The musselwatch

To monitor the concentration of a pollutant in waterways, it sometimes makes sense to determine the concentration in mussels, , which have been exposed in such waterways, rather than to determine the concentration in the water, . The first argument is that of bio-availability. Not all of the chemically determined pollutant in the water is actually available to organisms, due to a variety of chemical forms in which the pollutant is present (e.g. ligands). Therefore tissue concentrations provide information which is more relevant to the problem of pollution. Another argument is that the concentration of pollutant might have some sharp peeks, which are easy to miss in infrequent determinations of concentrations in the water. Mussels, in some way, integrate the external concentration in time. Therefore it should still be possible to observe a trace of such peeks in tissue concentrations at infrequent sampling, if, at least, the mussels do not close their valves during such peeks. Let us study this argument in more detail. Suppose that the tissue concentration follows a simple one-compartment process, i.e.

where stands for the change of in time. For other symbols, see Table 2.1. So, if we know in sufficient detail, and if it is sufficiently smooth, we can reconstruct through

(2.2)

For the present purpose, we need an explicit expression of in terms of , however, because we want to study the effects of rapidly changing water concentrations. The solution is found from (2.1) to be

If is actually constant, (2.3) reduces to

Now we will approximate the continuous function by a step function which changes only at discrete, equidistant time points . That is, is constant over a time interval , at value . The tissue concentration at the end of the interval is given by

with and . Now we assume that the values represent trials taken independently from some probability density function. This is reasonable for the situation in a river, where well-mixed water surrounding the mussel is completely replaced in an interval of length $d$. The schedule (2.5) is known as a (first-order) stochastic difference equation or an autoregression process, because the new value for is a weighted sum of the old value and an independent random variable. Alternatively, it can be expressed in a so-called moving average scheme:

Table 2.1: List for frequently used symbols

The expected value for will be

(2.7)

Ultimately, i.e. for large i, we have

So, the ultimate mean tissue concentration is just the mean external concentration times the bio-accumulation factor . This result corresponds with the deterministic situation if $c$ is constant. Then , as can be seen directly from (2.4). The variance of the tissue concentrations is found from (2.6) to be

(2.9)

Ultimately we have

So, for the ratio of the coefficients of variation, CV, we have

This ratio is larger than 1, which means that . The quotient is increasing in r, so when is large (i.e. is large compared with $d$), the behaviour of the tissue concentration will be much more smooth than the behaviour of the concentration in the water. The subsequent tissue concentrations are correlated, as opposed to the concentrations in the water. This is expressed by the so-called autocovariance function , or the autocorrelation function (both considered as functions in h), given by

and

Show this by writing in terms of using (2.5) several times. We can also study the interdependence of and by looking at the (cross-)correlation function corr() which in this case can easily be derived to be

The crosscorrelation function in Fig.2.1 shows how the concentration in tissue lags behind concentration fluctuations in the water. The value has been chosen in (2.13) and (2.14). This smoothing also results in a gradually decreasing autocorrelation function of Q.

Figure 2.1: Left: A random sample of the time path of the concentration fluctuations in water and tissue, if the first show no memory. Right: The autocorrelation functions of concentrations in water, $\Diamond$, and tissue, $\Box$ and their cross correlation function, $\circ$.

Usually, the behaviour of water concentrations shows more 'memory'. Suppose, we have a lake of constant volume , with an inflow and an outflow of water at rate $v$ per unit of time. Writing $T$ for , which is known as the residence time, we have for the concentration $c$ in the lake and in the inflowing water, assuming a one-compartment process:

(2.15)

The solution, in analogy with (2.3), is:

(2.16)

We will approximate the same way we did . That is, is constant over an interval at value . Then we have

This is again an auto-regression scheme if the 's are independent and identically distributed. Ultimately we have, analogously with (2.8) and (2.10), and . So the (ultimate) mean and variance of do not depend on $i$. The main difference with the river situation is that subsequent values for are now correlated. The autocovariance function (see (2.12)) is given by

We can express the variance of in terms of that for using (2.6):

Now, we arrive at the following ratio for the coefficients of variation of the water and the tissue concentrations:

(2.20)

which reduces, as expected, to the river situation (2.11) for . The ratio is larger than 1. It is an increasing function in r, but decreasing in s. So we can conclude that the tissue concentrations will show less variation than the external concentrations indeed, and that the reduction can be significant if the process governing the external concentrations has little memory and if the tissue concentrations are following the external concentrations slowly. The time behaviour of the tissue concentration and its relation with the water concentration is further illustrated by the autocovariance and the cross-covariance functions. To obtain these functions, we first write (2.17) into the moving-average scheme

(2.21)

which we substitute into (2.6), obtaining for large i

(2.22)

Straightforward expansion gives

and, for , we have

The auto- and cross-covariance functions contain information about the interdependence of the variables. We started to study the case of independent concentrations in the water, which results in an autocovariance function, which starts at for time lag , but drops to zero for . The cross-covariance function in Fig.2.1 shows how the concentration in tissue lags behind concentration fluctuations in the water. The values and has been chosen in (2.23), (2.24) and (2.18). This smoothing also results in a gradually decreasing autocovariance function. When the concentration in the water has some memory, like that modelled in (2.17), the autocovariance function of concentrations in the water drops gradually, as expected. See Fig.2.2, where has been chosen 0.6. The cross-covariance function now shows first an increase. However, the way the tissue follows the concentrations in the water, was exactly the same as in Fig.2.1. It is therefore very difficult to interprete covariance functions without a model for the way the variables depend on each other.

Figure 2.2: Left: A random sample of the time path of the concentration fluctuations in water and tissue, if the first shows memory according to the moving average process. The autocovariance functions of concentrations in water, $\Diamond$, and tissue, $\Box$ and their cross-covariance function, $\circ$.