Chapter 11

Write each system of differential equations in matrix form. $$ \begin{array}{l} \frac{d x_{1}}{d t}=2 x_{1}+3 x_{2} \\ \frac{d x_{2}}{d t}=-4 x_{1}+x_{2} \end{array} $$

Suppose that the densities of two species evolve in accordance with the Lotka- Volterra model of interspecific competition. Assume that species 1 has intrinsic rate of growth \(r_{1}=2\) and carrying capacity \(K_{1}=20\) and that species 2 has intrinsic rate of growth \(r_{2}=3\) and carrying capacity \(K_{2}=15 .\) Furthermore, assume that 20 individuals of species 2 have the same effect on species 1 as 4 individuals of species 1 have on themselves and that 30 individuals of species 1 have the same effect on species 2 as 6 individuals of species 2 have on themselves. Find a system of differential equations that describes this situation.

Write each system of differential equations in matrix form. $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}+x_{2} \\ \frac{d x_{2}}{d t}=-2 x_{2} \end{array} $$

Suppose the densities of two species evolve in accordance with the Lotka- Volterra model of interspecific competition. Assume that species 1 has intrinsic rate of growth \(r_{1}=4\) and carrying capacity \(K_{1}=17\) and that species 2 has intrinsic rate of growth \(r_{2}=1.5\) and carrying capacity \(K_{2}=32 .\) Furthermore, assume that 15 individuals of species 2 have the same effect on species 1 as 7 individuals of species 1 have on themselves and that 5 individuals of species 1 have the same effect on species 2 as 7 individuals of species 2 have on themselves. Find a system of differential equations that describes this situation.

The point \((0,0)\) is always an equilibrium. Use the analytical approach to investigate its stability \(\frac{d x_{1}}{d t}=-x_{1}-x_{2}+x_{1}^{2}\) \(\frac{d x_{2}}{d t}=x_{2}-x_{1}^{2}\)

Write each system of differential equations in matrix form. $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{3}-2 x_{1} \\ \frac{d x_{2}}{d t}=-x_{1} \end{array} $$

The point \((0,0)\) is always an equilibrium. Use the analytical approach to investigate its stability \(\frac{d x_{1}}{d t}=x_{1}+x_{1}^{2}-2 x_{1} x_{2}+x_{2}\) \(\frac{d x_{2}}{d t}=x_{1}\)

Write each system of differential equations in matrix form. $$ \begin{array}{l} \frac{d x_{1}}{d t}=2 x_{2}-3 x_{1}-x_{3} \\ \frac{d x_{2}}{d t}=-x_{1}+x_{2} \end{array} $$

The point \((0,0)\) is always an equilibrium. Use the analytical approach to investigate its stability \(\frac{d x_{1}}{d t}=3 x_{1} x_{2}-x_{1}+x_{2}\) \(\frac{d x_{2}}{d t}=x_{2}^{2}-x_{1}\)

Consider $$ \begin{array}{l} \frac{d x_{1}}{d t}=-x_{1}+2 x_{2} \\ \frac{d x_{2}}{d t}=x_{1} \end{array} $$ Determine the direction vectors associated with the following points in the \(x_{1}-x_{2}\) plane, and graph the direction vectors in the \(x_{1}-\) \(x_{2}\) plane: \((1,0),(0,1),(-1,0),(0,-1),(1,1),(0,0)\), and \((-2,1)\)

Use the graphical approach to classify the following Lotka-Volterra models of interspecific competition according to "coexistence," "founder control," "species 1 excludes species 2," or "species 2 excludes species 1." \(\frac{d N_{1}}{d t}=N_{1}\left(1-\frac{N_{1}}{20}-\frac{N_{2}}{5}\right)\) \(\frac{d N_{2}}{d t}=2 N_{2}\left(1-\frac{N_{2}}{15}-\frac{N_{1}}{3}\right)\)

Consider $$ \begin{array}{l} \frac{d x_{1}}{d t}=2 x_{1}-x_{2} \\ \frac{d x_{2}}{d t}=-x_{2} \end{array} $$ Determine the direction vectors associated with the following points in the \(x_{1}-x_{2}\) plane, and graph the direction vectors in the \(x_{1^{-}}\) \(x_{2}\) plane: \((2,0),(1.5,1),(1,0),(0,-1),(1,1),(0,0)\), and \((-2,-2)\).

