Chapter 7

For the predator-prey model \(\left\\{\begin{array}{l}x^{\prime}=0.4 x-0.1 x^{2}-0.2 x y \\ y^{\prime}=-0.5 y+0.1 x y\end{array}\right.\) show that the species cannot coexist. If the death rate 0.5 of species \(Y\) could be reduced, determine how much it would have to decrease before the species can coexist.

Suppose the income tax structure is as follows: the first \(\$ 30,000\) is taxed at \(15 \%,\) the remainder is taxed at \(28 \% .\) Compute the \(\operatorname{tax} T_{1}\) on an income of \(\$ 40,000 .\) Now, suppose that inflation is \(5 \%\) and you receive a cost of living \((5 \%)\) raise to \(\$ 42,000\) Compute the \(\operatorname{tax} T_{2}\) on this income. To compare the taxes, you should adjust the \(\operatorname{tax} T_{1}\) for inflation (add \(5 \%\) ).

For the differential equation \(y^{\prime}=\frac{x^{2}+7 x+3}{y^{2}}\) used in exercises 41 and \(42,\) notice that \(y^{\prime}(x)\) does not exist at any \(x\) for which \(y(x)=0 .\) Given the solution of example \(2.4,\) this occurs if \(x^{3}+\frac{21}{2} x^{2}+9 x+3 c=0 .\) Find the values \(c_{1}\) and \(c_{2}\) such that this equation has three real solutions if and only if \(c_{1}

For the general predator-prey model \(\left\\{\begin{array}{l}x^{\prime}=b x-c x^{2}-k_{1} x y \\ y^{\prime}=-d y+k_{2} x y\end{array}\right.\) show that the species can coexist if and only if \(b k_{2}>c d\).

Suppose that the value of a \(\$ 40,000\) asset decreases at a constant percentage rate of \(10 \% .\) Find its worth after (a) 10 years and (b) 20 years. Compare these values to a \(\$ 40,000\) asset that is depreciated to no value in 20 years using linear depreciation.

In the predator-prey model of exercise 45, the prey could be a pest insect that attacks a farmer's crop and the predator, a natural predator (e.g.., a bat) of the pest. Assume that \(c=0\) and the coexistence equilibrium point is stable. The effect of a pesticide would be to reduce the birthrate \(b\) of the pest. It could also potentially increase the death rate \(d\) of the predator. If this happens, state the effect on the coexistence equilibrium point. Is this the desired effect of the pesticide?

Suppose that the value of a $$ 400,000\( asset decreases at a constant percentage rate of \)40 % .\( Find its worth after (a) 5 years and (b) 10 years. Compare these values to a $$ 40,000\) asset that is depreciated to no value in 10 years using linear depreciation.

One of the mysteries in population biology is how populations regulate themselves. The most famous myth involves lemmings diving off of cliffs at times of overpopulation. It is true that lemming populations rise and fall dramatically, for whatever reason (not including suicide). Animal ecologists draw graphs to visualize the rises and falls of animal populations. Instead of graphing population versus time, ecologists graph the logarithm of population versus time. To understand why, note that a population drop from 1000 to 500 would represent the same percentage decrease as a drop from 10 to \(5 .\) Show that the slopes of the drops are different, so that these drops would appear to be different on a population/time graph. However, show that the slopes of the drops in the logarithms (e.g., In 1000 to In 500 ) are the same. In general, if a population were changing at a constant percentage rate, what would the graph of population versus time look like? What would the graph of the logarithm of population versus time look like?

It has been conjectured that half the people who have ever lived are still alive today. To see whether this is plausible, assume that humans have maintained a constant birthrate \(b\) and death rate \(d .\) Show that the statement is true if and only if \(b \geq 2 d\)

Relate to money investments. A house mortgage is a loan that is to be paid over a fixed period of time. Suppose \(\$ 150,000\) is borrowed at \(8 \%\) interest. If the monthly payment is \(\$ P\), then explain why the equation \(A^{\prime}(t)=0.08 A(t)-12 P, A(0)=150,000\) is a model of the amount owed after \(t\) years. For a 30 -year mortgage, the payment \(P\) is set so that \(A(30)=0 .\) Find \(P .\) Then, compute the total amount paid and the amount of interest paid.

Relate to money investments. A person contributes \(\$ 10,000\) per year to a retirement fund continuously for 10 years until age 40 but makes no initial payment and no further payments. At \(8 \%\) interest, what is the value of the fund at age \(65 ?\)

An Internet site reports that the antidepressant drug amitriptyline has a half-life in humans of \(31-46\) hours. For a dosage of \(150 \mathrm{mg},\) compare the amounts left in the bloodstream after one day for a person for whom the half-life is 31 hours versus a person for whom the half-life is 46 hours. Is this a large difference?

It is reported that Prozac "has a half-life of 2 to 3 days but may be found in your system for several weeks after you stop taking it. What percentage of the original dosage would remain after 2 weeks if the half-life is 2 days? How much would remain if the half-life is 3 days?

Relate to money investments. An endowment is seeded with \(\$ 1,000,000\) invested with interest compounded continuously at \(10 \% .\) Determine the amount that can be withdrawn (continuously) annually so that the endowment lasts thirty years.

