Chapter 11

In Problems \(1-4\), write each system of differential equations in matrix form. \(\begin{aligned} & \frac{d x_{1}}{d t}=2 x_{1}+3 x_{2} \\ \frac{d x_{2}}{d t} &=-x_{1}+x_{2} \end{aligned}\)

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." \(\frac{d N_{1}}{d t}=2 N_{1}\left(1-\frac{N_{1}}{10}-0.2 \frac{N_{2}}{10}\right)\) \(\frac{d N_{2}}{d t}=5 N_{2}\left(1-\frac{N_{2}}{15}-0.5 \frac{N_{1}}{15}\right)\)

The point \(\mathbf{( 0 , 0 )}\) is always an equilibrium. Determine whether it is stable or unstable. \(\frac{d x_{1}}{d t}=-x_{1}+x_{2}+x_{1}^{2}\) \(\frac{d x_{2}}{d t}=x_{2}+x_{1}^{3}\)

Write each system of differential equations in matrix form. \(\begin{aligned} & \frac{d x_{1}}{d t}=x_{1}+x_{2} \\ \frac{d x_{2}}{d t} &=-x_{1} \end{aligned}\)

The point \(\mathbf{( 0 , 0 )}\) is always an equilibrium. Determine whether it is stable or unstable. \(\frac{d x_{1}}{d t}=x_{1}+x_{1}^{2}-2 x_{1} x_{2}+3 x_{2}\) \(\frac{d x_{2}}{d t}=-x_{1}\)

Write each system of differential equations in matrix form. \(\frac{d x_{1}}{d t}=x_{3}-2 x_{1}\) \(\frac{d x_{2}}{d t}=-x_{1}+x_{3}\) \(\frac{d x_{3}}{d t}=x_{1}+x_{2}+x_{3}\)

Write each system of differential equations in matrix form. \(\begin{aligned} & \frac{d x_{1}}{d t}=2 x_{2}-3 x_{1}-x_{3} \\ & \frac{d x_{2}}{d t}=x_{2}-2 x_{1} \\ & \frac{d x_{3}}{d t}=5 x_{1}+x_{3} \end{aligned}\)

The point \(\mathbf{( 0 , 0 )}\) is always an equilibrium. Determine whether it is stable or unstable. \(\frac{d x_{1}}{d t}=\ln \left(1+x_{1}+x_{2}\right)\) \(\frac{d x_{2}}{d t}=x_{1}-x_{2}\)

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 \((-1,1)\)

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

The point \(\mathbf{( 0 , 0 )}\) is always an equilibrium. Determine whether it is stable or unstable. \(\frac{d x_{1}}{d t}=-2 \sin x_{1}\) \(\frac{d x_{2}}{d t}=-x_{2} e^{x_{1}}\)

Consider $$ \begin{array}{l} \frac{d x_{1}}{d t}=2 x_{1}-x_{2} \\ \frac{d x_{2}}{d t}=-x_{2} \end{array} $$ 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: \((1,0),(0,1),(-1,0),(0,-1),(1,1),(0,0)\), and \((-2,-2)\)

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

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

Consider $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}+x_{2} \\ \frac{d x_{2}}{d t}=3 x_{1}-x_{2} \end{array} $$ 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 \((1,2) .\)

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 determine the stability of each equilibrium. \(\frac{d x_{1}}{d t}=2 x_{1}-x_{1}^{2}-2 x_{2} x_{1}\) \(\frac{d x_{2}}{d t}=x_{2}-2 x_{2}^{2}-x_{1} x_{2}\)

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. \(\frac{d N_{1}}{d t}=N_{1}\left(1-\frac{N_{1}}{25}-0.1 \frac{N_{2}}{25}\right)\) \(\frac{d N_{2}}{d t}=N_{2}\left(1-\frac{N_{2}}{28}-1.2 \frac{N_{1}}{28}\right)\)

Find all equilibria of each system of differential equations and determine the stability of each equilibrium. \(\frac{d x_{1}}{d t}=4 x_{1}\left(1-x_{1}\right)-2 x_{1} x_{2}\) \(\frac{d x_{2}}{d t}=x_{2}\left(2-x_{2}\right)-x_{2}\)

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 \(\mathrm{co}^{-}\) exist: Species 1 reaches an equilibrium of about 180 and species 2 reaches an equilibrium of about \(80 .\) If their densities follow the Lotka-Volterra equation of interspecific competition, find \(\alpha_{12}\) and \(\alpha_{21}\).

