Chapter 17

Find \(r\) and \(M\) so that the solution \(p(t)=\frac{M p_{0}}{p_{0}+\left(M-p_{0}\right) e^{-r t}} \quad\) of the logistic equation \(\quad p^{\prime}=r \times p \times\left(1-\frac{p}{M}\right)\) approximates the data $$\begin{array}{|r|r|r|} \hline {\mathrm{pH} 6.25} \\\\\hline \text { Time } & \text { Index } & \text { Population } \\ \text { (min) } & t & \text { Density } \\\\\hline 0 & 0 & 0.022 \\ 16 & 1 & 0.036 \\\32 & 2 & 0.060 \\\48 & 3 & 0.101 \\ 64 & 4 & 0.169 \\\80 & 5 & 0.266 \\\96 & 6 & 0.360 \\\112 & 7 & 0.510 \\\128 & 8 & 0.704 \\\144 & 9 & 0.827 \\\160 &10& 0.928 \\\\\hline\end{array}$$

Draw the direction field for \(y^{\prime}(t)=\sqrt{y(t)}\) and decide whether the equilibrium solution \(y(t)=0\) is stable.

Show that each solution satisfies the initial condition and the differential equation. $$\begin{aligned} &\text { Solution } \quad \text { Initial Condition } \quad \text { Differential Equation }\\\ &\text { a. } \quad y(t)=e^{2 t}+e^{t} \quad y(0)=2 \quad y^{\prime}(t)-y(t)=e^{2 t} \end{aligned}$$ $$\text { b. } \quad y(t)=\frac{1}{3} e^{t}+\frac{2}{3} e^{-2 t}, \quad y(0)=1, \quad y^{\prime}(t)+2 y(t)=e^{t}$$ c. \(\quad y(t)=t e^{t} \quad y(0)=0 \quad y^{\prime}(t)-y(t)=e^{t}\) d. \(\quad y(t)=\frac{t^{2}}{3}+\frac{1}{t}, \quad y(1)=\frac{4}{3}, \quad t \times y^{\prime}(t)+y(t)=t^{2}\) e. \(\quad y(t)=\sqrt{t+1} \quad y(0)=1 \quad y(t) \times y^{\prime}(t)=\frac{1}{2}\) f. \(\quad y(t)=\sqrt{1+t^{2}}, \quad y(0)=1, \quad y(t) \times y^{\prime}(t)=t\) \(\begin{array}{lll}\text { g. } y(t)=\sqrt{4+t^{2}} & y(0)=2 & y(t) \times y^{\prime}(t)=t\end{array}\) \(\begin{array}{lll}\text { Solution } & \text { Initial } & \text { Differential Equation }\end{array}\) Condition h. \(y(t)=\frac{1}{t+1}, \quad y(0)=1, \quad y^{\prime}(t)+(y(t))^{2}=0\) i. \(\quad y(t)=0.5+0.5 e^{-0.2 \sin t} \quad y(0)=1, \quad y^{\prime}(t)+0.2(\cos t) y(t)=0.1 \cos t\) j. \(\quad y(t)=\tan t, \quad y(0)=0, \quad y^{\prime}(t)=1+(y(t))^{2}\) k. \(y(t)=3\) \(y(0)=3, \quad y^{\prime}(t)=(y(t)-1) \times(y(t)-3) \times(y(t)-5)\) l. \(y(t)=5\) \(y(0)=5, \quad y^{\prime}(t)=(y(t)-1) \times(y(t)-3) \times(y(t)-5)\)

Write an equation that describes the temperature of an egg after it is uncovered (the adult bird leaves the nest to feed). Assume that the rate of change of the temperature of the egg is proportional to difference between the air temperature and the egg temperature.

Show that the variables are not separable in the equation \(y^{\prime}(t)=t+y .\) That is, there are not two functions, \(g(t)\) and \(h(y),\) which for all \(t\) and \(y \quad t+y=g(t) \times h(y)\) A procedure is to assume two such functions, \(g(t)\) and \(h(y)\) exist and then show that the following equations are incompatible. \(\begin{array}{lllll}t=0 & y=0 & t=0 & y=1 & t=1 & y=0\end{array}\) \(g(0) \times h(0)=0+0=0 \quad g(0) \times h(1)=0+1=1 \quad g(1) \times h(0)=1+0=1\) Show that \(g(0) \times h(0)=0, g(0) \times h(1)=1\) and \(g(1) \times h(0)=1\) are incompatible.

