Chapter 8

In Problems 13 and 14 we assumed that the per colony extinction rate was proportional to \(p .\) This means that the per colony extinction rate goes to 0 for small \(p .\) This may not be realisticsubpopulations may still go extinct even if they are not competing among themselves. One way to model this is to say that the per colony extinction rate is a function \(m(p)\) of \(p\). In Problems 15 and 16 we will assume that \(m(p)=a+b p\) for some constants \(a, b>0 .\) That is, the extinction rate increases with \(p\) because of competition between subpopulations, but \(m(p)\) does not vanish as \(p \rightarrow 0\). Then our model for proportion of occupied sites must be modified to: $$ \frac{d p}{d t}=c p(1-p)-(a+b p) p $$ where \(c, a, b\) are all positive constants. Assuming that the subpopulations obey the differential Equation (8.60) and the coefficients are \(a=1, b=2\), but \(c\) is allowed to take any value: (a) Find the equilibrium values of \(p\) (your answer will depend on the unknown coefficient \(c\) ). (b) What are the conditions on \(c\) for \(p\) to have a nontrivial equilibrium, that is, an equilibrium in which \(p \in(0,1] ?\) (c) Show that if your condition from (b) is met, then the nontrivial equilibrium is also stable.

In Problems , solve each autonomous differential equc tion. $$ \frac{d x}{d t}=-2 x, \text { where } x(1)=3 $$

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d t}=y^{2} t+y^{2} $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d y}{d t}=y(y-2) $$

In Problems , solve each autonomous differential equc tion. $$ \frac{d x}{d t}=1-3 x, \text { where } x(1)=-2 $$

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d t}=y-y t $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d y}{d t}=y^{2}-y $$

Interactions between different subpopulations need not be competitive. In fact, different subpopulations may share resources, and the presence of many subpopulations may provide a pool of genetic diversity that helps the population of organisms to react to changing conditions. We will model cooperation between subpopulations by again assuming that the extinction rate depends on \(p\), but now \(m(p)=a-b p\), where a and \(b\) are both positive constants. So \(m(p)\) decreases as \(p\) increases. Ourmodel for the number of subpopulations then becomes: $$ \frac{d p}{d t}=c p(1-p)-(a-b p) p $$ We will analyze this model in Problems 17 and \(18 .\) (a) Find the equilibrium values of \(p\) (your answer will depend on the constant \(c\) ). You may assume \(c>1\). (b) What is the condition on \(c\) for \(p\) to have a nontrivial equilibrium (i.e., an equilibrium in which \(\hat{p} \in(0,1])\) ? (c) Show that if your condition from (b) is met, then the nontrivial equilibrium is also stable.Assume that the number of subpopulations obeys (8.61) with \(a=2, b=1\), and \(c\) some unknown (positive) constant.

In Problems , solve each autonomous differential equc tion. $$ \frac{d h}{d s}=2 h+1, \text { where } h(0)=4 $$

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d t}=\cos t $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d y}{d t}=y^{2}-2 y-8 $$

In Problems , solve each autonomous differential equc tion. $$ \frac{d N}{d t}=5-N, \text { where } N(2)=2 $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d x}{d t}=x-x^{3} $$

To study the effects of habitat destruction on a single species, we modify the Levins model in the following way: We assume that a fraction \(D\) of patches is permanently destroyed. Consequently, only patches that are vacant and undestroyed can be successfully colonized. A fraction \(1-p(t)-D\) of patches is both vacant and undestroyed where \(p(t)\) is the fraction of occupied patches. Then: $$ \frac{d p}{d t}=c p(1-p-D)-m p $$ (a) Explain in words the meaning of the different terms in \((8.62)\) (b) Assume that \(m=0.2, c=2\), and \(D=0.2\). Show that \((8.62)\) predicts a nontrivial equilibrium value for \(p(t)\) and that this equilibrium is stable.

whose size at time \(t\) is denoted by \(N(t)\), grows according to $$ \frac{d N}{d t}=0.3 N \quad \text { with } N(0)=20 $$ Solve this differential equation, and find the size of the population at time \(t=5\).