Find the corresponding compartment diagram for each system of differential equations. \(\frac{d x_{1}}{d t}=-0.4 x_{1}+0.3 x_{2}\) \(\frac{d x_{2}}{d t}=0.1 x_{1}-0.5 x_{2}\)

We assume that the diagonal elements \(a_{i i}\) of the community matrix of an ecosystem containing two species in equilibrium are negative. Explain why this assumption implies that species \(i\) exhibits self-regulation.

Find all equilibria of each system of differential equations and determine the stability of each equilibrium. \(\begin{aligned} & \frac{d x_{1}}{d t}=2 x_{1}\left(5-x_{1}-x_{2}\right) \\ & \frac{d x_{2}}{d t}=3 x_{2}\left(7-3 x_{1}-x_{2}\right) \end{aligned}\)

Find all equilibria of each system of differential equations and determine the stability of each equilibrium. \(\frac{d x_{1}}{d t}=x_{1} x_{2}-2 x_{2}\) \(\frac{d x_{2}}{d t}=x_{1}+x_{2}\)

Use a graphing calculator to sketch solution curves of the given Lotka- Volterra predator-prey model in the N-P plane. That is, you should plot the level curves of the associated function \(f(N, P) .\) \(\frac{d N}{d t}=2 N-P N\) \(\frac{d P}{d t}=2 P N-\frac{1}{2} P\) passing through the points: (a) \((N(0), P(0))=(2,2)\) (b) \((N(0), P(0))=(3,3)\) (c) \((N(0), P(0))=(4,4)\)

Find the corresponding compartment diagram for each system of differential equations. \(\frac{d x_{1}}{d t}=-0.2 x_{1}+0.1 x_{2}\) \(\frac{d x_{2}}{d t}=-0.1 x_{2}\)

The classical Lotka-Volterra model of predation is given by $$ \begin{array}{l} \frac{d N}{d t}=r N-a N P \\ \frac{d P}{d t}=b N P-d P \end{array} $$ where \(N=N(t)\) is the prey density at time \(t\) and \(P=P(t)\) is the predator density at time \(t .\) The constants \(a, b, d\), and \(r\) are all positive. (a) Find the nontrivial equilibrium \((\hat{N}, \hat{P})\) with \(\hat{N}>0\) and \(\hat{P}>0\). (b) Find the community matrix corresponding to the nontrivial equilibrium. (c) Explain each entry of the community matrix found in (b) in terms of how individuals in this community affect each other.

Find all equilibria of each system of differential equations and determine the stability of each equilibrium. \(\frac{d x_{1}}{d t}=x_{1}-x_{2}\) \(\frac{d x_{2}}{d t}=x_{1} x_{2}-x_{2}\)

The vector field of $$ \begin{array}{l} \frac{d x_{1}}{d t}=-x_{1}-x_{2} \\ \frac{d x_{2}}{d t}=-2 x_{2} \end{array} $$ is given in Figure \(11.31 .\) Sketch the solution curve that goes through the point \((3,4)\).

Use a graphing calculator to sketch solution curves of the given Lotka- Volterra predator-prey model in the N-P plane. That is, you should plot the level curves of the associated function \(f(N, P) .\) \(\frac{d N}{d t}=3 N-2 P N\) \(\frac{d P}{d t}=P N-P\) passing through the points: (a) \((N(0), P(0))=(1,3 / 2)\) (b) \((N(0), P(0))=(2,2)\) (c) \((N(0), P(0))=(3,1)\)

Find the corresponding compartment diagram for each system of differential equations. \(\frac{d x_{1}}{d t}=-0.2 x_{1}+1.1 x_{2}\) \(\frac{d x_{2}}{d t}=0.2 x_{1}-1.1 x_{2}\)

The modified Lotka-Volterra model of predation is given by $$ \begin{array}{l} \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right)-a N P \\ \frac{d P}{d t}=b N P-d P \end{array} $$ where \(N=N(t)\) is the prey density at time \(t\) and \(P=P(t)\) is the predator density at time \(t\). The constants \(a, b, d, r\) and \(K\) are positive. Assume that \(d / b0\) and \(\hat{P}>0\). (b) Find the community matrix corresponding to the nontrivial equilibrium. (c) Explain each entry of the community matrix found in (b) in terms of how individuals in this community affect each other.

