Chapter 10

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{rr}{-2} & {1} \\ {-1} & {-1}\end{array}\right]\)

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{-\frac{3}{2}} & {\frac{1}{2}} \\ {\frac{1}{2}} & {-\frac{3}{2}}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{1} \\ {2}\end{array}\right]\)

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{rr}{1} & {2} \\ {-2} & {1}\end{array}\right]\)

A Jacobian matrix and two equlibria are given. Determine if each is locally stable, unstable, or if the analysis is inconclusive. $$J=\left[ \begin{array}{cc}{-\frac{1}{1+x_{2}}} & {\frac{x_{1}}{\left(1+x_{2}\right)^{2}}} \\\ {-1+\frac{x_{2}}{\left(1+x_{1}\right)^{2}}} & {-\frac{1}{1+x_{1}}}\end{array}\right]$$ $$\begin{array}{l}{\text { (i) } \hat{x}_{1}=-2, \hat{x}_{2}=-2} \\ {\text { (ii) } \hat{x}_{1}=\frac{1}{2}, \hat{x}_{2}=-\frac{3}{4}}\end{array}$$

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{\frac{1}{2}} & {-\frac{3}{2}} \\ {-\frac{3}{2}} & {\frac{1}{2}}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{1} \\ {2}\end{array}\right]\)

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{rr}{1} & {2} \\ {2} & {-1}\end{array}\right]\)

Find all equilibria and determine their local stability properties. $$x^{\prime}=x(3-x-y), \quad y^{\prime}=y(2-x-y)$$

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{1} & {0} \\ {4} & {-1}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{3} \\ {2}\end{array}\right]\)

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{ll}{-1} & {2} \\ {-3} & {0}\end{array}\right]\)

Find all equilibria and determine their local stability properties. $$p^{\prime}=p(1-p-q), \quad q^{\prime}=q(2-3 p-q)$$

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{-1} & {-2} \\ {2} & {-2}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{1} \\ {5}\end{array}\right]\)

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{ll}{1} & {1} \\ {0} & {1}\end{array}\right]\)

Find all equilibria and determine their local stability properties. $$n^{\prime}=n(1-2 m), \quad m^{\prime}=m(2-2 n-m)$$

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{-3} & {4} \\ {-6} & {7}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{r}{-1} \\ {3}\end{array}\right]\)

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{rr}{2} & {-1} \\ {-1} & {2}\end{array}\right]\)

Find all equilibria and determine their local stability properties. $$x^{\prime}=x(2-x), \quad y^{\prime}=y(3-y)$$

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{0} & {1} \\ {-6} & {-5}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{-1} \\ {-2}\end{array}\right]\)

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{ll}{0} & {1} \\ {1} & {0}\end{array}\right]\)

Find all equilibria and determine their local stability properties. $$p^{\prime}=-p^{2}+q-1, \quad q^{\prime}=q(2-p-q)$$

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{-1} & {2} \\ {-3} & {-1}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{2} \\ {0}\end{array}\right]\)

Consider the system of linear differential equations \(d \mathbf{x} / d t=A \mathbf{x}+\mathbf{g},\) where \(\mathbf{g}\) is a vector of constants. Suppose that \(A\) is nonsingular. (a) What is the equilibrium of this system of equations? (b) Using \(\hat{\mathbf{x}}\) denote the equilibrium found in part \((a)\) define a new vector of variables \(\mathbf{y}=\mathbf{x}-\hat{\mathbf{x}} .\) What do the components of y represent? (c) Show that \(y\) satisfies the differential equation \(d \mathbf{y} / d t=A \mathbf{y} .\) This demonstrates how we can reduce a nonhomogeneous system of linear differential equations to a system that is homogenous by using a change of variables.

$$\begin{array}{l}{24-25 \text { Find all equilibria and determine their stability proper- }} \\ {\text { ties. Your answer might be a function of the constant } a \text { . }}\end{array}$$ $$x^{\prime}=-x y+y+a x, \quad y^{\prime}=2 y-x y$$

