Chapter 28, Problem 33P

The growth of floating, unicellular algae below a sewage treatment plant discharge can be modeled with the following simultaneous ODEs:

$\begin{array}{l}\frac{da}{dt}=-\left[K_g\left(n,p\right)-k_d-k_s\right]a\\ \frac{dn}{dt}=r_{nc}{k}_{h}c-r_{na}{k}_g\left(n,p\right)a\\ \frac{dp}{dt}{r}_{pc}{k}_{h}c-r_{pa}{k}_g\left(n,p\right)a\\ \frac{dc}{dt}=r_{ca}{k}_{a}a-k_{h}c\end{array}$

where $t=\text{traveltime }\left(\text{d}\right),a=\text{algal}$ chlorophyll concentration $\left(\mu g\text{A/L}\right),n=\text{inorganic}$ nitrogen concentration $\left(\mu g\text{N/L}\right),p=\text{inorganic}$ phosphorus concentration $\left(\mu gP/L\right),c=\text{detritus}$ concentration $\left(\mu gC/L\right),k_d=\text{algal}$ death rate $\left(l\text{d}\right),k_s\text{=algal }$ settling rate $\left(l\text{d}\right),k_h=\det \text{rital}$ hydrolysis rate $\left(l\text{d}\right),r_{nc}=\text{nitrogen}$ to-carbon ratio $\left(\mu g\text{N/}\mu \text{gC}\right),r_{pc}=\text{phosphorus-to-carbon}$ ratio $\left(\mu g\text{P/}\mu \text{gC}\right),r_{na}=\text{nitrogen-to}$ chlorophyll ratio $\left(\mu g\text{N/}\mu \text{gA}\right),r_{pa}=\text{Phophorus-to}$ chlorophyll ratio $\left(\mu g\text{p/}\mu \text{gA}\right),and{k}_g\left(n,p\right)=\text{algal}$ growth rate $\left(l\text{d}\right)$, which can be computed with

$k_g\left(n,p\right)=k_g\min \left\{\frac{p}{k_{sp}+p},\frac{n}{k_{sn}+n}\right\}$

where $k_g=\text{the }$ the algal growth rate at excess nutrient levels $\left(l\text{d}\right),k_{sp}=\text{the}$ phosphorus half-saturation constant $\left(\mu g\text{P/L}\right),$ and $k_{sn}=\text{the}$ nitrogen half-saturation constant $\left(\mu g\text{N/L}\right)$. Use the ode45 and ode15s functions to solve these equations from $t=0$ to 50 d given the initial conditions $a=1,n=4000,p=800,\text{and}c=0$ Note that the parameters are $k_d=0.1,k_s=0.15,k_h=0.025,r_{nc}=0.18,r_{pc}=0.025,r_{na}=7.2,r_{pa}=1,$ $r_{ca}=40,k_g=0.5,k_{sp}=2,\text{and }{k}_{sn}=15$. Develop plots of both solutions and interpret the results.