The antibiotic ertapenem has a half-life of 4 hours in the human bloodstream. The dosage is 1 gm per day. Find and graph the amount in the bloodstream \(t\) hours after taking it \((0 \leq t \leq 24)\)

Relate to reversible bimolecular chemical reactions, where molecules \(\mathbf{A}\) and \(\mathbf{B}\) combine to form two other molecules C and D and vice versa. If \(x(t)\) and \(y(t)\) are the concentrations of \(\mathbf{C}\) and \(\mathbf{D}\), respectively and the initial concentrations of \(A, B, C\) and \(D\) are \(a, b, c\) and \(d,\) respectively, then the reaction is modeled by$$x^{\prime}(t)=k_{1}(a+c-x)(b+c-x)-k_{-1} x(d-c+x)$$ for rate constants \(k_{1}\) and \(k_{-1}\) If \(k_{1}=1, k_{-1}=0.625, a+c=0.4, b+c=0.6, c=d\) and \(x(0)=0.2,\) find the concentration \(x(t) .\) Graph \(x(t)\) and find the eventual concentration level.

A bank offers to sell a bank note that will reach a maturity value of $$ 10,000\( in 10 years. How much should you pay for it now if you wish to receive an \)8 \%\( return on your investment? (Note: This is called the present value of the bank note.) Show that in general, the present value of an item worth \) P\( in \)t\( years with constant interest rate \)r\( is given by \) P e^{-r t}$

Relate to reversible bimolecular chemical reactions, where molecules \(\mathbf{A}\) and \(\mathbf{B}\) combine to form two other molecules C and D and vice versa. If \(x(t)\) and \(y(t)\) are the concentrations of \(\mathbf{C}\) and \(\mathbf{D}\), respectively and the initial concentrations of \(A, B, C\) and \(D\) are \(a, b, c\) and \(d,\) respectively, then the reaction is modeled by$$x^{\prime}(t)=k_{1}(a+c-x)(b+c-x)-k_{-1} x(d-c+x)$$ for rate constants \(k_{1}\) and \(k_{-1}\) For the bimolecular reaction with \(k_{1}=0.6, k_{-1}=0.4\) \(a+c=0.5, b+c=0.6\) and \(c=d,\) write the differential equation for the concentration of C. For \(x(0)=0.2,\) solve for the concentration at any time and graph the solution.

Suppose that the value of a piece of land \(t\) years from now is $$\$ 40,000 e^{2 \sqrt{r}}$$. Given \(6 \%\) annual inflation, find \(t\) that maximizes the present value of your investment: $$\$ 40,000 e^{2 \sqrt{t}-0.06 t}$$

Relate to reversible bimolecular chemical reactions, where molecules \(\mathbf{A}\) and \(\mathbf{B}\) combine to form two other molecules C and D and vice versa. If \(x(t)\) and \(y(t)\) are the concentrations of \(\mathbf{C}\) and \(\mathbf{D}\), respectively and the initial concentrations of \(A, B, C\) and \(D\) are \(a, b, c\) and \(d,\) respectively, then the reaction is modeled by$$x^{\prime}(t)=k_{1}(a+c-x)(b+c-x)-k_{-1} x(d-c+x)$$ for rate constants \(k_{1}\) and \(k_{-1}\) For the bimolecular reaction with \(k_{1}=1.0, k_{-1}=0.4\) \(a+c=0.6, b+c=0.4\) and \(d-c=0.1,\) write the differential equation for the concentration of \(\mathbf{C}\). For \(x(0)=0.2\) solve for the concentration at any time and graph the solution.

Suppose that a business has an income stream of $$ P(t) .\( The present value at interest rate \)r\( of this income for the next \)T\( years is \)\int_{0}^{t} P(t) e^{-r t} d t .\( Compare the present values at \)5 \%\( for three people with the following salaries for 3 years: \)\begin{aligned} &\text { A: } \quad P(t)=60,000 ; \quad \text { B: } \quad P(t)=60,000+3000 t ; \quad \text { and }\\\ &\mathbf{C}: P(t)=60,000 e^{0.05 t} \end{aligned}$

In a second-order chemical reaction, one molecule each of substances \(A\) and \(B\) combine to produce one molecule of substance X. If \(a\) and \(b\) are the initial concentrations of \(\mathrm{A}\) and \(\mathrm{B}\), respectively, the concentration \(x\) of the substance X satisfies the differential equation \(x^{\prime}=r(a-x)(b-x)\) for some positive rate constant \(r .(\text { a) If } r=0.4, a=6, b=8 \text { and } x(0)=0,\) find \(x(t)\) and \(\lim _{t \rightarrow \infty} x(t) .\) Explain this answer in terms of the chemical process. (b) Repeat part (a) with \(r=0.6 .\) Graph the solutions and discuss differences and similarities.

In a second-order chemical reaction, if there is initially \(10 \mathrm{g}\) of substance A available and \(12 \mathrm{g}\) of substance \(\mathrm{B}\) available, then the amount \(x(t)\) of substance \(\mathrm{X}\) formed by time \(t\) satisfies the IVP \(x^{\prime}(t)=r(10-x)(12-x), x(0)=0 .\) Explain why, physically, it makes sense that \(0 \leq x<10 .\) Solve the IVP and indicate where you need this assumption.

