> | ||
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<!-- Introduction | <!-- Introduction | ||
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- | <h1> | + | <h1>Our model 101</h1> |
+ | </div> | ||
+ | <div> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/f/f0/ModSquadLogo_100.png" align="left"></img> | ||
+ | <p>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.</p> | ||
+ | <p>Our project can be split into four stages:</p> | ||
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+ | <img src="https://static.igem.org/mediawiki/2012/2/23/Process1.png" alt=""> | ||
+ | </div> | ||
+ | </div> | ||
+ | <div class="span8"> | ||
+ | <h3>1: Fish biomass data – collection and manipulation</h3> | ||
+ | <p>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.</p> | ||
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+ | <img src="https://static.igem.org/mediawiki/2012/9/9b/Process2.png" alt=""> | ||
+ | </div> | ||
+ | </div> | ||
+ | <div class="span8"> | ||
+ | <h3>2: Mathematical model</h3> | ||
+ | <p>We hypothesised a differential equation model which we believe incorporates the key features responsible for fish population growth and decline. | ||
+ | </p> | ||
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+ | <div class="span8"> | ||
+ | <h3>3: Parameter tuning</h3> | ||
+ | |||
+ | <p>We changed the parameters in our model until our model's predictions closely replicated the real world fish biomass data.</p> | ||
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+ | <img src="https://static.igem.org/mediawiki/2012/7/76/Process4.png" alt=""> | ||
+ | </div> | ||
+ | </div> | ||
+ | <div class="span8"> | ||
+ | <h3>4: Model predictions</h3> | ||
+ | <p>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.</p> | ||
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+ | <section id="data"> | ||
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+ | <h1>Fish biomass data – collection and manipulation</h1> | ||
+ | </div> | ||
+ | <div> | ||
+ | |||
+ | <div class="row"> | ||
+ | <div class="span6"> | ||
+ | <h2>Motivation</h2> | ||
+ | <p>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.</p> | ||
+ | <p>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.</p> | ||
+ | <h2>RAM database</h2> | ||
+ | <p>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.</p> | ||
+ | <ul class="thumbnails"> | ||
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+ | <a href="#ram-map" class="thumbnail" data-toggle="modal"> | ||
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+ | <img src="https://static.igem.org/mediawiki/2012/1/17/RAMmap.png" alt=""> | ||
+ | <h3>RAM Database coverage</h3> | ||
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+ | <p>Map from (Ricard et al., 2011)</p> | ||
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+ | </a> | ||
+ | </li> | ||
+ | </ul> | ||
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+ | </div> | ||
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+ | <div class="span6"> | ||
+ | <h2>Data manipulation</h2> | ||
+ | <p>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.</p> | ||
+ | <h2>Calculating total fish biomass</h2> | ||
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+ | <img src="https://static.igem.org/mediawiki/2012/0/05/PreExtrapDataSet_Graph_2.png"> | ||
+ | <p><span class="label label-info">Chart: All datasets prior to any manipulations</span></p> | ||
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+ | <h4><span class="badge badge">1/5</span> Choosing boundaries for the datasets</h4> | ||
+ | |||
+ | <p>The next step was to “combine” the individual species-specific time series into one, which would become our biomass data.</p> | ||
+ | <p>The time period coverage offered by different datasets varied significantly. Some datasets included biomass data from 1874 to 2005; while others gave information for the years 1990 to 2006.</p> | ||
+ | |||
+ | <p>FAO provided fish catch data from 1950, hence it was possible for us to run our model only from this date onwards. However, it was important that we could compare model predictions with ‘real’ data for as many years as possible, before we could use our model to predict future results. We, thus, chose 1950 as the baseline of our equation. The nature of our model, as a delay differential equation, meant that we had to have biomass data for 1-4 years (the average time for a fish to reach maturity) before 1950.</p> | ||
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+ | <img src="https://static.igem.org/mediawiki/2012/4/4d/PreExtrapDataSet_Graph.png"> | ||
+ | <p><span class="label label-info">Chart: Datasets spanning different time periods</span></p> | ||