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d t}=t^{3}+y t $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d x}{d t}=x^{5}-x $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d N}{d t}=N \ln (2 / N), N>0 $$

A reversible chemical reaction between chemicals \(A\) and \(B\) produces a product C: \(A+B \rightleftharpoons\). We modeled this reaction in Section 8.3.3 using a differential equation for the amount of \(C\) produced: $$ \frac{d x}{d t}=k_{A B}(a-x)(b-x)-k_{C} x $$ Here \(x(t)\) is the amount of \(C\) at time \(t, a\) is the initial amount of chemical \(A, b\) is the initial amount of \(B\), and \(k_{A B}\) and \(k_{C}\) are respectively the rate constants for the reaction that creates \(C\) and for the decay of \(C\) back into \(A\) and \(B\). Explain what each term in (8.63) represents and how the equation is derived.

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d x}=(x+1) y+(x+1) $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d N}{d t}=N^{3} e^{-N} $$

Assume that \(W(t)\) denotes the amount of radioactive material in a substance at time \(t .\) Radioactive decay is described by the differential equation $$ \frac{d W}{d t}=-\lambda W(t) \quad \text { with } W(0)=W_{0} $$ where \(\lambda\) is a positive constant called the decay constant. (a) Solve \((8.27)\). (b) Assume that \(W(0)=123 \mathrm{~g}\) and \(W(5)=20 \mathrm{~g}\) and that time is measured in minutes. Find the decay constant \(\lambda\) and determine the half-life of the radioactive substance. (Remember that the half-life of the substance is the time taken for \(W(t)\) to decrease to half of its initial value.)

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d x}=(x+1) y+(x+1) y^{2} $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d x}{d t}=\frac{x^{2}-x}{x^{2}+1} $$

A drug has first order elimination kinetics, meaning that a fixed fraction of drug is eliminated from the body in each unit of time. So if no further drug is absorbed into the patient's blood after time \(t=0\), the amount of drug in their blood will decay with time according to: $$ \frac{d M}{d t}=-k_{1} M $$ where \(k_{1}>0\) is the fraction of drug eliminated in one unit of time. (a) Assuming \(M(0)=M_{0}\), solve the differential equation. (b) According to your model, does \(M(t)\) ever reach 0 ? (c) Given that \(M_{0}=10\) and \(k_{1}=2\), calculate the time at which \(M(t)\) drops to \(M=1\)

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d x}=\frac{x y}{x+1} $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d x}{d t}=\frac{x+1}{x-1}, x \neq 1 $$

Fish Growth Denote by \(L(t)\) the length of a fish at time \(t\), and assume that the fish grows according to von Bertalanffy's equation $$ \frac{d L}{d t}=k(34-L(t)) \quad \text { with } L(0)=2 $$ (a) Solve the differential equation. (b) Use your solution in (a) to determine \(k\) under the assumption that \(L(4)=10 .\) Sketch the graph of \(L(t)\) for this value of \(k\) (c) Find the length of the fish when \(t=10\). (d) Find the asymptotic length of the fish; that is, find \(\lim _{t \rightarrow \infty} L(t)\)

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d x}=\frac{x}{y+1} $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d x}{d t}=\frac{x}{x+1}, x \neq-1 $$

Fish Growth Denote by \(L(t)\) the length of a certain fish at time \(t\), and assume that this fish grows according to von Bertalanffy's equation $$ \frac{d L}{d t}=k\left(L_{\infty}-L(t)\right) \quad \text { with } L(0)=1 $$ where \(k\) and \(L_{\infty}\) are positive constants. It is known that the asymptotic length is equal to 123 in. and that it takes the fish 27 months to reach half its asymptotic length. (a) Use this information to determine the constants \(k\) and \(L_{\infty}\) in (8.28). [Hint: Solve (8.28).] (b) Determine the length of the fish after 10 months. (c) How long will it take until the fish reaches \(90 \%\) of its asymptotic length?

