{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chap9_Sec4

# Chap9_Sec4 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

This preview shows pages 1–18. Sign up to view the full content.

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS 9

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
9.4 Models for Population Growth In this section, we will: Investigate differential equations used to model population growth. DIFFERENTIAL EQUATIONS
One of the models for population growth we considered in Section 9.1 was based on the assumption that the population grows at a rate proportional to the size of the population: NATURAL GROWTH dP kP dt =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Is that a reasonable assumption? NATURAL GROWTH
Suppose we have a population (of bacteria, for instance) with size P = 1000. At a certain time, it is growing at a rate of P’ = 300 bacteria per hour. NATURAL GROWTH

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Now, let’s take another 1,000 bacteria of the same type and put them with the first population. Each half of the new population was growing at a rate of 300 bacteria per hour. NATURAL GROWTH
We would expect the total population of 2,000 to increase at a rate of 600 bacteria per hour initially—provided there’s enough room and nutrition. So, if we double the size, we double the growth rate. NATURAL GROWTH

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
In general, it seems reasonable that the growth rate should be proportional to the size. NATURAL GROWTH
In general, if P ( t ) is the value of a quantity y at time t and, if the rate of change of P with respect to t is proportional to its size P ( t ) at any time, then where k is a constant. This is sometimes called the law of natural growth. LAW OF NATURAL GROWTH dP kP dt = Equation 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
If k is positive, the population increases. If k is negative, it decreases. LAW OF NATURAL GROWTH
Equation 1 is a separable differential equation. Hence, we can solve it by the methods of Section 9.3: where A (= ± e C or 0) is an arbitrary constant. LAW OF NATURAL GROWTH ln kt C C kt kt dP k dt P P kt C P e e e P Ae + = = + = = =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
To see the significance of the constant A , we observe that: P (0) = Ae 0 = A Thus, A is the initial value of the function. LAW OF NATURAL GROWTH
The solution of the initial-value problem is: LAW OF NATURAL GROWTH 0 (0) dP kP P P dt = = Equation 2 0 ( ) kt P t P e =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Another way of writing Equation 1 is: This says that the relative growth rate (the growth rate divided by the population size) is constant. Then, Equation 2 says that a population with constant relative growth rate must grow exponentially. LAW OF NATURAL GROWTH 1 dP k P dt =
We can account for emigration (or “harvesting”) from a population by modifying Equation 1—as follows. LAW OF NATURAL GROWTH

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
If the rate of emigration is a constant m , then the rate of change of the population is modeled by the differential equation See Exercise 13 for the solution and consequences of Equation 3. LAW OF NATURAL GROWTH dP kP m dt = - Equation 3
As we discussed in Section 9.1, a population often increases exponentially in its early stages, but levels off eventually and approaches its carrying capacity because of limited resources.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 71

Chap9_Sec4 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

This preview shows document pages 1 - 18. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online