Find the unique solutions to a. \(\quad y(0)=5 \quad y^{\prime}+2 y=0\) b. \(\quad y(0)=0 \quad y^{\prime}+2 y=0\) c. \(\quad y(0)=4 \quad y^{\prime}+3 y=t\) d. \(\quad y(1)=1 \quad y^{\prime}+3 y=t\) e. \(\quad y(0)=0 \quad y^{\prime}+0.2 y=e^{-0.2 t}\) f. \(y(0)=3 \quad y^{\prime}+t y=t\)

Consider the pair of differential equations $$u(0)=1 \quad u^{\prime}(t)=0.5 \times u(t)-0.2 \times u(t) \times v(t)$$ \(v(0)=2 \quad v^{\prime}(t)=0.1 \times u(t) \times v(t)-0.1 \times v(t) \quad 0 \leq t \leq 1\) This system is a predator prey system. We (including you!) will use Euler's method to approximate a solution on the time interval [0,1] with \(n=5\) subintervals.

Find the equilibrium points and for each determine whether or not it is stable. a. \(\quad y^{\prime}=y-1\) b. \(\quad y^{\prime}=-y+1\) c. \(y^{\prime}=y^{2}-1\) d. \(\quad y^{\prime}=1-y^{2}\) e. \(\quad y^{\prime}=e^{-y}-1\) f. \(\quad y^{\prime}=e^{y}-1\) g. \(\quad y^{\prime}=\sin y\) h. \(\quad y^{\prime}=-y+y^{2}\) i. \(\quad y^{\prime}=-y^{3}\) j. \(\quad y^{\prime}=y^{3}\) For parts i. and \(\mathrm{j}\). draw the phase plane with arrows to determine the question of stability.

Which of the following possible solutions satisfies the initial condition and the differential equation. Possible Solution \(\quad\) Initial Condition \(\quad\) Differential Equation a. \(\quad y(t)=e^{2 t}+2 e^{t} \quad y(0)=2\) \(y^{\prime}(t)-y(t)=e^{2 t}\) b. \(\quad y(t)=e^{2 t}+2 e^{t} \quad y(0)=3 \quad y^{\prime}(t)-y(t)=e^{2 t}\) \(\begin{array}{lll}\text { c. } y(t)=\sqrt{t+1} & y(0)=1 & y(t) \times y^{\prime}(t)=1\end{array}\) d. \(y(t)=t^{3}\) \(y(1)=1 \quad y^{\prime}(t) / y(t)=3 t\) e. \(\quad y(t)=\frac{1}{t}\) f. \(y(t)=t^{3}\) \(y(1)=1 \quad y^{\prime}(t) / y(t)=3 t\) g. \(\quad y(t)=t e^{t}\) \(y(0)=0 \quad y^{\prime}(t)-y(t)=e^{t}\)

Find the values of \(P(t)\) for which \(P^{\prime}(t)=0\) in each of the equations Verhulst \(\quad p^{\prime}(t)=r \times p(t) \times\left(1-\frac{P(t)}{M}\right)\) Ricker \(p^{\prime}(t)=r \times p(t) \times \frac{A e^{-p(t) / \beta}-1}{A-1}\) \(A>1 \quad p^{\prime}(t)=\alpha \times p(t) \times e^{-p(t) / \beta}-\gamma p(t)\) Beverton-Holt \(p^{\prime}(t)=\frac{r \times p(t)}{1+p(t) / \beta}\) Gompertz \(\quad p^{\prime}(t)=-r p(t) \ln (p(t) / \beta)\) Let \(M=\beta=1\) and \(A=3\) draw the graphs of \(U(p)\) vs \(p\) for each of the four models in Equations \(17.2-17.3,\) for \(0

Assume that \(1000 \mathrm{w} / \mathrm{m}^{2}\) of light is striking the surface of a lake and that \(40 \%\) of that light is reflected back into the atmosphere. Solve the initial value problem $$\begin{array}{r}I^{\prime}(x)=-K I(x) \\\I(0)=600\end{array} $$Suppose the light intensity at 10 meters is \(500 \mathrm{w} / \mathrm{m}^{2}\). Find the value of \(K\).