In Problems 25-28 consider the two-compartment model for two tanks with respective volumes \(V_{1}\) and \(V_{2}\). $$ \begin{array}{l} \frac{d C_{1}}{d t}=\frac{q}{V_{1}}\left(C_{\infty}-C_{1}\right) \\ \frac{d C_{2}}{d t}=\frac{q}{V_{2}}\left(C_{1}-C_{2}\right) \end{array} $$ where \(C_{1}(t)\) is the concentration in the first tank and \(C_{2}(t)\) is the concentration in the second tank, and \(q\) is the volume of water flowing between the two tanks in one unit of time. When we analyzed \((8.93)\) and \((8.94)\) in the main text we assumed that \(V_{1} \neq V_{2} .\) Now consider how the analysis must be modified if \(V_{1}=V_{2}\), and \(C_{1}(0)=C_{2}(0)=0 .\) (a) Show that \(C_{1}(t)=C_{\infty}\left(1-e^{-q t / V_{1}}\right)\) and \(C_{2}(t)=\) \(C_{\infty}\left(1-\left(1+\frac{q t}{V_{1}}\right) e^{-q t / V_{1}}\right)\) (b) Show that \(\lim _{t \rightarrow \infty} C_{1}(t)=C_{\infty}\) and \(\lim _{t \rightarrow \infty} C_{2}(t)=C_{\infty}\).

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d x}{d t}=\frac{x+1}{x}, x \neq 0 $$

In Problems 25-28 consider the two-compartment model for two tanks with respective volumes \(V_{1}\) and \(V_{2}\). $$ \begin{array}{l} \frac{d C_{1}}{d t}=\frac{q}{V_{1}}\left(C_{\infty}-C_{1}\right) \\ \frac{d C_{2}}{d t}=\frac{q}{V_{2}}\left(C_{1}-C_{2}\right) \end{array} $$ where \(C_{1}(t)\) is the concentration in the first tank and \(C_{2}(t)\) is the concentration in the second tank, and \(q\) is the volume of water flowing between the two tanks in one unit of time. Let \(C_{\infty}=0\), so that the fresh water is pumped into tank 1 and flushes solute from tank 1 into tank 2 . Now assume that \(C_{1}(0)=1\) and \(C_{2}(0)=0 .\) If \(q=1, V_{1}=1\), and \(V_{2}=2\), solve the pair of differential equations to find \(C_{1}(t)\) and \(C_{2}(t) .\) Sketch both functions of time.

A chemostat is a device that can be used to maintain a chemical at a particular concentration. Assume that the reaction \(A+B=C\) takes place in a chemostat that maintains \(A\) at a constant concentration \(a\). The chemical \(B\) has initial concentration \(b\) and is depleted by the reaction. (a) Explain why the concentration \(x(t)\) of \(C\) now obeys a differential equation: $$ d x / d t=k_{A B} a(b-x)-k_{C} x $$ (b) Find the equilibrium for \(x\) predicted by Equation (8.65). (c) Is the equilibrium that you found in part (b) stable or unstable?

In Problems 25-28 consider the two-compartment model for two tanks with respective volumes \(V_{1}\) and \(V_{2}\). $$ \begin{array}{l} \frac{d C_{1}}{d t}=\frac{q}{V_{1}}\left(C_{\infty}-C_{1}\right) \\ \frac{d C_{2}}{d t}=\frac{q}{V_{2}}\left(C_{1}-C_{2}\right) \end{array} $$ where \(C_{1}(t)\) is the concentration in the first tank and \(C_{2}(t)\) is the concentration in the second tank, and \(q\) is the volume of water flowing between the two tanks in one unit of time. Let \(C_{\infty}=0\), so that the fresh water is pumped into \(\operatorname{tank} 1\) and flushes solute from tank 1 into tank 2 . Now assume that \(C_{1}(0)=1\) and \(C_{2}(0)=0\). If \(q=1, V_{1}=3\), and \(V_{2}=1\), solve the pair of differential equations to find \(C_{1}(t)\) and \(C_{2}(t)\), and sketch both functions of time.

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d S}{d t}=\frac{1}{S}-\frac{1}{S^{5}}, S>0 $$

Let \(N(t)\) denote the size of a population at time \(t .\) Assume that the population exhibits exponential growth. (a) If you plot \(\log N(t)\) versus \(t\), what kind of graph do you get? (b) Find a differential equation that describes the growth of this population and sketch possible solution curves.

