On a Nonlinear Problem Arising from Interaction of Algae with Light

Abstract

We examine a mathematical model arising from the study of the interaction of algae with light. In fact, let us consider a population of algae suspended in water in sea or in artificial tanks. If n=n(z;t) measures the suspected concentration of the algae that, at time t are suspended in water at depth z (-∞<-b≤z≤0, b is the maximum depth of water), we conjecture the following evolution for n

$$\begin{array}{*{20}{c}} {{u_t} = {\Delta _y}\left( {u,v} \right) + u\;f\left( {x,u,v} \right)} \\ {{v_t} = {\Delta _j}\left( {u,v} \right) + v\;g\left( {x,u,v} \right)} \\ \end{array}$$

((1a))

In the first term of right hand side of (1a), $\frac{1}{\tau }$ represents the death rate of algae, whereas the second term of (1a) represents the growth of the algae concentration due to the absorption of light. This growth is thought to be proportional to the product of algae-concentration with a known function of total radiant energy flux (see equation (1b)). The third term in (1a) is a removal term. The radiant energy satisfies the Boltzmann equation [1].