Chapter 8

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. $$ \frac{d N}{d t}=N^{2}-N+1 \quad N>0 $$

We showed in Section \(8.3 .4\) that if \(bc / 2\), then under the greenbeard gene model \(x=1\) is a stable equilibrium and \(x=0\) is unstable. (b) What are the equilibria and their stability if \(b

$$ \text { In Problems } 35-38, \text { solve each differential equation. } $$ $$ \frac{d y}{d x}=y(1+y) $$

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. $$ \frac{d N}{d t}=1-N-N^{3} $$

$$ \text { In Problems } , \text { solve each differential equation. } $$$$ \frac{d y}{d x}=(1+y)^{2} $$

$$ \text { In Problems } , \text { solve each differential equation. } $$ $$ \frac{d y}{d x}=(1+y)^{3} $$

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. $$ \frac{d y}{d x}=y^{4}+y^{3}-1 $$

We will analyze the population dynamics that are predicted by (8.69), for different values of b and \(c\). Suppose \(b=c\), then \(\frac{d x}{d t}=\frac{k n b}{2} x(1-x)^{2}\). Assuming \(k=1\), \(n=1\), and \(b=1\), sketch the vector field plot for \(x\).

$$ \text { In Problems } , \text { solve each differential equation. } $$ $$ \frac{d y}{d x}=\left(1-y^{2}\right) $$

In Problems 39-48 you should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use \(a\) graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d y}{d t}=e^{y}-(1-y) $$

You should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d y}{d t}=y-h $$

You should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d y}{d x}=y^{2}-h $$

To model the spread of a disease in a population of size \(N\) we derived a differential equation model: $$ \frac{d I}{d t}=(k b-c) I-\frac{k b}{N} I^{2} $$ where \(I(t)\) is the number of infected individuals at time \(t\), and \(k, b\), and \(c\) are all positive coefficients. Assuming that I ( \(t\) ) is modeled by Equation \((8.70)\), you should locate the equilibria of the model, and find which of these equilibria are stable. Draw a vector field plot for each problem. \(k=1, b=1, c=0.5, N=50 .\)

Suppose that the size of a population at time \(t\) is denoted by \(N(t)\) and that \(N(t)\) satisfies the logistic 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 \(\lim _{t \rightarrow \infty} N(t)\).

You should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d y}{d x}=(y-1)(y+3)-h $$

Assume that a population, whose size is denoted by \(N(t)\), grows according to the logistic equation. Find the limiting growth rate for small \(N\) (i.e., find the constant \(r\) ) if the carrying capacity is \(100, N(0)=10\), and \(N(1)=30\)

You should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d y}{d x}=(y-2)(y+4)+h $$

. Let \(N(t)\) denote the size of a population at time \(t .\) Assume that the population grows according to the logistic equation. Assume also that the limiting growth rate for small \(N\) is 5 and that the carrying capacity is \(50 .\) (a) Find a differential equation that describes the growth of this population. (b) Without solving the differential equation in (a), sketch solution curves of \(N(t)\) as a function of \(t\) when (i) \(N(0)=10\), (ii) \(N(0)=40\), and (iii) \(N(0)=50\).

Logistic growth is described by the differential equation $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right) $$ We showed in Example 6 that the solution of this differential equation with initial condition \(N(0)=N_{0}\) is given by $$ N(t)=\frac{K}{1+\left(\frac{K}{N_{0}}-1\right) e^{-r t}} $$ (a) Show that $$ r=\frac{1}{t} \ln \left(\frac{K-N_{0}}{N_{0}}\right)+\frac{1}{t} \ln \left(\frac{N(t)}{K-N(t)}\right) $$ by solving \((8.29)\) for \(r\). (b) Equation \((8.30)\) can be used to estimate \(r\). Suppose we are studying a population that grows according to the logistic equation and find that \(N(0)=10, N(5)=22, N(100)=30\), and \(N(200)=30 .\) Estimate \(r .(\) Hint \(:\) First estimate \(K\) from the behavior of the solution for large \(t .\) )