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+ | <div class="carousel-caption"> | ||
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+ | <h4><span class="badge badge">2/5</span> Extrapolating datasets between 1932 and 2006 </h4> | ||
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+ | <p>The number of datasets present at 1932 was approximately the same as at 1945. Since the number of datasets influences the quality of extrapolation (refer to “How does extrapolation of datasets work?”), we could extrapolate just as well back to 1932 as we could to 1945.</p> | ||
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+ | <p>The upper boundary of extrapolation was set to 2006, since after 2006, the number of datasets with information decreased rapidly.</p> | ||
+ | </div> | ||
+ | </div> | ||
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+ | <div class="item"> | ||
+ | <!-- <img src="https://static.igem.org/mediawiki/2012/e/ec/Biomass_Chart_Smaller.png"> --> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/1/17/StSetsGraph-1.png"> | ||
+ | <p><span class="label label-info">Chart: The "standard" biomass sets</span></p> | ||
+ | |||
+ | <div class="carousel-caption"> | ||
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+ | <h4><span class="badge badge">3/5</span> How does extrapolation of datasets work?</h4> | ||
+ | <p>We termed those sets that had data for at least all years between 1932 and 2006 the “Standard Sets”. For the datasets requiring extrapolation, we assumed that the population of the species described by these datasets varied between 1932 and the first year of the dataset in a proportional way to the way in which the sum of the biomass of the standard sets varied during that time period. It is for this reason that it was extremely important, when searching for datasets, that we included as many as possible in our “Standard Sets” and our Standard Sets described species from many different temperate zones.</p> | ||
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+ | <img src="https://static.igem.org/mediawiki/2012/5/5d/Extrap_DataSets.png"> | ||
+ | <p><span class="label label-info">Chart: Extrapolated datasets</span></p> | ||
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+ | <div class="carousel-caption"> | ||
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+ | <h4><span class="badge badge">4/5</span> Combining the datasets</h4> | ||
+ | <p>Once the charts were extrapolated from 1932 to 2006, we had 234 individual data sets. In order to combine them, we summed up all dataset values at each year. Since the data we had is only a fraction of the total world fish population (although, a representative one), there was a need to upscale it. We referred to Villlie Christensen’s prediction of the 1950 biomass to carry out this task.</p> | ||
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+ | <!-- <img src="https://static.igem.org/mediawiki/2012/e/ec/Biomass_Chart_Smaller.png"> --> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/e/eb/Biomass_Chart_Smaller-1.png"> | ||
+ | <p><span class="label label-info">Chart: The final biomass-time evolution curve</span></p> | ||
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+ | <div class="carousel-caption"> | ||
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+ | <h4><span class="badge badge">5/5</span> Our final result</h4> | ||
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+ | <img src="https://static.igem.org/mediawiki/2012/6/65/Biomass-chart.png" alt=""> | ||
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+ | <p>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.</p> | ||
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</section> | </section> | ||
- | <!-- | + | <!-- Model |
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- | <h1> | + | <h1>Mathematical model</h1> |
+ | </div> | ||
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- | </ | + | <!--Text goes here --> |
+ | <div class="row"> | ||
+ | <div class="span6"> | ||
+ | <blockquote> | ||
+ | <p>Delay Differential Equations and Numerical Solution Approximation Methods - is it all really necessary?</p> | ||
+ | </blockquote> | ||
+ | <h2>Why model?</h2> | ||
+ | |||
+ | <p> In our project we sought to:</P | ||
+ | <ul> | ||
+ | <li>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. </li> | ||
+ | <li>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. </li> | ||
+ | </ul> | ||
+ | <br> | ||
+ | <p>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. </p> | ||
+ | |||
+ | </div> | ||
+ | <div class="span6"> | ||
+ | <h2>Why wet biomass?</h2> | ||