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{ll}{3} & {0} \\ {0} & {1}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{r}{-2} \\ {4}\end{array}\right]\)

Consider the system of linear differential equations $$\frac{d \mathbf{x}}{d t}=\left[ \begin{array}{rr}{-2} & {-1} \\ {2} & {1}\end{array}\right] \mathbf{x}$$ The system is nongeneric, that is, the determinant of the matrix of coefficients is zero. (a) There are an infinite number of equilibria, all lying on a line in the phase plane. What is the equation of this line? (b) Construct the phase plane for this system.

$$x^{\prime}=a x^{2}+a y-x, \quad y^{\prime}=x-y, \quad a \neq 0$$

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{0} & {-1} \\ {-1} & {0}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{2} \\ {1}\end{array}\right]\)

Consider an autonomous homogeneous system of lineardifferential equations with coefficient matrix $$A=\left[ \begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]$$ Suppose that det \(A=0 .\) Show that there are an infinite number of equilibria.

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{4} & {-2} \\ {3} & {1}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{0} \\ {1}\end{array}\right]\)

Consider the following homogeneous system of three linear differential equations: \(\begin{aligned} d x / d t &=3 x+2 y-z \\ d y / d t &=x-y-z \\ d z / d t &=y+3 z \end{aligned}\) Suppose that \(x+y=5\) at all times. Show that this system can be reduced to two nonhomogenous linear differential equations given by \(d x / d t=x-z+10\) \(d z / d t=-x+3 z+5\)

Competition-colonization models In Exercise 7.Review.23 a metapopulation model for two species was introduced. The equations were $$\begin{aligned} \frac{d p_{1}}{d t} &=c_{1} p_{1}\left(1-p_{1}\right)-m_{1} p_{1} \\ \frac{d p_{2}}{d t} &=c_{2} p_{2}\left(1-p_{1}-p_{2}\right)-m_{2} p_{2}-c_{1} p_{1} p_{2} \end{aligned}$$ $$\begin{array}{l}{\text { where } p_{i} \text { is the fraction of patches occupied by species } i} \\ {\text { and } c_{i} \text { and } m_{i} \text { are the species-specific rates of colonization }} \\ {\text { and extinction of patches, respectively. These equations }} \\ {\text { assume that any patch has at most one species, and spe- }} \\ {\text { cies } 2 \text { patches can be taken over by species } 1, \text { but not vice }} \\ {\text { versa. }}\end{array}$$ $$\begin{array}{l}{\text { (a) Suppose that } m_{1}=m_{2}=3, c_{1}=5, \text { and } c_{2}=30 . \text { Find }} \\ {\text { all equilibria. }} \\ {\text { (b) Calculate the Jacobian matrix. }}\end{array}$$ $$\begin{array}{l}{\text { (c) Determine the local stability properties of each equilib- }} \\ {\text { rium found in part (a) using the Jacobian from part (b). }} \\ {\text { (d) Are the species predicted to be able to coexist at a stable }} \\ {\text { equilibrium? }}\end{array}$$

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{2} & {-5} \\ {2} & {1}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{1} \\ {1}\end{array}\right]\)

Consider the following homogeneous system of four linear differential equations: \(\begin{aligned} d w / d t &=2 x+y-z \\ d x / d t &=3 x+z \\ d y / d t &=-y+2 z \\ d z / d t &=3 x-5 y \end{aligned}\) Suppose that \(x+z=2\) and \(y+w=3\) at all times. Show that this system can be reduced to two nonhomogeneous linear differential equations given by $$d w / d t=3 x-w+1$$ $$d x / d t=2 x+2$$

