(Solved):   Please don't just post a solution answered by others. Thanks. Question 1 The Gompertz ...

 

Please don't just post a solution answered by others. Thanks.

Question 1 The Gompertz model for a population is \[ \frac{d p}{d t}=-k p \log \left(\frac{p}{a}\right) \] where \( k \) and \( a \) are positive constants, and \( p(t) \) is the population size at time \( t \). Answer the following questions without solving for \( p \) in terms of \( t \). (a) Find \( \lim _{p \rightarrow 0^{+}}-k p \log \left(\frac{p}{a}\right) \). (b) Find the equilibrium solution(s) of the Gompertz ODE, or show that it does not have any. (c) Draw a phase plot for the Gompertz model ODE, and draw the family of solutions Indicate any equilibrium solution(s). d) Consider a cancer tumor whose growth is modelled by the Gompertz model. The tumor begins with a very small number of cells. Let \( p(t) \) be the number of cells (in millions) in the tumor at time \( t \). Find the maximum rate of change of \( p \), and the number of cells in the tumor when it occurs. Give your answers in terms of \( a \) and/or \( k \).