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.

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  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

Chart: All datasets prior to any manipulations

1/5 Choosing boundaries for the datasets

The next step was to “combine” the individual species-specific time series into one, which would become our biomass data.

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.

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.

Chart: Datasets spanning different time periods

2/5 Extrapolating datasets between 1932 and 2006

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.

The upper boundary of extrapolation was set to 2006, since after 2006, the number of datasets with information decreased rapidly.

Chart: The "standard" biomass sets

3/5 How does extrapolation of datasets work?

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.

Chart: Extrapolated datasets

4/5 Combining the datasets

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.

Chart: The final time-biomass evolution curve

5/5 Our final result

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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.

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.

Tuned parameters

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.