Relate to logistic growth with harvesting. Suppose that a population in isolation satisfies the logistic equation \(y^{\prime}(t)=k y(M-y) .\) If the population is harvested (for example, by fishing) at the rate \(R,\) then the population model becomes \(y^{\prime}(t)=k y(M-y)-R\) Suppose that a species of fish has population in hundreds of thousands that follows the logistic model with \(k=0.025\) and \(M=8 .\) Determine the long-term effect on population if the initial population is \(800,000[y(0)=8]\) and fishing removes fish at the rate of 20,000 per year.

The "Rule of \(72 "\) is used by many investors to quickly estimate how fast an investment will double in value. For example, at \(8 \%\) the rule suggests that the doubling time will be about \(\frac{72}{8}=9\) years. Calculate the actual doubling time. Explain why a "Rule of \(69^{\prime \prime}\) would be more accurate. Give at least one reason why the number 72 is used instead.

Relate to logistic growth with harvesting. Suppose that a population in isolation satisfies the logistic equation \(y^{\prime}(t)=k y(M-y) .\) If the population is harvested (for example, by fishing) at the rate \(R,\) then the population model becomes \(y^{\prime}(t)=k y(M-y)-R\) Determine the critical fishing level \(R_{c}\) such that there are two equilibrium points if and only if \(R

Relate to logistic growth with harvesting. Suppose that a population in isolation satisfies the logistic equation \(y^{\prime}(t)=k y(M-y) .\) If the population is harvested (for example, by fishing) at the rate \(R,\) then the population model becomes \(y^{\prime}(t)=k y(M-y)-R\) Solve the population model $$ \begin{aligned} P^{\prime}(t) &=0.05 P(t)[8-P(t)]-0.6 \\ &=0.4 P(t)[1-P(t) / 8]-0.6 \end{aligned}$$with \(P(0)>2\) and determine the limiting amount \(\lim _{t \rightarrow \infty} P(t)\) What happens if \(P(0)<2?\)

The resale value \(r(t)\) of a machine decreases at a rate proportional to the difference between the current price and the scrap value \(S\). Write a differential equation for \(r .\) If the machine sells new for \(\$ 14,000,\) is worth \(\$ 8000\) in 4 years and has a scrap value of \(\$ 1000,\) find an equation for the resale value at any time.

A granary is filled with \(6000 \mathrm{kg}\) of grain. The grain is shipped out at a constant rate of 1000 kg per month. Storage costs equal 2 cents per \(\mathrm{kg}\) per month. Let \(S(t)\) be the total storage charge for \(t\) months. Write a differential equation for \(S\) with \(0 \leq 1 \leq 6 .\) Solve the initial value problem for \(S(t) .\) What is the total storage bill for six months?

The population models \(P^{\prime}(t)=k P(t)\) and \(P^{\prime}(t)=k[P(t)]^{1.1}\) look very similar. The first is called exponential growth and is studied in detail in section \(7.1 .\) The second is sometimes called a doomsday model. Solve the general doomsday equation. Assuming that \(P(0)\) and \(k\) are positive, find the time at which the population becomes infinite.

Suppose that the thrust of a boat's propeller produces a constant acceleration, but that friction with water produces a deceleration that is proportional to the square of the speed of the boat.Write a differential equation for the speed \(v\) of the boat. Find equilibrium points and use a slope diagram to determine the eventual speed of the boat.

For the logistic equation \(y^{\prime}(t)=k y(M-y),\) show that a graph of \(\frac{1}{y} y^{\prime}\) as a function of \(y\) produces a linear graph. Given the slope \(m\) and intercept \(b\) of this line, explain how to compute the model parameters \(k\) and \(M\)

The downward velocity of a falling object is modeled by the differential equation \(\frac{d v}{d t}=32-0.4 v^{2} .\) If \(v(0)=0 \mathrm{ft} / \mathrm{s},\) the velocity will increase to a terminal velocity. The terminal velocity is an equilibrium solution where the upward air drag exactly cancels the downward gravitational force. Find the terminal velocity.

Suppose that \(f\) is a function such that \(f(x) \geq 0\) and \(f^{\prime}(x)<0\) for \(x>0 .\) Show that the area of the triangle with sides \(x=0, y=0\) and the tangent line to \(y=f(x)\) at \(x=a>0\) is \(A(a)=-\frac{1}{2}\left\\{a^{2} f^{\prime}(a)-2 a f(a)+[f(a)]^{2} / f^{\prime}(a)\right\\} .\) To find a curve such that this area is the same for any choice of \(a>0\) solve the equation \(\frac{d A}{d a}=0\)

The differential equation \(y^{\prime}=-a y \ln (y / b)\) (for positive constants \(a\) and \(b\) ) arises in the study of the growth of some animal tumors. Solve the differential equation and sketch several members of the family of solutions. What adjective (e.g., rapid, moderate, slow) would you use to characterize this type of growth?