The point \((0,0)\) is always an equilibrium. Use the analytical approach to investigate its stability \(\frac{d x_{1}}{d t}=-2 \sin x_{1}\) \(\frac{d x_{2}}{d t}=-x_{2} e^{x_{1}}\)

Consider $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}+3 x_{2} \\ \frac{d x_{2}}{d t}=-x_{1}+2 x_{2} \end{array} $$ Determine the direction vectors associated with the following points in the \(x_{1}-x_{2}\) plane, and graph the direction vectors in the \(x_{1}-\) \(x_{2}\) plane: \((1,0),(0,1),(-1,1),(0,-1),(-3,1),(0,0)\), and \((-2,1)\).

Use the eigenvalue approach to analyze all equilibria of the given Lotka- Volterra models of interspecific competition. \(\frac{d N_{1}}{d t}=3 N_{1}\left(1-\frac{N_{1}}{18}-1.3 \frac{N_{2}}{18}\right)\) \(\frac{d N_{2}}{d t}=2 N_{2}\left(1-\frac{N_{2}}{20}-0.6 \frac{N_{1}}{20}\right)\)

In Problems \(7-12\), find all equilibria of each system of differential equations and use the analytical approach to determine the stability of each equilibrium. \(\frac{d x_{1}}{d t}=-x_{1}+2 x_{1}\left(1-x_{1}\right)\) \(\frac{d x_{2}}{d t}=-x_{2}+5 x_{2}\left(1-x_{1}-x_{2}\right)\)

Consider $$ \begin{array}{l} \frac{d x_{1}}{d t}=-x_{2} \\ \frac{d x_{2}}{d t}=x_{1}+x_{2} \end{array} $$ Determine the direction vectors associated with the following points in the \(x_{1}-x_{2}\) plane, and graph the direction vectors in the \(x_{1}-\) \(x_{2}\) plane: \((1,0),(0,1),(-1,0),(0,-1),(1,1),(0,0)\), and \((-2,-2)\).

Use the eigenvalue approach to analyze all equilibria of the given Lotka- Volterra models of interspecific competition. \(\frac{d N_{1}}{d t}=4 N_{1}\left(1-\frac{N_{1}}{12}-0.3 \frac{N_{2}}{12}\right)\) \(\frac{d N_{2}}{d t}=5 N_{2}\left(1-\frac{N_{2}}{15}-0.2 \frac{N_{1}}{15}\right)\)

Find all equilibria of each system of differential equations and use the analytical approach to determine the stability of each equilibrium. $$ \begin{array}{l} \frac{d x_{1}}{d t}=-x_{1}+3 x_{1}\left(1-x_{1}-x_{2}\right) \\ \frac{d x_{2}}{d t}=-x_{2}+5 x_{2}\left(1-x_{1}-x_{2}\right) \end{array} $$

Use the eigenvalue approach to analyze all equilibria of the given Lotka- Volterra models of interspecific competition. \(\frac{d N_{1}}{d t}=N_{1}\left(1-\frac{N_{1}}{35}-3 \frac{N_{2}}{35}\right)\) \(\frac{d N_{2}}{d t}=3 N_{2}\left(1-\frac{N_{2}}{40}-4 \frac{N_{1}}{40}\right)\)

Find all equilibria of each system of differential equations and use the analytical approach to determine the stability of each equilibrium. $$ \begin{array}{l} \frac{d x_{1}}{d t}=4 x_{1}\left(1-x_{1}\right)-2 x_{1} x_{2} \\ \frac{d x_{2}}{d t}=x_{2}\left(2-x_{2}\right)-x_{2} \end{array} $$