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

In Problems 13-18, find the general solution of each system of differential equations and sketch the lines in the direction of the eigenvectors. Indicate on each line of eigenvectors the direction in which the solution would move if it starts on that line. (Figure 11.32) $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} 1 & 3 \\ 5 & 3 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$

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.

Use a graphing calculator to study the following example of the Fitzhugh- Nagumo model: $$ \begin{array}{l} \frac{d V}{d t}=-V(V-0.3)(V-1)-w \\ \frac{d w}{d t}=V-c w \end{array} $$ Explain whether or not the model predicts multiple equilibria for the following values of \(c\) : (a) \(c=2\), (b) \(c=16\).

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.

Assume that $$ \begin{array}{l} \frac{d N}{d t}=5 N-P N \\ \frac{d P}{d t}=P N-P \end{array} $$ (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.

Use a graphing calculator to study the following example of the Fitzhugh- Nagumo model: $$ \begin{array}{l} \frac{d V}{d t}=-V(V-0.6)(V-1)-w \\ \frac{d w}{d t}=V-c w \end{array} $$ Explain whether or not the model predicts multiple equilibria for the following values of \(c\) : (a) \(c=8\), (b) \(c=20\), (c) \(c=50\).

Biological Control Agent 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 (11.75). 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.

Find the corresponding compartment diagram for each system of differential equations. \(\frac{d x_{1}}{d t}=-1.2 x_{1}\) \(\frac{d x_{2}}{d t}=0.3 x_{1}-0.2 x_{2}\)

Assume the following example of the Fitzhugh-Nagumo model: $$ \begin{array}{l} \frac{d V}{d t}=-V(V-1 / 2)(V-1)-w \\ \frac{d w}{d t}=V-c w \end{array} $$ Show that the model predicts multiple equilibria provided \(c \geq 16\). What happens if \(c<16 ?\)

An insulin pump is used to treat type I diabetes by continuously infusing insulin into the fat in a patient's abdomen or thigh. We will model this flow by a two-compartment model. We identify the fat into which insulin is pumped as the first compartment, and the patient's blood as the second compartment. Assume that the pump infuses \(0.5 \mathrm{IU}\) of insulin into the fat each hour. In one hour \(10 \%\) of this insulin is eliminated from the fat (i.e., passes from the body without entering the patient's blood), and \(70 \%\) is absorbed into blood. In the patient's blood \(80 \%\) of insulin is metabolized (i.e., used up) by the patient's tissues each hour. (a) Draw a compartment diagram (like Figure \(11.40\) ) showing the flow of insulin between fat and blood. (b) Write down a system of differential equations for \(x_{1}(t)\) and \(x_{2}(t)\). (c) Find all equilibria for the system of differential equations that you wrote down in (b). Are these equilibria stable or unstable?

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?

Find the corresponding compartment diagram for each system of differential equations. \(\frac{d x_{1}}{d t}=-0.2 x_{1}+0.4 x_{2}\) \(\frac{d x_{2}}{d t}=0.2 x_{1}-0.4 x_{2}\)

Assume the following example of the Fitzhugh-Nagumo model: $$ \begin{array}{l} \frac{d V}{d t}=-V(V-3 / 5)(V-1)-w \\ \frac{d w}{d t}=V-c w \end{array} $$ Find the smallest value of \(c\) for which the model predicts the existence of multiple equilibria.

Find the corresponding compartment diagram for each system of differential equations. \(\frac{d x_{1}}{d t}=-0.2 x_{1}\) \(\frac{d x_{2}}{d t}=-0.3 x_{2}\)

Find the general solution of each system of differential equations and sketch the lines in the direction of the eigenvectors. Indicate on each line of eigenvectors the direction in which the solution would move if it starts on that line. (Figure 11.37) $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{cc} 7 & 4 \\ 2 & 5 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$

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 \((11.78)\) 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.