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The logistic equation is the autonomous differential equation: dN =r· N [ 1 dt (1-#) K Where r and K are both positive constants. The logistic equation is used to model the size of a population subject to constrained growth. (1) In this problem we will discuss the meaning of constants r and K. (a) When the population size N is much smaller than K, explain why the logistic dN = r· N. In that dt equation behaves like the exponential growth equation: discuss how r can be thought of as the "initial growth rate" for a population subject to constrained growth. (b) When N is very close to K, explain why the logistic equation behaves like the sense, dN = 0. In this sense, discuss how K can be thought of dt differential equation: as the "carrying capacity" or "limiting capacity" for a population subject to constrained growth. (c) Assume that N(0) = No, where No is much smaller than K. Imagine a function that satisfies the differential equation in (a) for small values of t, and that has as a limit the solution to the differential equation in (b). Sketch the graph of such a function, be sure to label No and K in your graph.

The logistic equation is the autonomous differential equation: dN =r· N [ 1 dt (1-#) K Where r and K are both positive constants. The logistic equation is used to model the size of a population subject to constrained growth. (1) In this problem we will discuss the meaning of constants r and K. (a) When the population size N is much smaller than K, explain why the logistic dN = r· N. In that dt equation behaves like the exponential growth equation: discuss how r can be thought of as the "initial growth rate" for a population subject to constrained growth. (b) When N is very close to K, explain why the logistic equation behaves like the sense, dN = 0. In this sense, discuss how K can be thought of dt differential equation: as the "carrying capacity" or "limiting capacity" for a population subject to constrained growth. (c) Assume that N(0) = No, where No is much smaller than K. Imagine a function that satisfies the differential equation in (a) for small values of t, and that has as a limit the solution to the differential equation in (b). Sketch the graph of such a function, be sure to label No and K in your graph.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 16EQ
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The logistic equation is the autonomous differential equation: N(1 - *). dN = r. dt K Where r and K are both positive constants. The logistic equation is used to model the size of a population subject to constrained growth. (1) In this problem we will discuss the meaning of constants r and K. (a) When the population size N is much smaller than K, explain why the logistic dN = r· N. In that dt equation behaves like the exponential growth equation: sense, discuss how r can be thought of as the "initial growth rate" for a population subject to constrained growth. (b) When N is very close to K, explain why the logistic equation behaves like the dN differential equation: dt 0. In this sense, discuss how K can be thought of as the "carrying capacity" or “limiting capacity" for a population subject to constrained growth. (c) Assume that N(0) = No, where No is much smaller than K. Imagine a function that satisfies the differential equation in (a) for small values of t, and that has as a limit the solution to the differential equation in (b). Sketch the graph of such a function, be sure to label No and K in your graph.
Transcribed Image Text:The logistic equation is the autonomous differential equation: N(1 - *). dN = r. dt K Where r and K are both positive constants. The logistic equation is used to model the size of a population subject to constrained growth. (1) In this problem we will discuss the meaning of constants r and K. (a) When the population size N is much smaller than K, explain why the logistic dN = r· N. In that dt equation behaves like the exponential growth equation: sense, discuss how r can be thought of as the "initial growth rate" for a population subject to constrained growth. (b) When N is very close to K, explain why the logistic equation behaves like the dN differential equation: dt 0. In this sense, discuss how K can be thought of as the "carrying capacity" or “limiting capacity" for a population subject to constrained growth. (c) Assume that N(0) = No, where No is much smaller than K. Imagine a function that satisfies the differential equation in (a) for small values of t, and that has as a limit the solution to the differential equation in (b). Sketch the graph of such a function, be sure to label No and K in your graph.
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