Show that the variables are not separable in the equation $$\text { a. } \quad y^{\prime}(t)=\ln (t \times y) \quad \text { b. } \quad y^{\prime}(t)=\ln (t+y)$$

A differential equation with initial condition and its analytic solution are shown. i. Show that the analytic solution satisfies the initial condition and the differential equation. ii. Use Euler's method and the trapezoid methods to approximate the solution to the differential equation on the interval shown and using the step size shown. iii. Plot the solution and the Euler's and trapezoid approximations on a single \(t, y\) plane. a. \(y(0)=1 \quad y^{\prime}(t)=y^{2} \quad y(t)=(1-t)^{-1} \quad 0 \leq t \leq 0.4 \quad h=0.1\) b. \(y(0)=2 \quad y^{\prime}(t)=-y^{2} \quad y(t)=(t+0.5)^{-1} \quad 0 \leq t \leq 0.4 \quad h=0.1\) c. \(y(0)=1 \quad y^{\prime}(t)=t \times y \quad y(t)=e^{t^{2} / 2} \quad 0 \leq t \leq 1 \quad h=0.2\) d. \(y(0)=1 \quad y^{\prime}(t)=\sqrt{y} \quad y=(t / 2+1)^{2} \quad 0 \leq t \leq 1 \quad h=0.2\)

The special case of \(y^{\prime}=f(t, y)\) in which \(f(t, y)=F(t)(f\) is independent of \(y)\) has a familiar solution from the Fundamental Theorem of Calculus I. Check by substitution that $$y(t)=y_{a}+\int_{a}^{t} F(x) d x \quad \text { solves } \quad y(a)=y_{a} \quad \text { and } \quad y^{\prime}(t)=F(t)$$ The differential equation $$y(a)=y_{a} \quad y^{\prime}(t)=F(t)$$ has therefore been completely solved. Henceforth we will consider that \(f\) is dependent on \(y\) and possibly also on \(t\).

Let \(m\left(p_{0}\right)\) be \(p^{\prime}(0)\) in the Gompertz model, where $$p^{\prime}(t)=-r p(t) \ln (p(t) / \beta), \quad t \geq 0, \quad p(0)=p_{0}$$ Then $$m\left(p_{0}\right)=-r p_{0} \ln \left(p_{0} / \beta\right)$$ For \(p_{0} / \beta \ll 1\) it follows from Explore 17.1 .1 that \(m\left(p_{0}\right)=p^{\prime}(0) \gg r p_{0} .\) Show, however, that $$\lim _{p_{0} \rightarrow 0} m\left(p_{0}\right)=0$$

Find an implicit or explicit expression for \(y(t)\) for each equation. Then use the given data point to evaluate the constant \(C\) of integration. The following derivative formulas will be helpful. \([\ln (y-1)]^{\prime}=\frac{1}{y-1} \quad\left[\ln \left(t^{2}+1\right)\right]^{\prime}=\frac{2 t}{t^{2}+1} \quad[\ln (1-y)]^{\prime}=-\frac{1}{y-1}\) a. \(\quad y^{\prime}=\frac{t}{y} \quad y(0)=2\) b. \(\quad y^{\prime}=\frac{y}{t} \quad y(1)=1\) c. \(\quad y^{\prime}=\frac{t-1}{y} \quad y(2)=1\) d. \(\quad y^{\prime}=\frac{t}{y\left(t^{2}+1\right)} \quad y(0)=1\) e. \(\quad y^{\prime}=y-1 \quad y(0)=1.5\) f. \(\quad y^{\prime}=1-y \quad y(0)=0.5\) g. \(\quad y^{\prime}=y^{2}-1 \quad y(0)=2\) h. \(\quad y^{\prime}=e^{t-y} \quad y(0)=1.5\)

Use the Euler and trapezoid methods to compute the solutions to the following differential equations with initial conditions on the intervals shown and using the step sizes shown. a. \(y(0)=4 \quad y^{\prime}(t)=t-\sqrt{y} \quad 0 \leq t \leq 1 \quad h=0.2\) b. \(y(0)=0.5 \quad y^{\prime}(t)=y /(1+y) \quad 0 \leq t \leq 1 \quad h=0.1\) c. \(y(0)=0.5 \quad y^{\prime}(t)=-\ln y \quad 0 \leq t \leq 1 \quad h=0.1\) d. \(y(0)=0.15 \quad y^{\prime}(t)=y(y-0.1)(1-y) \quad 0 \leq t \leq 1 \quad h=0.4\) e. \(y(0)=0 \quad y^{\prime}(t)=y(y-0.1)(1-y) \quad 0 \leq t \leq 1 \quad h=0.4\) f. \(\quad y(0)=0.05 \quad y^{\prime}(t)=y(y-0.1)(1-y) \quad 0 \leq t \leq 1 \quad h=0.4\)