In Problems 25-28 consider the two-compartment model for two tanks with respective volumes \(V_{1}\) and \(V_{2}\). $$ \begin{array}{l} \frac{d C_{1}}{d t}=\frac{q}{V_{1}}\left(C_{\infty}-C_{1}\right) \\ \frac{d C_{2}}{d t}=\frac{q}{V_{2}}\left(C_{1}-C_{2}\right) \end{array} $$ where \(C_{1}(t)\) is the concentration in the first tank and \(C_{2}(t)\) is the concentration in the second tank, and \(q\) is the volume of water flowing between the two tanks in one unit of time. Let \(C_{\infty}=0\), so that the fresh water is pumped into \(\tan \mathrm{k} 1\) and flushes solute from \(\operatorname{tank} 1\) into tank 2 . Now assume that \(C_{1}(0)=1\) and \(C_{2}(0)=0 .\) If \(q=1\) and \(V_{1}=V_{2}=1\), solve the pair of differential equations to find \(C_{1}(t)\) and \(C_{2}(t)\), and sketch both functions of time.

Use the partial-fraction method to solve $$ \frac{d y}{d x}=y(1+y) $$ where \(y(0)=2\).

Consider a two-compartment model where, instead of having a separate reservoir feeding into tank 1, the two tanks are separated by two pipes, one of which carries water from tank 1 to tank 2 , at rate \(q\), and the other carries water from tank 2 to \(\operatorname{tank} 1\), at the same rate \(q\). A schematic and diagram of the flows is given in Figure \(8.59\). (a) Explain why, although there is no net flow between the tanks, we would expect the concentrations in the tanks to change over time. (b) Explain why the change in concentrations over time can be modeled using differential equations: $$ \begin{array}{l} \frac{d C_{1}}{d t}=\frac{q}{V_{1}}\left(C_{2}-C_{1}\right) \\ \frac{d C_{2}}{d t}=\frac{q}{V_{2}}\left(C_{1}-C_{2}\right) \end{array} $$ (c) To solve the differential equations in \((8.95)\) start by assuming that \(V_{1}=V_{2} .\) Then define \(C(t)=\frac{1}{2}\left(C_{1}+C_{2}\right)\) and by deriving a differential equations for \(d C / d t\) and explain why \(C(t)\) is constant. (d) Using the fact that \(C(t)\) is a constant, eliminate \(C_{2}(t)\) from the equation for \(\frac{d C_{1}}{d t}\). Solve the equation you then obtain, and write down expressions for \(C_{1}(t)\) and \(C_{2}(t)\). (e) Use the expression from part (d) to explain why, no matter what the starting values for \(C_{1}(0)\) and \(C_{2}(0)\) are, we expect \(C_{1}(t)\) and \(C_{2}(t)\) to converge to the same limit as \(t \rightarrow \infty\). (f) To solve the differential equations in \((8.95)\) in the most general case \(\left(V_{1} \neq V_{2}\right)\), let \(C(t)=\frac{V_{1} C_{1}+V_{2} C_{2}}{V_{1}+V_{2}}\) (the weighted average of the concentrations in the two tanks). Explain why \(C(t)\) is a constant. (g) Using the fact that \(C(t)\) is a constant, eliminate \(C_{2}(t)\) from the equation for \(\frac{d C_{1}}{d t} .\) Solve the equation you then obtain, and write down expressions for \(C_{1}(t)\) and \(C_{2}(t)\). (h) Use the expression from part (g) to explain why, no matter what the starting values for \(C(t)\) and \(C_{2}(t)\) are, we expect \(C_{1}(t)\) and \(C_{2}(t)\) to converge to the same limit as \(t \rightarrow \infty\).