+ | <p>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:</p> | ||
+ | <ul> | ||
+ | <li>Most relevant data for fish population modelling, for example - recruitment rate, is expressed in terms of biomass. Therefore, we avoided unnecessary conversions and errors.</li> | ||
+ | <li>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. </li> | ||
+ | </ul> | ||
+ | <h2>Why adult fish?</h2> | ||
+ | <p>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.</p> | ||
+ | |||
+ | <p>We sought to model fish biomass throughout time but to model <em>all</em> biomass would have required us to take into account the <em>growth</em> 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.</p> | ||
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+ | </div> | ||
+ | </div> | ||
+ | <!-- And ends here --> | ||
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+ | <div class="span12"> | ||
+ | <h2>The mathematics</h2> | ||
+ | <p>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.</p> | ||
+ | <br> | ||
+ | |||
+ | <div class="well span11"> | ||
+ | <h4>Our mathematical model</h4> | ||
+ | <p>$$\textrm{Biomass (this year)} - \textrm{Biomass (last year)} = \textrm{Recruits} - \textrm{Natural Deaths} - \textrm{Fish Caught}$$</p> | ||
+ | <p>$$\frac{dB}{dt}=r w e^{-\delta_J \tau}(1-\frac{B(t-\tau)}{k})B(t-\tau) -\delta_M B(t) - F(t) B(t)$$</p> | ||
+ | </div> | ||
+ | <div class="accordion span12" id="accordion2"> | ||
+ | <h3>Equation explained</h3> | ||
+ | <div class="accordion-group"> | ||
+ | <div class="accordion-heading"> | ||
+ | <a class="accordion-toggle" data-toggle="collapse" data-parent="#accordion2" href="#collapseOne"> | ||
+ | Parameter definitions | ||
+ | </a> | ||
+ | </div> | ||
+ | <div id="collapseOne" class="accordion-body collapse"> | ||
+ | <div class="accordion-inner"> | ||
+ | <table class="table table-hover"> | ||
+ | <thead> | ||
+ | <tr> | ||
+ | <th>Parameter</th> | ||
+ | <th>Explanation</th> | ||
+ | <th>Units</th> | ||
+ | </tr> | ||
+ | </thead> | ||
+ | <tbody> | ||
+ | <tr> | ||
+ | <td>$\frac{dB}{dt}$</td> | ||
+ | <td>Biomass(this year)-Biomass(last year), when the time scale over which you are calculating these yearly changes is large</td> | ||
+ | <td>Tonnes per year</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$B(t)$</td> | ||
+ | <td>Biomass at time t</td> | ||
+ | <td>Tonnes</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$r$</td> | ||
+ | <td>Number of junior fish produced by 1kg of mature adult fish per year</td> | ||
+ | <td>Per kg</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$w$</td> | ||
+ | <td>Average mass of a mature fish</td> | ||
+ | <td>kg</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$\tau$</td> | ||
+ | <td>Average time for a junior fish to reach maturity (gain ability to breed)</td> | ||
+ | <td>Years</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$\delta_J$</td> | ||
+ | <td>Juvenile natural mortality rate (fraction of junior fish that die to natural causes in a year)</td> | ||
+ | <td>Per year</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$k$</td> | ||
+ | <td>Carrying capacity of fish population (maximum size population can reach before competition for resourses causes population to decrease)</td> | ||
+ | <td>Tonnes</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$\delta_M$</td> | ||
+ | <td>Natural mortality rate (fraction of adult fish that die due to natural causes in a year)</td> | ||
+ | <td>Per year</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$F(t)$</td> | ||
+ | <td>Fishing mortality rate (fraction of adult fish that die due to being caught at time t)</td> | ||
+ | <td>Per year</td> | ||
+ | </tr> | ||
+ | </tbody> | ||
+ | </table> | ||
+ | </div> <!-- Accordion1 end --> | ||
+ | </div> | ||
+ | </div> | ||
+ | <div class="accordion-group"> | ||
+ | <div class="accordion-heading"> | ||
+ | <a class="accordion-toggle" data-toggle="collapse" data-parent="#accordion2" href="#collapseTwo"> | ||
+ | In depth explanation of terms in model | ||
+ | </a> | ||
+ | </div> | ||
+ | <div id="collapseTwo" class="accordion-body collapse"> | ||
+ | <div class="accordion-inner"> | ||
+ | <table class="table table-hover"> | ||
+ | <thead> | ||
+ | <tr> | ||
+ | <th>Term from model</th> | ||
+ | <th>Physical meaning</th> | ||
+ | </tr> | ||
+ | </thead> | ||
+ | <tbody> | ||
+ | <tr> | ||
+ | <td>$rB(t-\tau)$</td> | ||
+ | <td>Maximum number of junior fish that could reach maturity at time t (if no natural death present)</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$e^{-\delta_J \tau}$</td> | ||
+ | <td>Fraction of junior fish that survive to reach maturity</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$r w e^{-\delta_J \tau}B(t-\tau)$</td> | ||