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{ll}{3} & {-4} \\ {1} & {-3}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{2} \\ {3}\end{array}\right]\)

$$\begin{array}{c}{29-31 \text { Consumer resource models often have the following }} \\ {\text { general form }} \\ {R^{\prime}=f(R)-g(R, C) \quad C^{\prime}=\varepsilon g(R, C)-h(C)} \\ {\text { where } R \text { is the number of individuals of the resource and } C \text { is }} \\ {\text { the number of consumers. The function } f(R) \text { gives the rate of }} \\\ {\text { replenishment of the resource, } g(R, C) \text { describes the rate of }}\end{array}$$ $$ \begin{array}{l}{\text { replenishment of the resource, } g(R, C) \text { describes the rate of }} \\ {\text { consumption of the resource, and } h(C) \text { is the rate of loss of the }} \\ {\text { consumer. The constant } \varepsilon, \text { where } 0<\varepsilon<1, \text { is the conversion }} \\\ {\text { efficiency of resources into consumers. Find all equilibria of the }} \\\ {\text { following examples and determine their stability properties. }}\end{array}$$ $$\begin{array}{l}{\text { A chemostat is an experimental consumer-resource }} \\\ {\text { system. If the resource is not self-reproducing, then it can }} \\\ {\text { be modeled by choosing } f(R)=\theta, g(R, C)=b R C, \text { and }} \\ {h(C)=\mu C . \text { Suppose } \theta=2, b=1, \varepsilon=1, \text { and } \mu=1}\end{array}$$

In Exercise 10.1 .24 we considered the nongeneric system of differential equations $$\frac{d \mathbf{x}}{d t}=\left[ \begin{array}{rr}{-2} & {-1} \\ {2} & {1}\end{array}\right] \mathbf{x}$$ Theorem 2 applies to this system, and we can obtain the general solution \((8)\) in the usual way. Do so.

Second-order linear differential equations take the form $$y^{\prime \prime}(t)+p(t) y^{\prime}(t)+q(t) y(t)=g(t)$$ where \(p, q,\) and \(g\) are continuous functions of \(t .\) Suppose we have initial conditions \(y(0)=a\) and \(y^{\prime}(0)=b .\) Show that this equation can be rewritten as a system of two first-order linear differential equations having the form \(\frac{d \mathbf{x}}{d t}=\left[ \begin{array}{cc}{0} & {1} \\ {-q(t)} & {-p(t)}\end{array}\right] \mathbf{x}+\left[ \begin{array}{c}{0} \\\ {g(t)}\end{array}\right]\) \(\begin{aligned} \text { with } & \mathbf{x}(0)=\left[ \begin{array}{l}{a} \\\ {b}\end{array}\right] \\ \text { where } x_{1}(t)=y(t) \text { and } x_{2}(t) &=y^{\prime}(t) \end{aligned}\)

Metapopulation dynamics Example 2 presents a model for a population of deer mice that is split into two patches through habitat fragmentation. The model is $$\frac{d x_{\mathrm{A}}}{d t}=-x_{\mathrm{A}}+2 x_{\mathrm{B}} \quad \frac{d x_{\mathrm{B}}}{d t}=3 x_{\mathrm{A}}-3 x_{\mathrm{B}}$$ where \(x_{\mathrm{A}}\) and \(x_{\mathrm{B}}\) are the population sizes in patches \(\mathrm{A}\) and \(\mathrm{B},\) respectively. (a) Construct the phase plane, including the nullclines. (b) Describe what happens to the population in each patch as \(t \rightarrow \infty\) if both start with nonzero sizes.

$$\begin{array}{l}{\text { A model for self-reproducing resources with limited }} \\ {\text { growth is obtained by choosing } f(R)=r R(1-R / K)} \\ {g(R, C)=b R C, \text { and } h(C)=\mu C . \text { Suppose } r=2, K=5} \\ {b=1, \varepsilon=1, \text { and } \mu=1}\end{array}$$

Gene regulation Genes produce molecules called mRNA that then produce proteins. High levels of protein can inhibit the production of mRNA, resulting in a feedback that regulates gene expression. Using \(m\) and \(p\) to denote the amounts of mRNA and protein in a cell times (X 10^{2} copies cell), \right. a simple model of gene regulation is \(\begin{aligned} d m / d t &=1-p-m \\ d p / d t &=m-p \end{aligned}\) Construct the phase plane, including the nullclines. [Hint: This system is nonhomogeneous.]

