Chapter 8

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). \(\frac{d y}{d t}=y^{2}-1\) (a) \(y(0)=-1\), (b) \(y(0)=-1 / 2\), (c) \(y(0)=1 / 2\), (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. $$ \frac{d y}{d t}=\frac{t+1}{t v+t v^{3}} \text { where } y(1)=1 \text { . } $$

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). \(\frac{d y}{d t}=(y+3)(1-y)\) (a) \(y(0)=-1\), (b) \(y(0)=-1 / 2\), (c) \(y(0)=-2\), (d) \(y(0)=2\).

The per capita growth rate of a population of cells varies over the course of a day. Assume that time \(t\) is measured in hours and $$ \frac{d N}{d t}=2\left(1-\cos \frac{2 \pi t}{24}\right) N $$ if \(N(0)=5\), find the number of cells after one day (that is, find \(N(24))\)

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). \(\frac{d N}{d t}=N(N-1)(5-N)\) (a) \(N(0)=1\), (b) \(N(0)=1 / 2\), (c) \(N(0)=3 / 2\), (d) \(N(0)=7\).

Bite strength varies as animals grow, which may mean that the animal's diet must change. Christiansen and Adolfsson (2005) studied the relationship between the strength of animal teeth with skull size in carnivores from the cat and dog families. They found that tooth strength \(S\), and skull length \(L\), were related in a power law: $$ S=C L^{2.85} $$ where \(C\) is some constant. Find the relationship between the relative rates of growth of \(S\) and \(L\) (i.e., between \(\frac{1}{S} \frac{d S}{d t}\) and \(\left.\frac{1}{L} \frac{d L}{d t}\right)\).

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). \(\frac{d N}{d t}=(N-1)(N+1)(N-4)\) (a) \(N(0)=0\), (b) \(N(0)=2\), (c) \(N(0)=6\), (d) \(N(0)=-2\).

Homeostasis Sterner and Elser (2002) studied the relationship between the amount of nitrogen in an animal's body and the amount of nitrogen present in the food that it eats. Many animals maintain homeostasis (balance), that is, they control their own nitrogen content. As the amount of nitrogen present in their food increases, the amount of nitrogen in the animal's body increases more slowly. If the amount of nitrogen in the animal is \(N\) and the amount of nitrogen in its food is \(F\), Sterner and Elser argue that: $$ \frac{1}{N} \frac{d N}{d t}=\frac{\sigma}{F} \frac{d F}{d t} $$ where \(\sigma\) is a constant. (a) Show that if \(\sigma=1\), then \(N \propto F ;\) that is, the nitrogen content of the animal increases in proportion to its food. This is called absence of homeostasis. (b) If \(\sigma=0\), then \(N\) is a constant, independent of \(F\). This is called homeostasis (the animal maintains a balanced amount of nitrogen, independent of its food). (c) Show that if \(0<\sigma<1\), then, if \(F\) doubles, \(N\) also increases but by a factor less than 2 .

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). \(\frac{d y}{d x}=(y-1)(y-2)(y-5)\) (a) \(y(0)=0\), (b) \(y(0)=4\) (c) \(y(0)=3 / 2\), (d) \(y(0)=6\).

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). \(\frac{d y}{d x}=-y-y^{3}\) (a) \(y(0)=0\), (b) \(y(0)=1\), (c) \(y(0)=2\), (d) \(y(0)=3\).

One of the key ideas for sketching solutions from vector field plots is that a solution curve must be monotonic, that is, \(x(t)\) is either increasing or decreasing or constant but cannot switch from one behavior to another. We showed that a solution \(x(t)\) could not start by increasing and then switch to decreasing. Suppose that \(x(t)\) is a solution of the differential equation \(\frac{d x}{d t}=g(x)\) and that \(x(t)\) starts off decreasing with time. Show that \(x(t)\) cannot switch to increasing.

For Problems \(77-88\) find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d y}{d t}=2-3 y $$

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d y}{d t}=y-2 $$

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d y}{d t}=y(2-y)(y-3) $$

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d y}{d t}=y(y-1)(y-2) $$

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d N}{d t}=N \ln \left(\frac{2}{N}\right) \quad N>0 $$

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d N}{d t}=\frac{N-1}{N+1} \quad N \geq 0 $$

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable.$$ \frac{d y}{d x}=\frac{y^{2}-y}{y^{2}+1} $$

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d y}{d x}=\frac{1}{y^{3}}-\frac{1}{y} \quad, \quad y>0 $$

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d x}{d t}=x e^{-x} $$

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d x}{d t}=e^{-x}-e^{-2 x} $$

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. \(\frac{d x}{d t}=h x-x^{2}\), where \(h\) is a constant and (a) \(h>0\), (b) \(h<0\)

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. \(d x / d t=h x-x^{3}\), where \(h\) is a constant and (a) \(h>0\), (b) \(h<0\)

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 $$ \frac{d N}{d t}=g(N) $$ where $$ g(N)=N\left(1-\frac{N}{50}\right)-\frac{9 N}{5+N} $$ The first term on the right-hand side describes the logistic growth; the second term describes the effect of predation. (a) Make the vector field plot for this differential equation. (b) Find all equilibria of \((8.38)\). (c) Use your vector field plot in (a) to determine the stability of the equilibria you found in (b). (d) Repeat your analysis from part (c) but now use the method of eigenvalues to determine the stability of the equilibria you found in (b).

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 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 and use the graphical approach to discuss their stability, when \(h=0.1\). (b) Show that if \(h

A population whose growth is affected by the Allee effect is modeled using the differential equation: $$ \frac{d N}{d t}=r N(N-a)\left(1-\frac{N}{K}\right) $$ where \(r, a, k\) are all positive constants and \(a