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.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.