Continuous infusion of penicillin. Suppose a patient recovering from surgery is to be administered penicillin intravenously at a constant rate of 5 grams per hour. The patient's kidneys will remove penicillin at a rate proportional to the serum penicillin concentration. Let \(P(t)\) be the penicillin concentration \(t\) hours after infusion is begun. Then a simple model of penicillin pharmacokinetics is \(\begin{array}{l}\text { Net Rate of Increase } & \text { Clearance } & \text { Infusion }\end{array}\) $$P^{\prime}(t)=-K \times P(t)+5$$ $$\frac{\mathrm{gm}}{\mathrm{hr}} \quad \frac{1}{\mathrm{hr}} \times \operatorname{gm} \quad \frac{\mathrm{gm}}{\mathrm{hr}}$$ The proportionality constant, \(K,\) must have units \(\frac{1}{\mathrm{hr}}\) in order for the units on the equation to balance. We initially assume that \(K=2.5 \frac{1}{\mathrm{hr}}\) which is in the range of physiological reality. It is reasonable to assume that there was no penicillin in the patient at time \(t=0,\) so that \(P(0)=0\). a. Draw the phase plane for the differential equation $$P(0)=0 \quad P^{\prime}(t)=-2.5 P(t)+5$$ b. Find the equilibrium point of \(P^{\prime}=-2.5 P+5\). c. Is the equilibrium point stable? d. Show that the units of the equilibrium point are grams. e. Suppose the patient's kidneys are impaired and only operating at \(60 \%\) of normal. Then \(K=1.5\) instead of \(2.5 .\) What effect does this have on the equilibrium point.

Argue that \(y=\tan t \quad\) is the only solution to \(\quad y(0)=0, \quad y^{\prime}(t)=1+y^{2}(t)\) Use the integral formula, \(\int \frac{y^{\prime}(\tau)}{1+y^{2}(\tau)} d \tau=\arctan y(\tau)+C\).

Release of nitrogen in the tissue of a SCUBA diver as she ascends from deep water has been compared to the release of carbon dioxide in a Coca-Cola \(\odot\) when it is opened. Write a mathematical model descriptive of release of carbon dioxide in a Coca-Cola \(\odot\). From your model, write a differential equation descriptive of the partial pressure of carbon dioxide in a Coca-Cola \(^{\odot} t\) minutes after opening the Coca-Cola \(^{\odot}\).

Solve for \(p\) in $$\frac{p}{M-p}=\frac{p_{0}}{M-p_{0}} e^{r t}$$ to obtain Equation 17.31 , $$p(t)=\frac{M p_{0}}{p_{0}+\left(M-p_{0}\right) e^{-r t}} "$$ It will be useful to first solve for \(p\) in $$\frac{p}{M-P}=K \quad\left(K \quad \text { replaces } \quad \frac{p_{0}}{M-p_{0}} e^{r t}\right)$$ You should get $$p=M \times \frac{K}{1+K}$$ Then substitute $$K=\frac{p_{0}}{M-p_{0}} e^{r t} $$and simplify. As a final step, divide numerator and denominator by \(e^{r t} .\)

Suppose \(y^{\prime}=f(y)\) has three and only three equilibrium points, \(e_{1}, e_{2},\) and \(e_{3},\) where \(f\)and \(f^{\prime}\) are continuous and \(f^{\prime}\left(e_{1}\right) \neq 0, f^{\prime}\left(e_{2}\right) \neq 0,\) and \(f^{\prime}\left(e_{3}\right) \neq 0 .\) Argue that one of \(e_{1}, e_{2},\) and \(e_{3}\) is stable.

Use the Extended Mean Value Theorem 14.5 .2 to show that the graphs of $$E(t)=15.3\left(1.035-0.050 \sin \left(\frac{2 * \pi}{11200}(t+4200)\right)\right)$$ and $$E_{t_{0}}(t)=E\left(t_{0}\right) e^{-\frac{\ln 2}{5730}\left(t-t_{0}\right)}$$ can not intersect at two points for \(-10000 \leq t_{0} \leq 0\) and \(t_{0} \leq t \leq 0\). Argue as follows. They obviously intersect at \(\left(t_{0}, E\left(t_{0}\right) .\right.\) Argue that: a. \(E^{\prime}(t) \geq-0.000429\) for \(t_{0} \leq t\) b. \(E_{t_{0}}^{\prime}(t) \leq-0.00045\) for \(t_{0} \leq t\). c. Suppose the graphs intersect at another point, \(\left(t_{1}, E\left(t_{1}\right)\right)\) with \(t_{0} \leq t_{1}\). Then at some time \(\tau\) between \(t_{0}\) and \(t_{1}\) the slopes of the two graphs are equal. This leads to a contradiction.

