Please write a Matlab code as the solution to the following task.

In lectures we considered the problem of phosphorous exchange between the upper and lower ocean. Over time, the phosphorous in the upper ocean dissipates into the lower

ocean. This diffusion effect helps life survive. In this problem you will explore the effects of change in ocean temperature on the retention of phosphorous in these layers

Given $m_1,m_2$ represent the concentration of phosphorous in tera$-$moles in the lower and upper ocean respectively, the rate of change of these concentrations w$.$r$.$t$.$ time are

governed by the following pair of smultaneous differential equations.

$\frac{d{m}_1}{dt}=-k_{12}{m}_1+k_{21}{m}_2$;$\frac{d{m}_2}{dt}=k_{12}{m}_1-k_{21}{m}_2$

Here $t$ is measured in years, and $k_{12}=0\mathrm{.}58,k_{21}=1\mathrm{.}87$ are coefficients which express the diffusion rate between the layers of ocean.

Assume that at $t=0$ years, $m_1(0)=87\mathrm{.}5$Tmol, and $M=280$Tmol. Phosphorous doesn't dissappear over time, it just moves around. Hence the total mass of phosphorous in

the ocean is a constant and so $m_2(0)=M-m_1(0).$

Task $1$: Write two functions called lowerocean and upper$0$cean which take the samle arguments $($k$12,k21,{?}^{\text{'}}m1,m2$ and return the derivatives $\frac{del{m}_1}{\text{d}\text{e}\text{l}\text{t}}$ and $\frac{del{m}_2}{\text{d}\text{e}\text{l}\text{t}}$ respectively.

Task $2$: Solve for $m_1(t)$ and $m_2(t)$ over a time interval of $0$ to $5$ years in steps of $3$ months, using Euler's method. Store your output data in variables $m1.3$ and $m2.$ e$3.$ Calculate

the concentrations of phosphorous in the lower and upper ocean at $t=3$ years, assign these to variables $m1$ pe and $m2$ pe respectively.

Task $3$: Solve for $m_1(t)$ and $m_2(t)$ over a time interval of $0$ to $5$ years in steps of $1$ month, using Euler's method. Store your output data in variables $m11\_$e$1$ and $m2\_$e$1.$

Task $4$ : Render these estimates in Figure $1$ as follows.

Plot m$1\_$e$3$ and m$2\_$e$3$ using a blue line with $+$ markers for each point. No need to change linewidths or markersizes

Plot m$1\_$e$1$ and m$2\_$e$1$ using a green line with $"."$ markers $($a dot$)$ for each point. Use markersize $8$ so you can see the points

Task $6$ : The ocean temperature changes by $1$ deg Celcius. This causes $k_{12}$ to increase by $5\%.$ Repeat Task $3$ but store your output in variables $m{1}_{?}$e$1\_$hot and $m2\_$e$1\_$hot.

Render these new estimates in Figure $2$ as follows.

Plot m$1\_$e$1$ and m$2\_$e$1$ using a black line with x markers for each point.

Plot $m{1}_{?}$e$1\_$hot and $m{2}_{e}1$dots hot using a red line with $"."$ markers for each point and markersize of $8$

Task $7$ : Using your results from Task $6,h=1$ month, calculate the percentage change in phosphorous in the lower and upper ocean after $3$ years. Assign these values to the

variable $m_1\_$change$\_$hot and $m2\_$change$\_$hot. Use the following calculation to measure percentage change $100|m_{\text{h}\text{o}\text{t}}-m\frac{|}{m},$ where $m_{\text{h}\text{o}\text{t}}$ is the value of $m$ after the change and

$m$ is the value before the change. Apply this to changes in both $m_1,m_2.$

Hints: You might find it easier to write your Euler's solution as a function. But you'll have to wrestle with qetting those functions to call your two qradient functions.