Use the eigenvalue approach to analyze all equilibria of the given Lotka- Volterra models of interspecific competition. \(\frac{d N_{1}}{d t}=N_{1}\left(1-\frac{N_{1}}{25}-0.1 \frac{N_{2}}{25}\right)\) \(\frac{d N_{2}}{d t}=N_{2}\left(1-\frac{N_{2}}{28}-1.2 \frac{N_{1}}{28}\right)\)

Find all equilibria of each system of differential equations and use the analytical approach to determine the stability of each equilibrium. $$ \begin{array}{l} \frac{d x_{1}}{d t}=2 x_{1}\left(5-x_{1}-x_{2}\right) \\ \frac{d x_{2}}{d t}=3 x_{2}\left(7-3 x_{1}-x_{2}\right) \end{array} $$

Suppose that two species of beetles are reared together in one experiment and separately in another. When species 1 is reared alone, it reaches an equilibrium of about \(200 .\) When species 2 is reared alone, it reaches an equilibrium of about \(150 .\) When both of them are reared together, they seem to be able to coexist: Species 1 reaches an equilibrium of about 180 and species 2 reaches an equilibrium of about \(80 .\) If their densities follow the LotkaVolterra equation of interspecific competition, find \(\alpha_{12}\) and \(\alpha_{21}\).

Find all equilibria of each system of differential equations and use the analytical approach to determine the stability of each equilibrium. $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}-x_{2} \\ \frac{d x_{2}}{d t}=x_{1} x_{2}-x_{2} \end{array} $$

Suppose that two species of beetles are reared together. Species 1 wins if there are initially 100 individuals of species 1 and 20 individuals of species \(2 .\) But species 2 wins if there are initially 20 individuals of species 1 and 100 individuals of species \(2 .\) When the beetles are reared separately, both species seem to reach an equilibrium of about \(120 .\) On the basis of this information and assuming that the densities follow the Lotka-Volterra model of interspecific competition, can you give lower bounds on \(\alpha_{12}\) and \(\alpha_{21} ?\)

Find all equilibria of each system of differential equations and use the analytical approach to determine the stability of each equilibrium. $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1} x_{2}-x_{2} \\ \frac{d x_{2}}{d t}=x_{1}+x_{2} \end{array} $$

Find the general solution of each given system of differential equations and sketch the lines in the direction of the eigenvectors. Indicate on each line the direction in which the solution would move if it starts on that line. $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} 1 & 3 \\ 5 & 3 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$

Use a graphing calculator to sketch solution curves of the given Lotka- Volterra predator-prey model in the \(N-\) P plane. Also graph \(N(t)\) and \(P(t)\) as functions of \(t\) \(\frac{d N}{d t}=2 N-P N\) \(\frac{d P}{d t}=\frac{1}{2} P N-P\) with initial conditions (a) \((N(0), P(0))=(2,2)\) (b) \((N(0), P(0))=(3,3)\) (c) \((N(0), P(0))=(4,4)\)

For which value of \(a\) has $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{2}\left(x_{1}+a\right) \\ \frac{d x_{2}}{d t}=x_{2}^{2}+x_{2}-x_{1} \end{array} $$ a unique equilibrium? Characterize its stability.

Find the general solution of each given system of differential equations and sketch the lines in the direction of the eigenvectors. Indicate on each line the direction in which the solution would move if it starts on that line. $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} 2 & 1 \\ 4 & -1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$

Use a graphing calculator to sketch solution curves of the given Lotka- Volterra predator-prey model in the \(N-\) P plane. Also graph \(N(t)\) and \(P(t)\) as functions of \(t\) \(\frac{d N}{d t}=3 N-2 P N\) \(\frac{d P}{d t}=P N-P\) with initial conditions (a) \((N(0), P(0))=(1,3 / 2)\) (b) \((N(0), P(0))=(2,2)\) (c) \((N(0), P(0))=(3,1)\) In Problems 15 and 16 , we investigate the Lotka-Volterra predatorprey model.

