1gillman_m_an_introduction_to_mathematical_models_in_ecology

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observed processes within the study system – for example, levels of survival or migration – and test them against observed data. Both of these modelling activities will be described in this book.

Ecological and evolutionary patterns in space and time are intrinsically mathematical. That is, events at one point in time or space can be related to events at previous points with the use of mathematical operators. This idea is illustrated by the Fibonacci sequence which starts with the numbers 1 and 1 and then continues by adding the previous two numbers to produce the sequence 1, 1, 2, 3, 5, 8 and so on. The sequence was named after Leonardo of Pisa who was known as Fibonacci (c. 1170–1250) and published in his Liber Abaci in 1202. This simple arithmetic sequence was originally used to predict changes in rabbit numbers and so is an early, perhaps the earliest, example of an ecological model. Although the underlying assumptions about how rabbits reproduce were naïve – for example, that female rabbits always produce one pair of rabbits every month, which contains one male and one female – they did consider appropriate biological issues of sex ratio and delayed reproduction (female rabbits were assumed to start reproducing after 2 months). In the model the first two numbers of the sequence (1, 1) represent one pair of rabbits in months 1 and 2. At the start of month 3 one new pair is born so the total is 2. At the start of month 4 another pair is born to the first pair making the total 3 and at the start of month 5 the original pair again reproduces as does the second pair (total of 5). The model sequence generates an increase with time which, at least in the short term, looks something like population increase (Fig. 1.1; we will see similar examples of population increase later).

This very simple model of rabbit population dynamics can be contrasted with modern complex models which incorporate features such as disease

Fig. 1.1 Increase in population size represented by the Fibonacci sequence increase. The fit of the curve through the Fibonacci sequence is discussed in Section 1.4.

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Asia, North and Central America and Africa. The authors’ main interest was in the fraction of the total net primary production (NPP) below ground and its contribution to global carbon cycling. Here we will use their data on aboveand below-ground NPP and mean temperature and rainfall at the different sites (Fig. 1.3). Note that the units of NPP are g of dry matter (DM) m−2 that accumulated over a fixed period of time; that is, it is a measure of the growth of the grassland species above and below ground.

Several trends are suggested from the best-fit lines in Fig. 1.3. Belowground NPP appears to decline with increasing temperature and rainfall

Fig. 1.3 Relationship of aboveand below-ground net primary production (NPP) with temperature and rainfall. The lines of best fit are determined by linear regression. DM, dry matter.

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Fig. 1.4 Relationship between rainfall and temperature.

whereas above-ground NPP appears to increase with temperature and rainfall. The distribution of points in the two graphs looks rather similar, which makes us suspect that temperature and rainfall are closely correlated (Fig. 1.4)

In fact the correlation coefficient is 0.777, which is statistically significant (P < 0.01). This high and significant correlation means that either rainfall or temperature can be used for the analysis. Notice that we have not fitted a line through the points in Fig. 1.4. This is because we do not wish to imply that one variable is dependent on the other.

Although trends are suggested in Fig. 1.3 we need to run the regression analysis to check the significance. The null hypothesis is that there is no change in the NPP with either rainfall or temperature. The results for aboveground NPP and temperature are F = 1.435, P = 0.256. Therefore the slope is not significantly different from zero. The regression analysis also provides an estimate of the values of m and c (gradient and intercept; the gradient is sometimes indicated by the letter b). However, this estimation is less useful when the regression is non-significant. In this case the value of m is 8.014 and the intercept is 272.39. Therefore, the regression equation would be:

Equation 1.1 could be used for predicting the value of above-ground NPP for a given value of temperature, but we would be wary of using it because the regression is not significant.

In fact, none of the four regressions in Fig. 1.3 are significant! This seems odd as we might expect temperature and/or rainfall to have a major effect on biomass. It may of course be that the non-significance is the correct result and that grassland NPP accumulation, when measured across different geographical areas, is independent of climate. An inspection of the data suggests

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Fig. 1.5 Residuals of below-ground NPP from regression against rainfall.

that some of the problems may lie in the large amount of variation in the data around the regression. We can consider this by examining the residuals; that is, the size of the difference of the observed values from the predicted value on the regression line (Fig. 1.5)

The pattern of residuals of below-ground NPP with respect to the rainfall shows a large scatter of values (positive and negative) between approximately 200 and 700 mm of rainfall, with small residuals for the higher values thereafter. This is a common distribution of residuals and a source of concern as regression analyses require that the residuals follow the same distribution pattern across the full range of independent values. A consequence of this deviant pattern is that the larger values of rainfall have a large effect on the regression. Indeed, it is possible to have a completely non-significant scatter of points at low values and for one point with a high value (for both variables) to generate a significant positive regression. Generally this means that more samples are needed in the intermediate range. In this case there is one large value with a mean temperature above 25°C and rainfall of nearly 1600 mm (you can see this clearly on the plot of rainfall against temperature, Fig. 1.4). Not surprisingly this occurs in a tropical region (Thailand). If we remove this value the regressions are even further away from significance.

Whereas the NPP above or below ground did not show a significant response to the climate variables, the fraction of below-ground NPP was significant (Fig. 1.6).

This study illustrates the importance of exploring your data. Although above-ground and below-ground NPP show no significant effects, the fraction of NPP below ground declines significantly with increasing temperature. Alternatively, we could also use the fraction of NPP above ground (what slope would you expect for this regression?). The same effect is seen with rainfall, which is expected as rainfall and temperature are positively correlated. The relationship of fraction of below-ground NPP with temperature is given by the equation:

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Fig. 1.6 (a) Relationship of fraction of below-ground NPP with temperature. We can also see that the distribution of the residuals with respect to temperature (b) is much better than in Fig. 1.5.

Therefore for each increase of 1°C of mean temperature there is predicted to be a corresponding decrease of 0.01225 in the fraction of below-ground NPP. The high significance of the regression (P < 0.001) gives us greater confidence in using equation 1.2 over equation 1.1. Furthermore the residuals appear to be well behaved along the full range of independent values. However, there still appears to be a reasonable amount of scatter around the predicted line. The amount of variation explained by the regression is summarized by the value of r2, the correlation coefficient squared. It has a particular meaning in regression analysis where it is known as the coefficient of determination. The r2 indicates the fraction of variation in the dependent variable explained by variation in the independent variable. Therefore, in this example, an r2 of 0.651 means that 0.651 (or 65.1%) of the variation in fraction of belowground NPP is explained by variation in temperature. From a modelling

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Fig. 1.7 Percentage of methane flux plotted against the two independent variables (peat temperature and water-table depth).

First, we check that there is no relationship between the (assumed) independent variables. It is true in this case although we note that there is a curious shape to the data with little variation in the water table at high and low temperatures but large variation at intermediate values (Fig. 1.9). Whereas it is possible to analyse the relationship between each independent variable and the dependent variable separately, we suspect that it is a combination of factors that contribute to the methane flux and we wish to capture that information in one mathematical statement. Rather than y = mx + c we now have two independent variables so the overall equation will look like this:

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Fig. 1.8 Residuals from the linear regressions in Fig. 1.7.

Fig. 1.9 Relationship between temperature and water-table depth. The correlation is not significant (r = 0.02, P > 0.9).