Chapter 8

Suppose that $$\frac{d y}{d x}=y(2-y)$$ (a) Find the equilibria of this differential equation. (b) Graph \(d y / d x\) as a function of \(y\), and use your graph to discuss the stability of the equilibria. (c) Compute the eigenvalues associated with each equilibrium, and discuss the stability of the equilibria.

Solve each pure-time differential equation. \(\frac{d y}{d x}=x+\sin x\), where \(y_{0}=0\) for \(x_{0}=0\)

Solve each pure-time differential equation. \(\frac{d y}{d x}=e^{-3 x}\), where \(y_{0}=10\) for \(x_{0}=0\)

Solve each pure-time differential equation. \(\frac{d y}{d x}=\frac{1}{x}\), where \(y_{0}=0\) when \(x_{0}=1\)

Suppose that $$\frac{d y}{d x}=y(2-y)(y-3)$$ (a) Find the equilibria of this differential equation. (b) Graph \(d y / d x\) as a function of \(y\), and use your graph to discuss the stability of the equilibria. (c) Compute the eigenvalues associated with each equilibrium, and discuss the stability of the equilibria.

Solve each pure-time differential equation. \(\frac{d y}{d x}=\frac{1}{1+x^{2}}\), where \(y_{0}=1\) when \(x_{0}=0\)

Assume that the size of a population evolves according to the logistic equation with intrinsic rate of growth \(r=1.5\). Assume that the carrying capacity \(K=100\). (a) Find the differential equation that describes the rate of growth of this population. (b) Find all equilibria, and, using the graphical approach, discuss the stability of the equilibria. (c) Find the eigenvalues associated with the equilibria, and use the eigenvalues to determine the stability of the equilibria. Compare your answers with your results in (b).

Solve each pure-time differential equation. \(\frac{d x}{d t}=\frac{1}{1-t}\), where \(x(0)=2\)

Suppose that \(N(t)\) denotes the size of a population at time \(t .\) The population evolves according to the logistic equation, but, in addition, predation reduces the size of the population so that the rate of change is given by $$\frac{d N}{d t}=N\left(1-\frac{N}{50}\right)-\frac{9 N}{5+N}$$ The first term on the right-hand side describes the logistic growth; the second term describes the effect of predation. (a) Set $$g(N)=N\left(1-\frac{N}{50}\right)-\frac{9 N}{5+N}$$ and graph \(g(N)\). (b) Find all equilibria of \((8.65)\). (c) Use your graph in (a) to determine the stability of the equilibria you found in (b). (d) Use the method of eigenvalues to determine the stability of the equilibria you found in (b).

Solve each pure-time differential equation. \(\frac{d x}{d t}=\cos (2 \pi(t-3))\), where \(x(3)=1\)

Solve each pure-time differential equation. \(\frac{d s}{d t}=\sqrt{3 t+1}\), where \(s(0)=1\)

Solve each pure-time differential equation. \(\frac{d h}{d t}=5-16 t^{2}\), where \(h(3)=-11\)

Suppose that a fish population evolves according to the logistic equation and that a fixed number of fish per unit time are removed. That is, $$\frac{d N}{d t}=r N\left(1-\frac{N}{K}\right)-H$$ Assume that \(r=2\) and \(K=1000\). (a) Find possible equilibria, and discuss their stability when \(H=\) \(100 .\) (b) What is the maximal harvesting rate that maintains a positive population size?

Suppose that the volume \(V(t)\) of a cell at time \(t\) changes according to $$\frac{d V}{d t}=1+\cos t \quad \text { with } V(0)=5$$ Find \(V(t)\).

Suppose that a fish population evolves according to a logistic equation and that fish are harvested at a rate proportional to the population size. If \(N(t)\) denotes the population size at time \(t\), then $$\frac{d N}{d t}=r N\left(1-\frac{N}{K}\right)-h N$$ Assume that \(r=2\) and \(K=1000\). (a) Find possible equilibria, use the graphical approach to discuss their stability when \(h=0.1\), and find the maximal harvesting rate that maintains a positive population size. (b) Show that if \(h

Suppose that the amount of phosphorus in a lake at time \(t\), denoted by \(P(t)\), follows the equation $$\frac{d P}{d t}=3 t+1 \quad \text { with } P(0)=0$$ Find the amount of phosphorus at time \(t=10\).