You should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d x}{d t}=x^{2}-h x $$

In this question we will interpret the recovery rate, \(c\), that appears in the model. Assume that a population of infected individuals is quarantined (that is, they are unable to transmit the disease to others, or to catch it again once they recover). (a) Explain why under these assumptions we expect: $$ \frac{d I}{d t}=-c I $$ (b) Assuming \(I(0)=I_{0}\), find \(I(t)\) by solving \((8.71)\). (c) How long will it take for the number of infected individuals to decrease from \(I_{0}\) to \(I_{0} / 2 ?\) (d) Assume that it takes 7 days for the number of infected individuals to decrease from 50 to \(25 .\) Calculate the recovery rate \(c\) for this disease.

Population genetics is the study of how the frequency of particular traits changes within a population over time. We are studying a gene that comes in two alleles (i.e., variants) \(A\) and \(a\). The \(A\) allele makes individuals reproduce a little faster than the \(a\) allele. So we expect the \(A\) alleles to take over the population with time. Suppose that a proportion \(p\) of all individuals within the population carry the \(A\) allele (with the remaining proportion, \(1-p\), carrying the \(a\) allele). If the \(A\) allele boosts reproduction rate by an amount \(s\) it can be shown under some assumptions that the proportion of \(A\) -allele individuals obeys a differential equation $$ \frac{d p}{d t}=\frac{1}{2} s p(1-p) $$ (a) Use separation of variables and partial fractions to find the solution of \((8.31)\), assuming \(p(0)=p_{0}\). (b) Show that if \(p_{0} \neq 0\), then \(\lim _{t \rightarrow \infty} p(t)=1\). Explain why this behavior makes sense biologically. (c) Suppose \(p_{0}=0.1\) and \(s=0.01\); how long will take until \(p(t)=0.5 ?\)

You should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d x}{d t}=x^{3}-h x $$

In Problems 46-54, solve each differential equation with the given initial condition. $$ \frac{d y}{d x}=2 \frac{y}{x}, \text { with } y(1)=1 $$

You should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d x}{d t}=x\left(x^{2}-1\right)-h $$

Handwashing One way to control the spread of a disease is to run public health programs that educate people on how to limit their exposure to the disease. For example, frequent handwashing can prevent people from picking up a virus after touching surfaces that it may live on. (a) Explain why in our model such efforts to control the disease would affect the value of the parameter \(k\), but would not affect \(b\) or \(c\). (b) Suppose that for a particular disease \(c=0.3 /\) day, and \(b=\) \(10 /\) day. What value must \(k\) remain below to prevent the disease from hecomino endemic?

In Problems, solve each differential equation with the given initial condition. $$ \frac{d y}{d x}=\frac{x+1}{y}, \text { with } y(0)=2 $$

In Problems, solve each differential equation with the given initial condition. $$ \frac{d y}{d x}=\frac{x y}{x+1}, \text { with } y(0)=1 . $$

For Problems \(49-56\) determine whether the equilibrium at \(x=0\) is stable, unstable, or semi-stable. $$ \frac{d x}{d t}=x^{3} $$

In Problems, solve each differential equation with the given initial condition. $$ \frac{d y}{d x}=(y+1) e^{-x}, \text { with } y(0)=2 $$

Determine whether the equilibrium at \(x=0\) is stable, unstable, or semi- stable. $$ \frac{d x}{d t}=-x^{5} $$

In Problems, solve each differential equation with the given initial condition. $$ \frac{d y}{d x}=\frac{y^{2}}{x}, \text { with } y(1)=1 . $$

Determine whether the equilibrium at \(x=0\) is stable, unstable, or semi- stable. $$ \frac{d x}{d t}=x^{4} $$

In Problems, solve each differential equation with the given initial condition. $$ \frac{d y}{d x}=\frac{y+1}{x-1}, \text { with } y(2)=5 $$

