Exponential growth and the logistic equation are not the only mathematical models used to describepopulation growth. The Gompertz equation, which describes the behavior of a population that isrestricted to a confined space with limited resources, has been particularly successfully in modelingtumor growth. The Gompertz equation is given by:dyK= ry ln%3Ddtwhere r and K are positive constants.(a) Find the critical points of the Gompertz equation.(b) Draw the phase line and classify each critical point as a stable or an unstable equilibrium.(c) Use the phase line to determine the long-term behavior of the solution that satisfies y(0) = 5.(d) Solve the Gompertz equation using the substitution v = ln (). Express your solution as anexplicit function y = y(t).Hint: Use the fact that In () = – In ( 5).KK(e) Find the particular solution that satisfies y(0)and compute lim y(t). Is this result2consistent with your result from (c)?

Answered: Exponential growth and the logistic…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Related questions

Concept explainers

Question

Exponential growth and the logistic equation are not the only mathematical models used to describe
population growth. The Gompertz equation, which describes the behavior of a population that is
restricted to a confined space with limited resources, has been particularly successfully in modeling
tumor growth. The Gompertz equation is given by:
dy
K
= ry ln
%3D
dt
where r and K are positive constants.
(a) Find the critical points of the Gompertz equation.
(b) Draw the phase line and classify each critical point as a stable or an unstable equilibrium.
(c) Use the phase line to determine the long-term behavior of the solution that satisfies y(0) = 5.
(d) Solve the Gompertz equation using the substitution v = ln (). Express your solution as an
explicit function y = y(t).
Hint: Use the fact that In () = – In ( 5).
K
K
(e) Find the particular solution that satisfies y(0)
and compute lim y(t). Is this result
2
consistent with your result from (c)?

Expert Solution

Step 1

Disclaimer: As the question contains multiple parts, unless stated otherwise, only first two parts are answered.

Given a first-order autonomous differential equation
$y' = f (y)$ the values of y such that f(y) = 0 are called the critical points of the differential equation.

We have y'= $\frac{d y}{d t}$ =f(y)= $r y \ln (\frac{K}{y})$

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.

Similar questions