Solve the given autonomous differential equations. \(\frac{d y}{d x}=3 y\), where \(y_{0}=2\) for \(x_{0}=0\)

(Adapted from Crawley, 1997) Denote plant biomass by \(V\), and herbivore number by \(N .\) The plant-herbivore interaction is modeled as $$ \begin{array}{l} \frac{d V}{d t}=a V\left(1-\frac{V}{K}\right)-b V N \\ \frac{d N}{d t}=c V N-d N \end{array} $$ (a) Suppose the herbivore number is equal to \(0 .\) What differential equation describes the dynamics of the plant biomass? Can you explain the resulting equation? Determine the plant biomass equilibrium in the absence of herbivores. (b) Now assume that herbivores are present. Describe the effect of herbivores on plant biomass; that is, explain the term \(-b V N\) in the first equation. Describe the dynamics of the herbivoresthat is, how their population size increases and what contributes to decreases in their population size. (c) Determine the equilibria (1) by solving $$ \frac{d V}{d t}=0 \quad \text { and } \quad \frac{d N}{d t}=0 $$ and (2) graphically. Explain why this model implies that "plant abundance is determined solely by attributes of the herbivore," as stated in Crawley (1997).

Solve the given autonomous differential equations. \(\frac{d y}{d x}=2(1-y)\), where \(y_{0}=2\) for \(x_{0}=0\)

Solve the given autonomous differential equations. \(\frac{d x}{d t}=-2 x\), where \(x(1)=5\)

Suppose that a tank holds 1000 liters of water, and \(2 \mathrm{~kg}\) of salt is poured into the tank. (a) Compute the concentration of salt in g liter \(^{-1}\). (b) Assume now that you want to reduce the salt concentration. One method would be to remove a certain amount of the salt water from the tank and then replace it by pure water. How much salt water do you have to replace by pure water to obtain a salt concentration of \(1 \mathrm{~g}\) liter \(^{-1}\) ? (c) Another method for reducing the salt concentration would be to hook up an overflow pipe and pump pure water into the tank. That way, the salt concentration would be gradually reduced. Assume that you have two pumps, one that pumps water at a rate of 1 liter \(\mathrm{s}^{-1}\), the other at a rate of 2 liter \(\mathrm{s}^{-1} .\) For each pump, find out how long it would take to reduce the salt concentration from the original concentration to \(1 \mathrm{~g}\) liter \(^{-1}\) and how much pure water is needed in each case. (Note that the rate at which water enters the tank is equal to the rate at which water leaves the tank.) Compare the amount of water needed using the pumps with the amount of water needed in part (b).

Solve the given autonomous differential equations. \(\frac{d h}{d s}=2 h+1\), where \(h(0)=4\)

Solve the given autonomous differential equations. \(\frac{d N}{d t}=5-N\), where \(N(2)=3\)

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

Suppose that you follow the size of a population over time. When you plot the size of the population versus time on a semilog plot (i.e., the horizontal axis, representing time, is on a linear scale, whereas the vertical axis, representing the size of the population, is on a logarithmic scale), you find that your data fit a straight line which intercepts the vertical axis at 1 (on the log scale) and has slope \(-0.43 .\) Find a differential equation that relates the growth rate of the population at time \(t\) to the size of the population at time \(t\).

Assume that \(W(t)\) denotes the amount of radioactive material in a substance at time \(t\). Radioactive decay is then 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 \((8.42)\). (b) Assume that \(W(0)=123 \mathrm{gr}\) and \(W(5)=20 \mathrm{gr}\) and that time is measured in minutes. Find the decay constant \(\lambda\) and determine the half-life of the radioactive substance.

Denote by \(p=p(t)\) the fraction of occupied patches in a metapopulation model, and assume that $$\frac{d p}{d t}=0.5 p(1-p)-1.5 p \quad \text { for } t \geq 0$$ (a) Set \(g(p)=0.5 p(1-p)-1.5 p .\) Graph \(g(p)\) for \(p \in[0,1]\). (b) Find all equilibria of \((8.69)\) that are in \([0,1] .\) Use your graph in (a) to determine their stability. (c) Use the eigenvalue approach to analyze the stability of the equilibria that you found in (b).

Suppose that a population, whose size at time \(t\) is given by \(N(t)\), grows according to $$\frac{d N}{d t}=\frac{1}{100} N^{2}, \quad \text { with } N(0)=10$$ (a) Solve \((8.43)\). (b) Graph \(N(t)\) as a function of \(t\) for \(0 \leq t<10\). What happens as \(t \rightarrow 10 ?\) Explain in words what this means.

Denote by \(p=p(t)\) the fraction of occupied patches in a metapopulation model, and assume that $$\frac{d p}{d t}=c p(1-p)-p^{2} \quad \text { for } t \geq 0$$ where \(c>0 .\) The term \(p^{2}\) describes the density-dependent extinction of patches; that is, the per-patch extinction rate is \(p\), and a fraction \(p\) of patches are occupied, resulting in an extinction rate of \(p^{2}\). The colonization of vacant patches is the same as in the Levins model. (a) Set \(g(p)=c p(1-p)-p^{2}\) and sketch the graph of \(g(p)\). (b) Find all equilibria of \((8.70)\) in \([0,1]\), and determine their stability. (c) Is there a nontrivial equilibrium when \(c>0 ?\) Contrast your findings with the corresponding results in the Levins model.