+ | <td>Biomass contributed to stock of adult fish biomass at time t due to junior fish reaching maturity at that point</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$\delta_M B(t) $</td> | ||
+ | <td>Adult fish biomass lost from stock at time t due to natural death</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$F(t) B(t)$</td> | ||
+ | <td>Adult fish biomass lost from stock at time t due to fishing</td> | ||
+ | </tr> | ||
+ | </tbody> | ||
+ | </table> | ||
+ | </div> <!-- Accordion2 end --> | ||
+ | </div> | ||
+ | </div> | ||
+ | </div> <!-- Table end --> </div> | ||
+ | </div> | ||
+ | </div> | ||
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- | |||
- | |||
- | |||
- | |||
- | |||
- | |||
</section> | </section> | ||
+ | <!-- Model tuning | ||
+ | ================================================== --> | ||
- | <!-- Data | + | <section id="tuning"> |
+ | <div class="page-header"> | ||
+ | <h1>Parameter tuning</h1> | ||
+ | </div> | ||
+ | <div> | ||
+ | |||
+ | <div class="row"> | ||
+ | <div class="span6"> | ||
+ | <h2>Defining our model</h2> | ||
+ | |||
+ | <p>Content with the formulation of our model, we then sought to assign values to the parameters involved (‘parameter’ refers to, for example, “r”, “w” or “k”). Further, we looked for values which enabled our model to make predictions throughout time that resembled our biomass data. We could then use the tuned differential equation to make well grounded future biomass estimates.</p> | ||
+ | <h2>Data collection and initial values for parameters</h2> | ||
+ | <p>We located values for the recruitment rate (r), the mass of an adult fish (w), the time for a fish to reach maturity ($\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.</p> | ||
+ | <p>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.</p> | ||
+ | <p>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.</p> | ||
+ | |||
+ | <h2>Tuning and refining our model</h2> | ||
+ | <p>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.</p> | ||
+ | <p>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:</p> | ||
+ | <p>$$\frac{dB}{dt}=r w e^{-(\delta_J +FJ(t)) \tau}(1-\frac{B(t-\tau)}{k})B(t-\tau) -\delta_M B(t) - F(t) B(t)$$</p> | ||
+ | <p> | ||
+ | $$ | ||
+ | FJ(t) = \left\{ | ||
+ | \begin{array}{lr} | ||
+ | FJ & : t \leq 1978 \\ | ||
+ | 0 & : t \geq 1979 \\ | ||
+ | \end{array} | ||
+ | \right. | ||
+ | $$ | ||
+ | </p> | ||
+ | <p>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.</p> | ||
+ | |||
+ | </div> | ||
+ | <div class="span6"> | ||
+ | <div id="myCarousel2" class="carousel slide"> | ||
+ | <!-- Carousel items --> | ||
+ | |||
+ | <div class="carousel-inner"> | ||
+ | <div class="active item"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/d/db/Model_Tuning-2.png"> | ||
+ | <p><span class="label label-info">Chart: Model prediction and actual biomass values </span></p> | ||
+ | |||
+ | <div class="carousel-caption"> | ||
+ | <h4><span class="badge badge">1/2</span> Parameter tuning</h4> | ||
+ | |||
+ | </div> | ||
+ | |||
+ | </div> | ||
+ | |||
+ | <div class="item"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/1/16/Catch_Comparisons-2.png"> | ||
+ | <p><span class="label label-info">Chart: Model catch prediction and real-world catch values</span></p> | ||
+ | |||
+ | |||
+ | <div class="carousel-caption"> | ||
+ | <h4><span class="badge badge">2/2</span> Catch Predictions and Catch Data Points</h4> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | |||
+ | </div><!-- Carousel nav --> | ||
+ | |||
+ | <a class="carousel-control left" href="#myCarousel2" data-slide="prev">‹</a> <a class="carousel-control right" href="#myCarousel2" data-slide="next">›</a> | ||
+ | </div> <!-- Carousel --> | ||
+ | <script type="text/javascript"> | ||
+ | $('#myCarousel2').carousel({ | ||
+ | interval: 500000000000 | ||
+ | }) | ||
+ | </script> | ||
+ | <div class="well"> | ||
+ | <h2>Browse the data</h2> | ||
+ | <p> | ||
+ | <form action="https://docs.google.com/folder/d/0By6Sb8tMgPIaLVZCNkVmVXl3WE0/edit"> | ||
+ | <button type="submit" class="btn btn-large btn-info" href="https://docs.google.com/folder/d/0By6Sb8tMgPIaLVZCNkVmVXl3WE0/edit"></img>Modelling data</button></form><!-- Wikitext kills our usual buttons, so here's a terrible hack with a form button --> | ||
+ | </p> | ||
+ | |||
+ | <p>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.</p> | ||
+ | </div> | ||
+ | |||
+ | <h2>A sensible result?</h2> | ||
+ | <p>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.</hp> | ||
+ | <p>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.</p> | ||
+ | <p>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.</p> | ||