Assume that \(a>0 .\) Find all point equilibria of $$ \begin{array}{l} \frac{d x_{1}}{d t}=1-a x_{1} x_{2} \\ \frac{d x_{2}}{d t}=a x_{1} x_{2}-x_{2} \end{array} $$ and characterize their stability.

$$ \text { Systems of Differential Equations } $$ $$ \begin{array}{l} \frac{d x_{1}}{d t}=-1.6 x_{1}+0.3 x_{2} \\ \frac{d x_{2}}{d t}=0.1 x_{1}-0.5 x_{2} \end{array} $$

Find the general solution of each given system of differential equations and sketch the lines in the direction of the eigenvectors. Indicate on each line the direction in which the solution would move if it starts on that line. $$ \left[\begin{array}{l} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{rr} -3 & 3 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$

Assume that $$ \begin{array}{l} \frac{d N}{d t}=N-4 P N \\ \frac{d P}{d t}=2 P N-3 P \end{array} $$ (a) Show that this system has two equilibria: the trivial equilibrium \((0,0)\), and a nontrivial one in which both species have positive densities. (b) Use the eigenvalue approach to show that the trivial equilibrium is unstable. (c) Determine the eigenvalues corresponding to the nontrivial equilibrium. Does your analysis allow you to infer anything about the stability of this equilibrium? (d) Use a graphing calculator to sketch curves in the \(N-P\) plane. Also, sketch solution curves of the prey and the predator densities as functions of time.

Assume that $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}\left(10-2 x_{1}-x_{2}\right) \\ \frac{d x_{2}}{d t}=x_{2}\left(10-x_{1}-2 x_{2}\right) \end{array} $$ (a) Graph the zero isoclines. (b) Show that \(\left(\frac{10}{3}, \frac{10}{3}\right)\) is an equilibrium, and use the analytical approach to determine its stability.

$$ \text { Systems of Differential Equations } $$ $$ \begin{array}{l} \frac{u x_{1}}{d t}=-1.2 x_{1} \\ \frac{d x_{2}}{d t}=0.3 x_{1}-0.2 x_{2} \end{array} $$

Find the general solution of each given system of differential equations and sketch the lines in the direction of the eigenvectors. Indicate on each line the direction in which the solution would move if it starts on that line. $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} -5 & 3 \\ -2 & 0 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$

Assume that $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}\left(10-x_{1}-2 x_{2}\right) \\ \frac{d x_{2}}{d t}=x_{2}\left(10-2 x_{1}-x_{2}\right) \end{array} $$ (a) Graph the zero isoclines. (b) Show that \(\left(\frac{10}{3}, \frac{10}{3}\right)\) is an equilibrium, and use the analytical approach to determine its stability.

Find the general solution of each given system of differential equations and sketch the lines in the direction of the eigenvectors. Indicate on each line the direction in which the solution would move if it starts on that line. $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} -2 & 0 \\ -3 & 1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$

Assume that \(N(t)\) denotes the density of an insect species at time \(t\) and \(P(t)\) denotes the density of its predator at time \(t\). The insect species is an agricultural pest, and its predator is used as a biological control agent. Their dynamics are given by the system of differential equations $$ \begin{array}{l} \frac{d N}{d t}=5 N-3 P N \\ \frac{d P}{d t}=2 P N-P \end{array} $$ (a) Explain why $$ \frac{d N}{d t}=5 N $$ describes the dynamics of the insect in the absence of the predator. Solve (11.85). Describe what happens to the insect population in the absence of the predator. (b) Explain why introducing the insect predator into the system can help to control the density of the insect. (c) Assume that at the beginning of the growing season the insect density is \(0.5\) and the predator density is \(2 .\) You decide to control the insects by using an insecticide in addition to the predator. You are careful and choose an insecticide that does not harm the predator. After you spray, the insect density drops to \(0.01\) and the predator density remains at \(2 .\) Use a graphing calculator to investigate the long-term implications of your decision to spray the field. In particular, investigate what would have happened to the insect densities if you had decided not to spray the field, and compare your results with the insect density over time that results from your application of the insecticide.

$$ \text { Systems of Differential Equations } $$ $$ \begin{array}{l} \frac{d x_{1}}{d t}=-0.2 x_{1} \\ \frac{d x_{2}}{d t}=-0.3 x_{2} \end{array} $$

