Chapter 2.3, Problem 3E

Interpretation Introduction

Interpretation:

The growth of the cancerous tumor is to be determined by Gompertz law $\dot{\text{N}}\text{= - aN ln}\left(\text{bN}\right)$, where $\text{N(t)}$ is the total number of cells in the tumor, $a$ and b are the parameter. Explain the parameter $a$ and b biologically. Sketch the vector field and N(t) vs t for various initial values.

Concept Introduction:

Gompertz model explains the growth of a cancerous tumor by using differential equation as $\dot{\text{N}}\text{= - aN ln}\left(\text{bN}\right)$.

When the vector field for flow represent on the real line, the points where the flow is towards them are called stable points, and the flow is away from them are called unstable points.

In the given problem, N=0 is an unstable fixed point since the vector field is away from it and $\text{N}=\frac{1}{b}$ is a stable fixed point since the vector field is towards it.