+ | |||
+ | |||
+ | |||
+ | <h2>Tuned parameters</h2> | ||
+ | <p> 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. </p> | ||
+ | <div class="accordion span5" id="accordion3"> | ||
+ | <div class="accordion-group"> | ||
+ | <div class="accordion-heading"> | ||
+ | <a class="accordion-toggle" data-toggle="collapse" data-parent="#accordion3" href="#collapseTwo1"> | ||
+ | Parameter tuned values | ||
+ | </a> | ||
+ | </div> | ||
+ | <div id="collapseTwo1" class="accordion-body collapse"> | ||
+ | <div class="accordion-inner"> | ||
+ | <table class="table table-hover"> | ||
+ | <thead> | ||
+ | <tr> | ||
+ | <th>Parameter</th> | ||
+ | <th>Initial estimate</th> | ||
+ | <th>Range for tuning</th> | ||
+ | <th>Step size for tuning</th> | ||
+ | <th>Final value</th> | ||
+ | </tr> | ||
+ | </thead> | ||
+ | <tbody> | ||
+ | <tr> | ||
+ | <td>$rw$</td> | ||
+ | <td>4.9</td> | ||
+ | <td>N/A</td> | ||
+ | <td>N/A</th> | ||
+ | <td>4.9</th> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$\delta_J$</td> | ||
+ | <td>N/A</td> | ||
+ | <td>0.7-2</td> | ||
+ | <td>0.05</th> | ||
+ | <td>0.7</th> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$\tau$</td> | ||
+ | <td>2.5</td> | ||
+ | <td>1-4</td> | ||
+ | <td>0.25</th> | ||
+ | <td>3.25</th> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$k$</td> | ||
+ | <td>N/A</td> | ||
+ | <td>5*10^9-11*10^9</td> | ||
+ | <td>1*10^9</th> | ||
+ | <td>5*10^9</th> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$\delta_M$</td> | ||
+ | <td>N/A</td> | ||
+ | <td>0.1-0.5</td> | ||
+ | <td>0.5</th> | ||
+ | <td>0.3</th> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$FJ$</td> | ||
+ | <td>N/A</td> | ||
+ | <td>0.01-0.3</td> | ||
+ | <td>0.01</th> | ||
+ | <td>0.02</th> | ||
+ | </tr> | ||
+ | </tbody> | ||
+ | </table> | ||
+ | </div> <!-- Accordion1 end --> | ||
+ | </div> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | |||
+ | </div> | ||
+ | </div> <!--row--> | ||
+ | </div> | ||
+ | </section> | ||
+ | |||
+ | <!-- Model predictions | ||
================================================== --> | ================================================== --> | ||
- | |||
- | |||
- | |||
- | |||
- | |||
- | + | <section id="predictions"> | |
+ | <div class="page-header"> | ||
+ | <h1>Model predictions</h1> | ||
+ | </div> | ||
+ | <div> | ||
+ | <div class="row-fluid"> | ||
+ | <div class="span12"> | ||
+ | <ul class="thumbnails"> | ||
+ | <li class="span12"> | ||
+ | <div class="thumbnail"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/9/9a/Fishpredictioncharts.png" alt=""> | ||
+ | <p class="label-set"> | ||
+ | <span class="label" style="background-color:#D2A3D2">Fish biomass depletion between 2006 and 2100</span> | ||
+ | <span class="label" style="background-color:#E6CCE6">An alternative outcome</span> | ||
+ | </p> | ||
+ | <h5>Two futures, one world: the choice is ours</h5> | ||
+ | <p>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.</p> | ||
+ | <p>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.</p> | ||
+ | </div> | ||
+ | </li> | ||
+ | </ul> | ||
+ | </div> | ||
+ | </div> <!--Row--> | ||
+ | |||
+ | <div class="row"> | ||
+ | <div class="span7"> | ||
+ | <h2>A fishy dilemma</h2> | ||
+ | <p>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.</p> | ||
+ | <h2>Changing the future</h2> | ||
+ | <p>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?</p> | ||
+ | <p>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 <b>replacing the need for wild fish in aquaculture</b>. 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.<p> | ||
+ | <p>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).</p> | ||
+ | <p>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!</p> | ||
+ | <h2>The cost of success</h2> | ||
+ | <p>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?</p> | ||
+ | <p>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 <i>((catch(t)+aquaculture(t)-(catch required to produce aquaculture)(t))/population(t)</i> 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.<p> | ||
+ | |||
+ | |||
+ | </div> | ||
+ | <div class="span5"> | ||
+ | <div id="myCarousel3" class="carousel slide"> | ||
+ | <!-- Carousel items --> | ||
+ | |||
+ | <div class="carousel-inner"> | ||
+ | <div class="active item"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/b/b5/Model_Prediction_Fish_Biomass-2.png"> | ||
+ | <p><span class="label label-info">Chart: From model – a biomass picture for the future</span></p> | ||
+ | |||
+ | <div class="carousel-caption"> | ||
+ | <h4><span class="badge badge">1/5</span> The Future for Fish</h4> | ||
+ | |||