Determine whether the equilibrium at \(x=0\) is stable, unstable, or semi- stable. $$ \frac{d x}{d t}=x^{3}-x^{5} $$

In Problems, solve each differential equation with the given initial condition.$$ \frac{d u}{d t}=\frac{\sin t}{u+1}, \text { with } u(0)=3 $$

Determine whether the equilibrium at \(x=0\) is stable, unstable, or semi- stable. $$ \frac{d x}{d t}=x^{3}+x^{4} $$

In Problems, solve each differential equation with the given initial condition.$$ \frac{d y}{d t}=\frac{t}{y}, \text { with } y(0)=1 . $$

Determine whether the equilibrium at \(x=0\) is stable, unstable, or semi- stable. $$ \frac{d x}{d t}=x^{2}-x^{3} $$

In Problems, solve each differential equation with the given initial condition. $$ \frac{d x}{d y}=\frac{1}{2} \frac{x}{y}, \text { with } x(3)=2 . $$

Determine whether the equilibrium at \(x=0\) is stable, unstable, or semi- stable. $$ \frac{d x}{d t}=\frac{x^{3}}{x-1} $$

In Problems 55-60 you will need to solve differential equations by separation of variables. In these problems it will not always be possible to solve explicitly for \(y\) in terms of \(t ;\) instead your solution may take the form of an implicit function relating the two variables. $$ \frac{d y}{d t}=\frac{y^{2}+y}{t-1} \text { where } y(0)=1 $$

Determine whether the equilibrium at \(x=0\) is stable, unstable, or semi- stable. $$ \frac{d x}{d t}=x^{3} e^{-x} $$

In Problems you will need to solve differential equations by separation of variables. In these problems it will not always be possible to solve explicitly for \(y\) in terms of \(t ;\) instead your solution may take the form of an implicit function relating the two variables. $$ \frac{d y}{d t}=\frac{y t}{\ln y} \text { where } y(1)=e $$

For Problems \(57-66\) draw the vector field plot of the differential equation. Then, using the given initial conditions, sketch the solutions (i.e., draw a graph showing the dependent variable as a function of the independent variable). \(\frac{d y}{d t}=3 y-2\) (a) \(y(0)=2\), (b) \(y(0)=0\).

In Problems you will need to solve differential equations by separation of variables. In these problems it will not always be possible to solve explicitly for \(y\) in terms of \(t ;\) instead your solution may take the form of an implicit function relating the two variables. $$ \frac{d y}{d t}=\frac{t+1}{y+y^{2}} \text { where } y(0)=1 $$

Draw the vector field plot of the differential equation. Then, using the given initial conditions, sketch the solutions (i.e., draw a graph showing the dependent variable as a function of the independent variable). \(\frac{d y}{d t}=1-y\) (a) \(y(0)=2\), (b) \(y(0)=-1\).

In Problems you will need to solve differential equations by separation of variables. In these problems it will not always be possible to solve explicitly for \(y\) in terms of \(t ;\) instead your solution may take the form of an implicit function relating the two variables. $$ \frac{d y}{d t}=\frac{t^{2}+1}{\cos y+\sin y} \text { where } y(0)=0 $$

Draw the vector field plot of the differential equation. Then, using the given initial conditions, sketch the solutions (i.e., draw a graph showing the dependent variable as a function of the independent variable). \(\frac{d y}{d t}=y(1-y)\) (a) \(y(0)=0\), (b) \(y(0)=1 / 2\), (c) \(y(0)=1 / 4\), (d) \(y(0)=2\).

In Problems you will need to solve differential equations by separation of variables. In these problems it will not always be possible to solve explicitly for \(y\) in terms of \(t ;\) instead your solution may take the form of an implicit function relating the two variables. $$ \frac{d y}{d t}=\sqrt{\frac{t+1}{y+1}} \text { where } y(0)=1 $$