Math Insight

1. The dynamical system is a model of disease spread, where $x$ is the number of susceptible individuals and $y$ is the number of infective individuals.
2. The dynamical system is a model of predator-prey dynamics, where $x$ is the size of the prey population and $y$ is the size of the predator population.

1. The drug is being consumed constantly at a rate $\alpha$. (Clearly, this is an approximation of the reality, where the drug would be consumed periodically.) This consumption increases $w$ at that rate.
2. Drug is absorbed from the digestive track into the blood stream at a rate proportional to the amount of drug in the digestive track. Denote this rate constant by $\beta$.
3. Drug in the digestive track is being excreted (via feces) at a rate proportional to the amount of drug in the digestive track. Denote this rate constant by $\gamma$.
4. Drug in the circulatory system is being excreted (via urine) at a rate proportional to the amount of drug in the circulatory system. Denote this rate constant by $\delta$.

Given these rules, write down a system of differential equations for $w$ and $x$.

We model the following mechanisms for nitrogen transfer among the three states.

1. Phytoplankton die at a rate proportional to their population size, with rate constant $a$. When they die, they return their nitrogen to the water, i.e., to free nitrogen.
2. Zooplankton die at a rate proportional to their population size, with rate constant $b$. When they die, they return their nitrogen to the water.
3. Phytoplankton consume nitrogen, changing free nitrogen to nitrogen inside phytoplankton. This consumption occurs a rate proportional to their population size and the amount of free nitrogen, with rate constant $c$.
4. Zooplankton consume phytoplankton, transferring the nitrogen to zooplankton nitrogen. This consumption occurs at a rate proportional to both population sizes, with rate constant $e$.

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 $e=0.5$.

1. Calculate the nullclines and draw them on the below phase plane.
2. Determine the equilibria and plot them on the above phase plane.
3. 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.

4. 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 $\lambda_2=a_{22}$.

5. 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?

1. From the phase plane, determine any equilibria, indicate them on the phase plane, and estimate their value.
2. For each segment of each nullcline, determine the direction that a solution $(v(t),w(t))$ must move when crossing the nullcline, and draw an arrow on the nullcline.
3. Sketch a plausible solution on the phase plane with initial conditions $(v(0),w(0)) = (-4,1)$.