Find the general solution of each given system of differential equations and sketch the lines in the direction of the eigenvectors. Indicate on each line the direction in which the solution would move if it starts on that line. $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} 5 & 2 \\ 1 & 6 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$

Assume that \(N(t)\) denotes prey density at time \(t\) and \(P(t)\) denotes predator density at time \(t\). Their dynamics are given by the system of equations $$ \begin{array}{l} \frac{d N}{d t}=4 N-2 P N \\ \frac{d P}{d t}=P N-3 P \end{array} $$ Assume that initially \(N(0)=3\) and \(P(0)=2\). (a) If you followed this predator-prey community over time, what would you observe? (b) Suppose that bad weather kills \(90 \%\) of the prey population and \(67 \%\) of the predator population. If you continued to observe this predator-prey community, what would you expect to see?

$$ \text { Systems of Differential Equations } $$ $$ \begin{array}{l} \frac{d x_{1}}{d t}=-x_{1} \\ \frac{d x_{2}}{d t}=x_{1}-0.5 x_{2} \end{array} $$

An unrealistic feature of the Lotka-Volterra model is that the prey exhibits unlimited growth in the absence of the predator. The model described by the following system remedies this shortcoming (in the model, we assume that the prey evolves according to logistic growth in the absence of the predator; the other features of the model are retained): $$ \begin{array}{l} \frac{d N}{d t}=3 N\left(1-\frac{N}{10}\right)-2 P N \\ \frac{d P}{d t}=P N-4 P \end{array} $$ (a) Explain why the prey evolves according to $$ \frac{d N}{d t}=3 N\left(1-\frac{N}{10}\right) $$ in the absence of the predator. Investigate the long-term behavior of solutions to (11.87). (b) Find all equilibria of (11.86), and use the eigenvalue approach to determine their stability. (c) Use a graphing calculator to sketch the solution curve of \((11.86)\) in the \(N-P\) plane when \(N(0)=2\) and \(P(0)=2 .\) Also, sketch \(N(t)\) and \(P(t)\) as functions of time, starting with \(N(0)=2\) and \(P(0)=2\).

An unrealistic feature of the Lotka-Volterra model is that the prey exhibits unlimited growth in the absence of the predator. The model described by the following system remedies this shortcoming (in the model, we assume that the prey evolves according to logistic growth in the absence of the predator; the other features of the model are retained): $$ \begin{array}{l} \frac{d N}{d t}=N\left(1-\frac{N}{K}\right)-4 P N \\ \frac{d P}{d t}=P N-5 P \end{array} $$ Here, \(K>0\) denotes the carrying capacity of the prey in the absence of the predator. In what follows, we will investigate how the carrying capacity affects the outcome of this predator-prey interaction. (a) Draw the zero isoclines of \((11.88)\) for (i) \(K=10\) and (ii) \(K=\) \(3 .\) (b) When \(K=10\), the zero isoclines intersect, indicating the existence of a nontrivial equilibrium. Analyze the stability of this nontrivial equilibrium. (c) Is there a minimum carrying capacity required in order to have a nontrivial equilibrium? If yes, find it and explain what happens when the carrying capacity is below this minimum and what happens when the carrying capacity is above this minimum.

Suppose that a drug is administered to a person in a single dose, and assume that the drug does not accumulate in body tissue, but is excreted through urine. Denote the amount of drug in the body at time \(t\) by \(x_{1}(t)\) and in the urine at time \(t\) by \(x_{2}(t) .\) If \(x_{1}(0)=6 \mathrm{mg}\) and \(x_{2}(0)=0\), find a system of differential equations for \(x_{1}(t)\) and \(x_{2}(t)\) if it takes 20 minutes for the drug to be at onehalf of its initial amount in the body.

Solve the given initial-value problem. $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} 3 & -2 \\ 0 & 1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ $$ \text { with } x_{1}(0)=1 \text { and } x_{2}(0)=1 \text { . } $$