Answer to Question #110253 in Quantum Mechanics for Gazal

Question

Formulate a one-dimensional model describing the dynamics of phytoplankton growth C(x, t)
in a water mass taking into account the following: D , its diffusion coefficient, a its rate of
growth, b its mortality rate due to sinking. Fixing the area of interest as 0 <[= x <=1 and the initial
concentration of phytoplankton as 20 moles / cm cube, find the concentration distribution of
phytoplankton in 0 <=x <=1 at any time t .

Answer 1

Initially, we have $C_0$ moles of phytoplankton. As time $t$ passes, their population increases exponentially:

$$C(t)=C_0e^{at}.$$

Of course, part of the population dies:

$$C(t)=C_0(e^{at}-e^{bt}).$$

The fact how these processes run depend on the diffusion coefficient:

$$C(t)=C_0D(e^{at}-e^{bt}).$$

The more space the population has, the more they can be:

$$C(x,t)=C_0Dx(e^{at}-e^{bt}).$$

With the initial concentration of phytoplankton as 20 mol/cm³, find the concentration distribution of

phytoplankton in 0 <=x <=1 at any time t:

$$C(t)=20D(e^{at}-e^{bt}).$$