Show that for $$ p(t)=\frac{M p_{0}}{p_{0}+\left(M-p_{0}\right) e^{-r t}} $$a. \(p(0)=p_{0}\) $$\text { b. } \lim _{t \rightarrow \infty} p(t)=M$$

There is 'conventional wisdom' among SCUBA divers that if you are going to make a dive that involves two depths, 'do the deep part first'. This problem and the next explores rationale for that wisdom. To be concrete, assume that \(K=0.071 / \mathrm{min}\) which corresponds to approximately 10 minute half- life for the compartment \(((\ln 2) / 0.07=9.9 \mathrm{~min})\). a. Assume a diver (d1) descends immediately to 10 meters and stays there for 15 minutes, then descends to 30 meters and stays there for 10 minutes. Let $$d_{1}(t)=\left\\{\begin{array}{ll}10 & \text { for } 0 \leq t \leq 15 \\\30 & \text { for } 15

Consider a modification of the Lotka-Volterra equations for competition between two species in which \(\alpha_{1,2}=0\). $$\begin{array}{l}p_{1}^{\prime}(t)=r_{1} \times p_{1}(t) \times\left(1-\frac{p_{1}(t)+0 \times p_{2}(t)}{M_{1}}\right)=r_{1} \times p_{1}(t) \times\left(1-\frac{p_{1}(t)}{M_{1}}\right) \\ p_{2}^{\prime}(t)=r_{2} \times p_{2}(t) \times\left(1-\frac{p_{2}(t)+\alpha_{2,1} p_{1}(t)}{M_{2}}\right) \end{array}$$ Thus population 1 is not affected by population 2 but population 2 is affected by population \(1 .\) Suppose that $$00\) then \(z(t) \rightarrow 0\) as \(t \rightarrow \infty\) c. What happens to the second population if \(M_{2}>\alpha_{2,1} M_{1} ?\)

Let \(M=10\) and \(r=0.1\) and plot the graphs of $$p(t)=\frac{M p_{0}}{p_{0}+\left(M-p_{0}\right) e^{-r t}}$$ for \(0 \leq t \leq 80\) and a. \(p_{0}=1\) $$\text { b. } p_{0}=12 \quad \text { c. } p_{0}=10$$

a. Show that Ricker's equation, $$p^{\prime}(t)=\alpha p e^{-p / \beta}-\gamma p$$ is equivalent to $$v^{\prime}(\tau)=v e^{-v}-\gamma_{0} v$$ with the substitutions, \(u(t)=p(t) / \beta, \tau=\alpha t,\) and \(\gamma_{0}=\gamma / \alpha\). b. Show that the Beverton-Holt equation, $$p^{\prime}(t)=\frac{r \times p}{1+p / \beta}$$ is equivalent to $$v^{\prime}(\tau)=\frac{v}{1+v}$$ with proper substitutions. c. Show that the Gompertz equation, $$p^{\prime}(t)=-r \ln \frac{p}{\beta}$$ with proper substitutions, is equivalent to an equation with no parameters.

Suppose population is described by Equation \(17.31, P(t)=p_{0} M /\left(p_{0}+\left(M-p_{0}\right) e^{-r t}\right)\) and \(0

Identify the stable and nonstable solutions of $$u^{\prime}(t)=u(t)(1-u(t))$$

Exercise 17.4.10 Suppose a marine fish population when not subject to harvest is reasonably modeled by $$u^{\prime}(t)=r \times u(t) \times(1-u(t))$$ with time measured in years. Suppose a harvest procedure is initiated, and that a fraction, \(h,\) of the existing population is harvested every year. The harvest is not a fixed amount each year, but depends on the number of fish available. The growth rate will be the difference between the natural birth-death process and the harvest and may be modeled by $$ u^{\prime}(t)=r \times u(t) \times(1-u(t))-h \times u(t) $$ a. Assume \(h=r\) (the harvest rate equals the low density growth rate) Substitute \(h=r\) in Equation 17.17, and simplify. Show that $$u(t)=\frac{1}{r t+1 / u_{0}} \quad \text { where } \quad u(0)=u_{0}$$ is a solution for this model. What will be the eventual annual fish harvest under this harvest strategy? b. Assume \(h=\frac{3}{4} r\) in Equation 17.17 and simplify. Draw a direction field or phase plane for this model. What will be the eventual annual fish harvest under this harvest strategy? c. Assume \(h=\frac{1}{2} r\) in Equation \(17.17,\) and simplify. Draw a direction field for this model. What will be the eventual annual fish harvest under this harvest strategy? d. Which of the three strategies will provide the largest long term harvest?

