We model the following mechanisms for nitrogen transfer among the three states.
We can summarize these transfers with the dynamical system
\begin{align*}
\diff{p}{t} &=-ap+cp(1-p-z)\\
\diff{z}{t} &= -bz+ezp
\end{align*}
Let the rate constants be $a=0.4$, $b=0.2$, $c=0.8$, and $e=0.5$.
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Calculate the nullclines and draw them on the below phase plane.
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Determine the equilibria and plot them on the above phase plane.
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The nullclines divide the biologically plausible phase plane ($p \ge 0$, $z \ge 0$ and $p+z \le 1$) into four regions. In each region, estimate the short-term dynamics by determining if $p$ and $z$ are increasing or decreasing. Sketch a direction arrow in each region.
The $z$-nullcline divides the biologically plausible portion of the $p$-nullcline into two pieces. The $p$-nullcline divides the biologically plausible portion of the $z$-nullcline into three pieces. On each nullcline piece, estimate the short-term dynamics by determining if $p$ and $z$ are increasing or decreasing. Sketch a direction arrow on each piece.
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Classify the equilibria.
You can use the fact that, if a matrix is of the form $A = \begin{bmatrix} a_{11} & a_{12}\\ 0 & a_{22}\end{bmatrix}$ or $A = \begin{bmatrix} a_{11} & 0\\ a_{21} & a_{22}\end{bmatrix}$, i.e., with a zero in the upper right and/or lower left, then its eigenvalues are the numbers on the diagonal, $\lambda_1=a_{11}$ and $\lambda_2=a_{22}$.
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Assume that we start with a nitrogen distribution with 10% free nitrogen, 80% of the nitrogen in phytoplankton, and 10% of the nitrogen in zooplankton. Sketch a solution trajectory $(p(t),z(t))$ on the above phase plane, starting with nitrogen distribution. You solution trajectory should be consistent with the direction arrows and the classification of the equilibria.
After a long time, what is the distribution of nitrogen among free nitrogen, nitrogen in phytoplankton, and nitrogen in zooplankton?