§7.2 Qualitative behavior of solutions to differential equations (continued)

Summary

Consider the differential equation

.

• Sketch a slope field on the axes at right. - Sketch the solutions whose initial values are $y(0)= -4, -3, \ldots, 4$. - What do your sketches suggest is the solution whose initial value is $y(0) = -1$? Verify that this is indeed the solution to this initial value problem. - By considering the differential equation and the graphs you have sketched, what is the relationship between $t$ and $y$ at a point where a solution has a local minimum?

Consider the situation from Exercise 7.1.3 of Section 7.1: Suppose that the population of a particular species is described by the function $P(t)$, where $P$ is expressed in millions. Suppose further that the population's rate of change is governed by the differential equation

where $f(P)$ is the function graphed below.

• Sketch a slope field for this differential equation. You do not have enough information to determine the actual slopes, but you should have enough information to determine where slopes are positive, negative, zero, large, or small, and hence determine the qualitative behavior of solutions. - Sketch some solutions to this differential equation when the initial population $P(0) \gt 0$. - Identify any equilibrium solutions to the differential equation and classify them as stable or unstable. - If $P(0) \gt 1$, what is the eventual fate of the species? if $P(0) \lt 1$? - Remember that we referred to this model for population growth as “growth with a threshold.” Explain why this characterization makes sense by considering solutions whose inital value is close to 1.

The population of a species of fish in a lake is $P(t)$ where $P$ is measured in thousands of fish and $t$ is measured in months. The growth of the population is described by the differential equation

. - Sketch a graph of $f(P) = P(6-P)$ and use it to determine the equilibrium solutions and whether they are stable or unstable. Write a complete sentence that describes the long-term behavior of the fish population. - Suppose now that the owners of the lake allow fishers to remove 1000 fish from the lake every month (remember that $P(t)$ is measured in thousands of fish). Modify the differential equation to take this into account. Sketch the new graph of $dP/dt$ versus $P$. Determine the new equilibrium solutions and decide whether they are stable or unstable. - Given the situation in part (b), give a description of the long-term behavior of the fish population. - Suppose that fishermen remove $h$ thousand fish per month. How is the differential equation modified? - What is the largest number of fish that can be removed per month without eliminating the fish population? If fish are removed at this maximum rate, what is the eventual population of fish?

Let $y(t)$ be the number of thousands of mice that live on a farm; assume time $t$ is measured in years.（This problem is based on an ecological analysis presented in a research paper by C.S. Hollings: The Components of Predation as Revealed by a Study of Small Mammal Predation of the European Pine Sawfly, Canadian Entomology 91: 283-320.） - The population of the mice grows at a yearly rate that is twenty times the number of mice. Express this as a differential equation. - At some point, the farmer brings $C$ cats to the farm. The number of mice that the cats can eat in a year is

thousand mice per year. Explain how this modifies the differential equation that you found in part a). - Sketch a graph of the function $M(y)$ for a single cat $C=1$ and explain its features by looking, for instance, at the behavior of $M(y)$ when $y$ is small and when $y$ is large. - Suppose that $C=1$. Find the equilibrium solutions and determine whether they are stable or unstable. Use this to explain the long-term behavior of the mice population depending on the initial population of the mice. - Suppose that $C=60$. Find the equilibrium solutions and determine whether they are stable or unstable. Use this to explain the long-term behavior of the mice population depending on the initial population of the mice. - What is the smallest number of cats you would need to keep the mice population from growing arbitrarily large?