Our focus has been on systems whose coefficient matrices have distinct eigenvalues. A simple example of a system with repeated eigenvalues is $$\frac{d \mathbf{x}}{d t}=\left[ \begin{array}{rr}{-1} & {0} \\ {0} & {-1}\end{array}\right] \mathbf{x}$$ (a) Show that \(x_{1}(t)=c_{1} e^{-t}\) and \(x_{2}(t)=c_{2} e^{-t}\) is a solution. The origin in this case is called a proper node. (b) Try obtaining this general solution by coefficient matrix. eigenvectors and eigenvalues of the coefficient matrix. Comment on anything unusual that occurs.

$$\begin{array}{c}{\text { The Kermack-McKendrick equations describe the out- }} \\ {\text { break of an infectious disease. Using } S \text { and } I \text { to denote the }} \\ {\text { number of susceptible and infected people in a population, }} \\ {\text { respectively, the equations are }} \\\ {S^{\prime}=-\beta S I \quad I^{\prime}=\beta S I-\mu I}\end{array}$$ $$\begin{array}{l}{\text { where } \beta \text { and } \mu \text { are positive constants representing the }} \\ {\text { transmission rate and rate of recovery. }} \\ {\text { (a) Verify that } \hat{I}=0, \text { along with any value of } S, \text { is an }} \\ {\text { equilibrium. }}\end{array}$$ $$\begin{array}{l}{\text { (b) Calculate the Jacobian matrix. }} \\ {\text { (c) Using your answer to part (b), determine how large } S} \\ {\text { must be to guarantee that the disease will spread when }} \\ {\text { rare. }}\end{array}$$

Prostate cancer During treatment, tumor cells in the prostate can become resistant through a variety of biochemical mechanisms. Some of these are reversible-the cells revert to being sensitive once treatment stops-and some are not. Using \(x_{1}, x_{2},\) and \(x_{3}\) to denote the fraction of cells that are sensitive, temporarily resistant, and permanently resistant, respectively, a simple model for their dynamics during treatment is \(\begin{aligned} d x_{1} / d t &=-a x_{1}-c x_{1}+b x_{2} \\ d x_{2} / d t &=a x_{1}-b x_{2}-d x_{2} \\ d x_{3} / d t &=c x_{1}+d x_{2} \end{aligned}\) Use the fact that \(x_{1}+x_{2}+x_{3}=1\) to reduce this to a non-homogeneous system of two linear differential equations for \(x_{1}\) and \(x_{3} .\)

A slightly more complicated system with repeated eigenvalues is $$\frac{d \mathbf{x}}{d t}=\left[ \begin{array}{rr}{-1} & {1} \\ {0} & {-1}\end{array}\right] \mathbf{x}$$ (a) Show that \(x_{1}(t)=c_{1} e^{-t}+c_{2} t e^{-t}\) and \(x_{2}(t)=c_{2} e^{-t}\) is a solution. The origin in this case is called an improper node. (b) Try obtaining this general solution by calculating the eigenvectors and eigenvalues of the coefficient matrix. Comment on anything unusual that occurs.

$$\begin{array}{l}{\text { The Michaelis-Menten equations describe a biochemical }} \\ {\text { reaction in which an enzyme E and substrate S bind to }} \\ {\text { form a complex } \mathrm{C} \text { . This complex can then either dissociate }} \\ {\text { back into its original components or undergo a reaction in }}\end{array}$$ $$\begin{array}{l}{\text { which a product } P \text { is produced along with the free enzyme: }} \\ {\mathrm{E}+\mathrm{S} \leftrightharpoons \mathrm{C} \rightarrow \mathrm{E}+\mathrm{P} \text { . This can be expressed by the }} \\ {\text { differential equations }}\end{array}$$ $$\begin{array}{l}{\frac{d x}{d t}=-k_{f} x y M+k_{r}(1-y) M} \\ {\frac{d y}{d t}=-k_{f} x y M+k_{r}(1-y) M+k_{c d}(1-y) M}\end{array}$$ $$\frac{d z}{d t}=k_{\operatorname{cat}}(1-y) M$$ $$\begin{array}{l}{\text { where } M \text { is the total number of enzymes (both free and }} \\ {\text { bound), } x \text { and } z \text { are the numbers of substrate and product }} \\ {\text { molecules, } y \text { is the fraction of the enzyme pool that is free, }} \\ {\text { and the } k_{i} \text { 's are positive constants. }}\end{array}$$ $$\begin{array}{l}{\text { (a) Although this is a system of three differential equa- }} \\ {\text { tions, } x \text { and } y \text { can be analyzed separately. Explain }} \\ {\text { why. }} \\ {\text { (b) Find the only equilibrium. }} \\ {\text { (c) Falculate the Jacobian matrix. }} \\ {\text { (d) Determine the local stability properties of the }} \\ {\text { equilibrium. }}\end{array}$$