+ | <p>Model's prediction of world fish biomass, as it changes throughout the years between 1950 and 2100; as well as biomass data. In this case, Fishing Mortality Rates are determined from a correlation between past fishing motality data and population data (refer to next chart). | ||
+ | </p> | ||
+ | </div> | ||
+ | |||
+ | </div> | ||
+ | |||
+ | <div class="item"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/4/46/Fishing_Mortality_Graph_Future.png"> | ||
+ | <p><span class="label label-info">Chart: Fishing mortality rates, projected into the future</span></p> | ||
+ | |||
+ | |||
+ | <div class="carousel-caption"> | ||
+ | <h4><span class="badge badge">2/5</span> Fishing Mortality Rates</h4> | ||
+ | |||
+ | <p>In order to run our model into the future, we had to obtain an estimate of how Fishing Mortality Rates would change with time. We found a strong correlation between past Fishing Mortality Rates and past population data (Pearson's r = 0.897; P-value < 0.00001), as might have been expected from the definition of Fishing Mortality Rate as Catch(t)/Biomass(t) at time t. The graph shows Fishing Mortality Rates at different times, when this correlation has been used to make the predictions. | ||
+ | </p> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | <div class="item"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/7/7c/Catch_Future-1.png"> | ||
+ | <p><span class="label label-info">Chart: Catch projected into the future</span></p> | ||
+ | |||
+ | <div class="carousel-caption"> | ||
+ | <h4><span class="badge badge">3/5</span> Catch: from Fishing Mortality Rates and Biomass Predictions | ||
+ | </h4> | ||
+ | |||
+ | <p>If Fishing Mortality values are those indicated in the previous graph and we run our model to calculate world fish biomass until 2100, we can obtain the model's predictions of catch at different times by multiplying the Fishing Mortality Rate (t) with the biomass (t). | ||
+ | </p> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | <div class="item"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/1/11/Fish_Survive_WooHoo.png"> | ||
+ | <p><span class="label label-info">Chart: From model – an alternative biomass picture for the future</span></p> | ||
+ | |||
+ | <div class="carousel-caption"> | ||
+ | <h4><span class="badge badge">4/5</span> An Alternative Future - for fish and for us | ||
+ | </h4> | ||
+ | |||
+ | <p>If we implement changes to our fishing habits, and reduce our predicted catch for the future (previous graph) by (0.7 x farmed fish produced in 2006) each year, our model can predict a very different outcome for fish in time. (We reduce catch by the specified amount each year as a result of the fact that we assume aquaculture output to remain at 2006 level and that for every one tonne of farmed fish produced, 0.7 tonnes of wild fish are required for feed. With our alternative omega 3 production, we think it is finally possible to produce this level of farmed fish with no input from our oceans). | ||
+ | </p> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | <div class="item"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/0/0c/Altered_Catch-1.png"> | ||
+ | <p><span class="label label-info">Chart: Catch and altered catch curves</span></p> | ||
+ | |||
+ | <div class="carousel-caption"> | ||
+ | <h4><span class="badge badge">5/5</span> Altered Catch | ||
+ | </h4> | ||
+ | |||
+ | <p>In order to provide fish with a future, we suggest reducing catch by (0.7 x farmed fish produced in 2006) at each year between 2006 and 2070, at which point we can no longer reduce catch by this amount (predicted catch is too low) and we recommend, instead, reducing it to zero.</p> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | |||
+ | </div><!-- Carousel nav --> | ||
+ | |||
+ | <a class="carousel-control left" href="#myCarousel3" data-slide="prev">‹</a> <a class="carousel-control right" href="#myCarousel3" data-slide="next">›</a> | ||
+ | </div> <!-- Carousel --> | ||
+ | <script type="text/javascript"> | ||
+ | $('#myCarousel3').carousel({ | ||
+ | interval: 500000000000 | ||
+ | }) | ||
+ | </script> | ||
+ | <ul class="thumbnails"> | ||
+ | <li class="span5"> | ||
+ | <a href="#how-much-omega" class="thumbnail" data-toggle="modal"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/3/31/How_much_Omega_3.png" alt=""> | ||
+ | <h5>Saving our oceans and feeding our world</h5> | ||
+ | <p>Omega-3 production levels for three different scenarios. Green: omega-3 required to maintain aquaculture output at 2006 level and provide farmed fish with fish-free feed. Purple: omega-3 necessary to maintain fish to population ratio that existed in 2000s. We will produce additional fish through the alternative aquaculture methods mentioned previously. Blue: omega-3 needed to provide world population with their recommended daily intake of omega-3 of 500mg per day.</p> | ||