a. The units on \(S^{\prime}, I^{\prime}\) and \(R^{\prime}\) in the SIR model are people/day. In order for the units to balance on the equations, what must be the units on \(\beta\) and \(\gamma ?\) b. Suppose the infection typically lasts seven days. What is an appropriate value of \(\gamma ?\) Note: The answer we would expect you to give for \(\gamma\) is \(1 / 7 \doteq 0.143 .\) We will find that a better answer is \(-\ln (1-1 / 7) \doteq 0.154\)

Parameter reduced population models are shown below. In the Ricker equation, find a condition on \(\gamma_{0}\) that will insure that there is a value of \(v\) for which the population is growing the fastest. In the Beverton-Holt equation show that there is no value of \(v\) for which the population is growing the fastest. In the Gompertz equation, find the value of \(v\) for which the population is growing the fastest. $$\text { a. } \quad v^{\prime}=v e^{-v}-\gamma_{0} v \text { Ricker }$$$$\begin{array}{l} \text { b. } \quad v^{\prime}=\frac{v}{1+v} \\\\\text { c. } v^{\prime}=-v \ln (v)\end{array}$$ Beverton-Holt Gompertz The three previous problems have important implications for wildlife management, at least conceptually. Suppose you are managing a wildlife population, salmon, for example, as a renewable resource, and wish to annually harvest as many salmon as possible. If you harvest too severely, the next years spawn will be low, and four years later the harvest will be limited. Your optimum strategy is to maintain the population at the level where the growth is the greatest.

Some population scientists have argued that population density can get so low that reproduction will be less than natural attrition and the total population will be lost. Named the Allee effect. after W. C. Allee who wrote extensivelv about it \(^{6}\), this mav be a basis for arguing. for example that marine fishing of a certain species (Atlantic cod on Georges Bank, for example \({ }^{7}\) ee Paul Greenberg, "Four Fish, the Future of the Last Wild Food, The Penguin Press, 2010 . The model is a lot more complicated than we present here.) should be suspended, despite the presence of a small residual population. How should we modify the logistic differential equation, \(u^{\prime}=u(1-u),\) to incorporate such a threshold? Assume a fixed area and uniform density throughout the area and a threshold number, \(\epsilon .\) If the population number is less than \(\epsilon\) the population will decline; if the population number is more than \(\epsilon\) the population will increase. a. Modify the direction field for \(u^{\prime}=u(1-u)\) to account for the Allee effect. That is, a \(u, t\) plane, draw the line \(u=1\) and a threshold line \(u=\epsilon\) where \(\epsilon=0.1,\) say. Arrows below \(u=\epsilon\) should point downward; arrows between \(u=\epsilon\) and \(u=1\) should point upwards. Draw enough direction field arrows to indicate the paths of solutions for a threshold model. b. Draw the the logistic phase plane graph and the phase plane graphs for the following three candidates of a threshold logistic differential equation where \(\epsilon=0.1\). \(u^{\prime}=f(u)=u \times(1-u) \quad\) Logistic \(u^{\prime}=f_{1}(u)=u^{\frac{2}{3}} \times(u-\epsilon)^{\frac{1}{3}} \times(1-u) \quad\) Candidate 1 \(u^{\prime}=f_{2}(u)=u \times \frac{u-\epsilon}{u+\epsilon} \times(1-u)\) Candidate 2 \(u^{\prime}=f_{3}(u)=u \times(u-\epsilon) \times(1-u)\) Candidate 3

Suppose immunity is not permanent in the SIR model, and recovered people become susceptible after six months. Modify the meaning of \(R\) and the SIR equations to account for this possibility.

Suppose a patient is administered penicillin by continuous infusion that enters the vascular pool of 2 liters at the rate of 2 gm/hour. Consider only the vascular pool, and write a model of penicillin amount in the vascular pool. Write an initial condition and a differential equation that is descriptive of the amount of penicillin in the serum as a function of time.