By breaking down each equation into two parts that you can sketch, determine how many equilib\mathrm{\\{} r i a ~ e a c h ~ d i f f e r e n t i a l ~ e q u a t i o n ~ h a s , ~ a n d ~ c l a s s i f y ~ t h e m ~ a s ~ s t a b l e ~ or unstable. You do not need to determine the location of the equilibria. $$ \frac{d y}{d t}=\ln y-e^{-y} \quad y>0 $$

By breaking down each equation into two parts that you can sketch, determine how many equilib\mathrm{\\{} r i a ~ e a c h ~ d i f f e r e n t i a l ~ e q u a t i o n ~ h a s , ~ a n d ~ c l a s s i f y ~ t h e m ~ a s ~ s t a b l e ~ or unstable. You do not need to determine the location of the equilibria. $$ \frac{d x}{d t}=\frac{1}{x}-\frac{x}{x+1} \quad x>0 $$

To derive the model for the growth or decline of the population of cooperators interacting in a snowdrift game, we modeled the proportion of cooperators using a model. $$ \frac{d x}{d t}=k n x(1-x)(-(b-c / 2) x+(b-c)) $$ where \(b>0\) represents the benefit of interaction if one player is a cooperator and \(c>0\) is the cost of cooperation. In Section 8.3.4 we analyzed Equation \((8.68)\) if \(b>c / 2 .\) Determine the equilibria and what their stability is if \(b=c / 2\).

Use the partial-fraction method to solve $$ \frac{d y}{d x}=y(y-2) $$ where \(y(0)=1\)

By breaking down each equation into two parts that you can sketch, determine how many equilib\mathrm{\\{} r i a ~ e a c h ~ d i f f e r e n t i a l ~ e q u a t i o n ~ h a s , ~ a n d ~ c l a s s i f y ~ t h e m ~ a s ~ s t a b l e ~ or unstable. You do not need to determine the location of the equilibria. $$ \frac{d x}{d t}=3 e^{-x^{2}}-x^{2} $$

Use the partial-fraction method to solve $$ \frac{d y}{d x}=(y+1)(y-2) $$ where \(y(0)=0\).

Filling Box Models In Problem 10 of Section \(8.3\) we analyzed the concentration in a tank whose volume changes over time because the inflows and outflows are not matched. For such a tank it can be shown that if the concentration in the inflow is \(C_{I}\), and the inflow and outflow rates are respectively \(q_{\text {in }}\) and \(q_{\text {out }}\), then both concentration, \(C(t)\), and volume of water in the tank, \(V(t)\), vary with time and can be modeled by a pair of differential equations: $$ \frac{d}{d t}(C V)=q_{\text {in }} C_{I}-q_{\text {out }} C $$ and $$ \frac{d V}{d t}=q_{\text {in }}-q_{\text {out }} . $$ (a) Show that the differential equations (8.96) imply that $$ \left(\left(q_{\mathrm{in}}-q_{\mathrm{out}}\right) t+V_{0}\right) \frac{d C}{d t}+q_{\mathrm{in}} C=q_{\mathrm{in}} C_{I} $$ where \(V_{0}\) is the initial volume of water in the tank. (b) Assuming that \(q_{\text {in }}=2, q_{\text {out }}=1, V_{0}=1, C(0)=0\), and \(C_{l}=1\), solve \((8.97)\) using integrating factors to find \(C(t)\). (c) Assuming that \(q_{\text {in }}=1, q_{\text {out }}=2, V_{0}=1, C(0)=0\), and \(C_{I}=1\), solve \((8.97)\) using integrating factors to find \(C(t) .\) What does your model predict will occur when \(t=1 ?\) Explain whether this answer makes sense given that \(V(1)=0\).

By breaking down each equation into two parts that you can sketch, determine how many equilib\mathrm{\\{} r i a ~ e a c h ~ d i f f e r e n t i a l ~ e q u a t i o n ~ h a s , ~ a n d ~ c l a s s i f y ~ t h e m ~ a s ~ s t a b l e ~ or unstable. You do not need to determine the location of the equilibria. $$ \frac{d x}{d t}=\frac{1}{2}-\frac{x^{2}}{x^{2}+1} $$

Use the partial-fraction method to solve $$ \frac{d y}{d t}=\frac{1}{2} y^{2}-2 y $$ where \(y(0)=-3\).