+ | </a> | ||
+ | </li> | ||
+ | </ul> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | |||
+ | </div> | ||
</section> | </section> | ||
+ | |||
+ | <section id="references"> | ||
+ | <div class="page-header"> | ||
+ | <h1>References</h1> | ||
+ | </div> | ||
+ | |||
+ | <p>Burd, A.C., 1986. Why Increase Mesh Sizes?, Lowestoft. | ||
+ | Christensen, Villy et al., 2009. Database-driven models of the world’s Large Marine Ecosystems. Ecological Modelling, 220(17), pp.1984–1996. Available at: http://dx.doi.org/10.1016/j.ecolmodel.2009.04.041 [Accessed July 26, 2012].<p> | ||
+ | <p>Food and Agriculture Organization of the United Nations (FAO), FIGIS - Fisheries Statistics - Global Production Statistics 1950-2010 . Available at: http://www.fao.org/figis/servlet/TabLandArea?tb_ds=Production&tb_mode=TABLE&tb_act=SELECT&tb_grp=COUNTRY&lang=en [Accessed September 22, 2012a].</p> | ||
+ | <p>Food and Agriculture Organization of the United Nations (FAO), Introduction to tropical fish stock assessment - Part 1: Manual – ESTIMATION OF GROWTH PARAMETERS. Available at: http://www.fao.org/docrep/W5449E/w5449e05.htm [Accessed September 23, 2012b].</p> | ||
+ | <p>Kris-Etherton, P.M. et al., 2007. Position of the American Dietetic Association and Dietitians of Canada: dietary fatty acids. Journal of the American Dietetic Association, 107(9), pp.1599–611. Available at: http://www.ncbi.nlm.nih.gov/pubmed/17936958 [Accessed September 11, 2012].</p> | ||
+ | <p>Ricard, D. et al., 2011. Examining the knowledge base and status of commercially exploited marine species with the RAM Legacy Stock Assessment Database. Fish and Fisheries, p.no–no. Available at: http://doi.wiley.com/10.1111/j.1467-2979.2011.00435.x [Accessed July 20, 2012].</p> | ||
+ | <p>Tacon, A.G.J. & Metian, M., 2008. Global overview on the use of fish meal and fish oil in industrially compounded aquafeeds: Trends and future prospects. Aquaculture, 285(1-4), pp.146–158. Available at: http://dx.doi.org/10.1016/j.aquaculture.2008.08.015 [Accessed July 20, 2012].</p> | ||
+ | <p>Tremblay-Boyer, L. et al., 2011. Modelling the effects of fishing on the biomass of the world’s oceans from 1950 to 2006. Marine Ecology. Available at: http://www.seaaroundus.org/researcher/dpauly/PDF/2011/JournalArticles/ModellingEffectsofFishingonBiomassofWorldsOceans.pdf [Accessed September 22, 2012].</p> | ||
+ | <p>United Nations Department of Economic and Social Affairs, World Population Prospects, the 2010 Revision. Available at: http://esa.un.org/wpp/Excel-Data/population.htm [Accessed September 23, 2012].</p> | ||
+ | </section> | ||
+ | |||
</div><!-- /container --> | </div><!-- /container --> | ||
+ | |||
+ | <!-- Chart modals --> | ||
+ | |||
+ | <div class="modal large hide" id="how-much-omega"> | ||
+ | <div class="modal-header"> | ||
+ | <button type="button" class="close" data-dismiss="modal">×</button> | ||
+ | <h3>Chart zoom: <em>Saving our oceans and feeding our world</em></h3> | ||
+ | </div> | ||
+ | <div class="modal-body"> | ||
+ | <center> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/3/31/How_much_Omega_3.png" alt=""> | ||
+ | </center> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | <div class="modal large hide" id="ram-map"> | ||
+ | <div class="modal-header"> | ||
+ | <button type="button" class="close" data-dismiss="modal">×</button> | ||
+ | <h3>Figure zoom: <em>RAM Database coverage</em></h3> | ||
+ | </div> | ||
+ | <div class="modal-body"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/1/17/RAMmap.png" alt="" > | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | <div class="modal large hide" id="biomass-curve"> | ||
+ | <div class="modal-header"> | ||
+ | <button type="button" class="close" data-dismiss="modal">×</button> | ||
+ | <h3>Figure zoom: <em>RAM Database coverage</em></h3> | ||
+ | </div> | ||
+ | <div class="modal-body"> | ||
+ | <img src="https://static.igem.org/mediawiki/2012/6/65/Biomass-chart.png" alt=""> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | <!-- End of chart modals --> | ||
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Latest revision as of 02:31, 27 September 2012
The mathematics of ω-3
Modelling the impact of alternative omega-3 production on the global fish population
Our model 101
![](https://static.igem.org/mediawiki/2012/f/f0/ModSquadLogo_100.png)
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:
![](https://static.igem.org/mediawiki/2012/2/23/Process1.png)
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.