Denote by \(L(t)\) the length of a fish at time \(t\), and assume that the fish grows according to the von Bertalanffy equation $$\frac{d L}{d t}=k(34-L(t)) \quad \text { with } L(0)=2$$ (a) Solve (8.44). (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 \(\lim _{t \rightarrow \infty} L(t)\)

Denote by \(L(t)\) the length of a certain fish at time \(t\), and assume that this fish grows according to the von Bertalanffy 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. A study showed that the asymptotic length is equal to 123 in and that it takes this fish 27 months to reach half its asymptotic length. (a) Use this information to determine the constants \(k\) and \(L_{\infty}\) in (8.45). [Hint: Solve (8.45).] (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?

Denote the size of a population at time \(t\) by \(N(t)\), and assume that $$\frac{d N}{d t}=2 N(N-10)\left(1-\frac{N}{100}\right) \quad \text { for } t \geq 0$$ (a) Find all equilibria of \((8.72)\). (b) Use the eigenvalue approach to determine the stability of the equilibria you found in (a). (c) Set $$g(N)=2 N(N-10)\left(1-\frac{N}{100}\right)$$ for \(N \geq 0\), and graph \(g(N) .\) Identify the equilibria of \((8.72)\) on your graph, and use the graph to determine the stability of the equilibria. Compare your results with your findings in (b). Use your graph to give a graphical interpretation of the eigenvalues associated with the equilibria.

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.

Denote the size of a population at time \(t\) by \(N(t)\), and assume that $$\frac{d N}{d t}=0.3 N(N-17)\left(1-\frac{N}{200}\right) \quad \text { for } t \geq 0$$ (a) Find all equilibria of \((8.73)\). (b) Use the eigenvalue approach to determine the stability of the equilibria you found in (a). (c) Set $$g(N)=0.3 N(N-17)\left(1-\frac{N}{200}\right)$$ for \(N \geq 0\), and graph \(g(N)\). Identify the equilibria of \((8.73)\) on your graph, and use the graph to determine the stability of the equilibria. Compare your results with your findings in (b). Use your graph to give a graphical interpretation of the eigenvalues associated with the equilibria.

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

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

Use the partial-fraction method to solve $$\frac{d y}{d x}=y(y-5)$$ where \(y_{0}=1\) for \(x_{0}=0\).

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

Use the partial-fraction method to solve $$\frac{d y}{d x}=2 y(3-y)$$ where \(y_{0}=5\) for \(x_{0}=1\).

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

Solve the given differential equations. $$ \frac{d y}{d x}=y(1+y) $$

Solve the given differential equations. $$ \frac{d y}{d x}=(1+y)^{2} $$

Solve the given differential equations. $$ \frac{d y}{d x}=(1+y)^{3} $$

Solve the given differential equations. $$ \frac{d y}{d x}=(3-y)(2+y) $$

(a) Use partial fractions to show that $$\int \frac{d u}{u^{2}-a^{2}}=\frac{1}{2 a} \ln \left|\frac{u-a}{u+a}\right|+C$$ (b) Use your result in (a) to find a solution of $$\frac{d y}{d x}=y^{2}-4$$ that passes through (i) \((0,0)\), (ii) \((0,2)\), and (iii) \((0,4)\).

Find a solution of $$\frac{d y}{d x}=y^{2}+4$$ that passes through \((0,2)\).

Suppose that the size of a population at time \(t\) is denoted by \(N(t)\) and that \(N(t)\) satisfies the differential equation $$\frac{d N}{d t}=0.34 N\left(1-\frac{N}{200}\right) \quad \text { with } N(0)=50$$ Solve this differential equation, and determine the size of the population in the long run; that is, find \(\lim _{t \rightarrow \infty} N(t)\).

Assume that the size of a population, denoted by \(N(t)\), evolves according to the logistic equation. Find the intrinsic rate of growth if the carrying capacity is \(100, N(0)=10\), and \(N(1)=20\).

Suppose that \(N(t)\) denotes the size of a population at time \(t\) and that $$\frac{d N}{d t}=1.5 N\left(1-\frac{N}{50}\right)$$ (a) Solve this differential equation when \(N(0)=10\). (b) Solve this differential equation when \(N(0)=90\). (c) Graph your solutions in (a) and (b) in the same coordinate system. (d) Find \(\lim _{t \rightarrow \infty} N(t)\) for your solutions in (a) and (b).

Suppose that the size of a population, denoted by \(N(t)\), satisfies $$\frac{d N}{d t}=0.7 N\left(1-\frac{N}{35}\right)$$ (a) Determine all equilibria by solving \(d N / d t=0\). (b) Solve \((8.46)\) for (i) \(N(0)=10\), (ii) \(N(0)=35\), (iii) \(N(0)=50\), and (iv) \(N(0)=0\). Find \(\lim _{t \rightarrow \infty} N(t)\) for each of the four initial conditions. (c) Compare your answer in (a) with the limiting values you found in (b).