Stability of Caribbean reefs Coral and macroalgae compete for space when colonizing Caribbean reefs. A modification of the model in Exercise 27 has been used to describe this process. The equations are $$\begin{aligned} \frac{d M}{d t} &=\gamma M(1-M)-\frac{g M}{1-C} \\ \frac{d C}{d t} &=r C(1-M-C)-\gamma C M-d C \end{aligned}$$ $$\begin{array}{l}{\text { where } M \text { is the fraction of the reef occupied by macro- }} \\ {\text { algae, } C \text { is the fraction occupied by coral, } r \text { is the coloniza- }} \\ {\text { tion rate of empty space by coral, } d \text { is the death rate of }} \\ {\text { coral, } \gamma \text { is the rate of colonization by macroalgae (in both }} \\\ {\text { empty space and space occupied by coral), and } g \text { is a }}\end{array}$$ constant governing the death rate of macroalgae. Notice that the per capita death rate of macroalgae decreases as coral cover increases. $$\begin{array}{l}{\text { (a) Suppose that } r=3, d=1, \gamma=2, \text { and } g=1 . \text { Find }} \\ {\text { all equilibria. There are five, but only four of them are }} \\ {\text { biologically relevant. }} \\ {\text { (b) Calculate the Jacobian matrix. }}\end{array}$$ $$\begin{array}{l}{\text { (c) Determine the local stability properties of the four }} \\ {\text { relevant equilibria found in part (a). }} \\ {\text { (d) In part (c) you should find two equilibria that are }} \\ {\text { locally stable. What do they represent in terms of the }} \\ {\text { structure of the reef? }}\end{array} $$

Jellyfish locomotion Jellyfish move by contracting an elastic part of their body, called a bell, that creates a high-pressure jet of water. When the contractive force stops, the bell then springs back to its natural shape. Jellyfish locomotion has been modeled using a second-order linear differential equation having the form \(m x^{\prime \prime}(t)+b x^{\prime}(t)+k x(t)=0\) where \(x(t)\) is the displacement of the bell at time \(t, m\) is the mass of the bell (in grams), \(b\) is a measure of the friction between the bell and the water (in units of \(N / m \cdot s ),\) and \(k\) is a measure of the stiffness of the bell (in units of \(N / m )\) Suppose that \(m=100 \mathrm{g}, b=0.1 \mathrm{N} / \mathrm{m} \cdot \mathrm{s},\) and \(k=1 \mathrm{N} / \mathrm{m}\) (a) Define the new variables \(z_{1}(t)=x(t)\) and \(z_{2}(t)=x^{\prime}(t),\) and show that the model can be expressed as a system of two first-order linear differential equations. (b) Construct the phase plane, including the nullclines, for the equations from part (a).

Fitzhugh-Nagumo equations Consider the following alternative form of the Fitzhugh-Nagumo equations: $$\frac{d v}{d t}=(v-a)(1-v) v-w \quad \frac{d w}{d t}=\varepsilon(v-w)$$ where \(\varepsilon>0\) and \(0<\)a\(<1\). $$\begin{array}{l}{\text { (a) Verify that the origin is an equilibrium. }} \\\ {\text { (b) Calculate the Jacobian matrix. }} \\ {\text { (c) Determine the local stability properties of the origin as }} \\ {\text { a function of the constants. }}\end{array}$$