Modelling Pesticides in the Food Chain
\[A= \left( \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \ldots & a_{mn} \end{array} \right) \]
.The number of plants of type i consumed by herbivore j each year is represented by the matrix
\[B= \left( \begin{array}{cccc} b_{11} & b_{12} & \ldots & b_{1p} \\ b_{21} & b_{22} & \ldots & b_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \ldots & b_{np} \end{array} \right) \]
.The entry
\[c_{ij}\]
of the product \[AB\]
is the amount of pesticide of type i consumed by a herbivore of species j each year.The number of herbivores of species i consumed by a carnivore of species j each year is represented by the matrix
\[C= \left( \begin{array}{cccc} c_{11} & c_{12} & \ldots & c_{1q} \\ c_{21} & c_{22} & \ldots & c_{2q} \\ \vdots & \vdots & \ddots & \vdots \\ c_{p1} & c_{p2} & \ldots & c_{pq} \end{array} \right) \]
.The entry
\[d_{ij}\]
in the product \[ABC\]
is the amount of pesticide of type i consumed each year by a carnivore of type j.If the vector
\[\mathbf{u}=\begin{pmatrix}u_1\\u_2\\ \vdots \\ u_n\end{pmatrix}\]
represent the number of plants of each type eaten by a human each year, let the vector \[\mathbf{v}=\begin{pmatrix}v_1\\v_2\\ \vdots \\ v_p\end{pmatrix}\]
represent the number of each species of herbivore eaten by a human each year, and \[\mathbf{v}=\begin{pmatrix}w_1\\w_2\\ \vdots \\ w_q\end{pmatrix}\]
represents the number of carnivores of each type eaten by a human each year, then \[A \mathbf{u}, \; AB \mathbf{v}, \; ABC \mathbf{w}\]
represents the amount of pesticide of each type eaten by humans each year from consuming these different foods. 