![](https://static.igem.org/mediawiki/2012/9/9b/Process2.png)
2: Mathematical model
We hypothesised a differential equation model which we believe incorporates the key features responsible for fish population growth and decline.
![](https://static.igem.org/mediawiki/2012/f/ff/Process3.png)
3: Parameter tuning
We changed the parameters in our model until our model's predictions closely replicated the real world fish biomass data.
![](https://static.igem.org/mediawiki/2012/7/76/Process4.png)
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.
Fish biomass data – collection and manipulation
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.
Mathematical model
Delay Differential Equations and Numerical Solution Approximation Methods - is it all really necessary?
Why model?
In our project we sought to:
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
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 |
Model predictions
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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.
References
Burd, A.C., 1986. Why Increase Mesh Sizes?, Lowestoft. Christensen, Villy et al., 2009. Database-driven models of the world’s Large Marine Ecosystems. Ecological Modelling, 220(17), pp.1984–1996. Available at: http://dx.doi.org/10.1016/j.ecolmodel.2009.04.041 [Accessed July 26, 2012].
Food and Agriculture Organization of the United Nations (FAO), FIGIS - Fisheries Statistics - Global Production Statistics 1950-2010 . Available at: http://www.fao.org/figis/servlet/TabLandArea?tb_ds=Production&tb_mode=TABLE&tb_act=SELECT&tb_grp=COUNTRY&lang=en [Accessed September 22, 2012a].
Food and Agriculture Organization of the United Nations (FAO), Introduction to tropical fish stock assessment - Part 1: Manual – ESTIMATION OF GROWTH PARAMETERS. Available at: http://www.fao.org/docrep/W5449E/w5449e05.htm [Accessed September 23, 2012b].
Kris-Etherton, P.M. et al., 2007. Position of the American Dietetic Association and Dietitians of Canada: dietary fatty acids. Journal of the American Dietetic Association, 107(9), pp.1599–611. Available at: http://www.ncbi.nlm.nih.gov/pubmed/17936958 [Accessed September 11, 2012].
Ricard, D. et al., 2011. Examining the knowledge base and status of commercially exploited marine species with the RAM Legacy Stock Assessment Database. Fish and Fisheries, p.no–no. Available at: http://doi.wiley.com/10.1111/j.1467-2979.2011.00435.x [Accessed July 20, 2012].
Tacon, A.G.J. & Metian, M., 2008. Global overview on the use of fish meal and fish oil in industrially compounded aquafeeds: Trends and future prospects. Aquaculture, 285(1-4), pp.146–158. Available at: http://dx.doi.org/10.1016/j.aquaculture.2008.08.015 [Accessed July 20, 2012].
Tremblay-Boyer, L. et al., 2011. Modelling the effects of fishing on the biomass of the world’s oceans from 1950 to 2006. Marine Ecology. Available at: http://www.seaaroundus.org/researcher/dpauly/PDF/2011/JournalArticles/ModellingEffectsofFishingonBiomassofWorldsOceans.pdf [Accessed September 22, 2012].
United Nations Department of Economic and Social Affairs, World Population Prospects, the 2010 Revision. Available at: http://esa.un.org/wpp/Excel-Data/population.htm [Accessed September 23, 2012].