(2) A common growth model for a population P(t) is so-called logistic growth: dP — k- Р. (М- Р) dt (2) For small times the growth dP/dtwill be roughly proportional to P, but for later times, the growth will slow down because M – P will approach zero. In the logistic equation the constant k is called the growth factor and M is called the carrying capacity. This differential equation can be solved as M P(t) 1+ ( -1) (3) M Po e-kMt where P(0) = Po- (a) Show that when we insert t = 0 in the general solution (3), we indeed find P(0) = Po. (b) Show that for large times, the general solution (3) yields lim→∞ P(t) = M (c) Now solve a special case of the logistic equation with k = 3, M = 2 and Po = 1. Use separation by variables and for the P-integral use partial fractions.

(2) A common growth model for a population P(t) is so-called logistic growth: dP — k- Р. (М- Р) dt (2) For small times the growth dP/dtwill be roughly proportional to P, but for later times, the growth will slow down because M – P will approach zero. In the logistic equation the constant k is called the growth factor and M is called the carrying capacity. This differential equation can be solved as M P(t) 1+ ( -1) (3) M Po e-kMt where P(0) = Po- (a) Show that when we insert t = 0 in the general solution (3), we indeed find P(0) = Po. (b) Show that for large times, the general solution (3) yields lim→∞ P(t) = M (c) Now solve a special case of the logistic equation with k = 3, M = 2 and Po = 1. Use separation by variables and for the P-integral use partial fractions.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: A population of bacteria growing exponentially can be modeled as P(t)= Pue, where t is the time in…

Q: Suppose I am interested in modeling how wages are determined. My model of interest is log(wage) = β0…

A: Hey, since there are multiple sub-part questions posted, we will answer first three sub part…

Q: Suppose that a population grows according to a logistic model with carrying capacity 6300 and k =…

Q: dy 2. When does the differential equation Cekt represent exponential growth? exponential dt decay?

A: We need to explain growth and decay.

Q: 24. A population grows according to the logistic growth model, with growth parameter r = 0.8.…

Q: a) The amount of antihistamine in the blood decays exponentially as the kidneys and liver do their…

A: Given: A(t)=A0(0.93)t A0=100%

Q: #2. Researchers have descried the maximum growth rate of a phytoplankton as a function of the cell…

Q: A biologist noticed that a small forest has a population of deer that naturally grows according to…

A: For part-1: The biologist noticed the logistic model to be y'=0.0005y300-y where y is the number of…

Q: A new fish species is introduced into a lake. Assuming that the population of fish quadruples in the…

A: Solved below

Q: If we are modeling a herd of clk, with an initial population of 100, in a region with a carrying…

A: Given that The population of elk satisfies a logistic growth model and Initial population = 100…

Q: If q'(t) is the growth rate of a population of a fish population in fish per month, what does…

Q: 7. A commercial fishery is estimated to have carrying capacity 40 tons of salmon. Suppose the annual…

Q: Biologists release two otters (a type of water animal) into a river. They believe that the river…

A: Detailed explanation:Part i: Finding α and βGiven:Carrying capacity K=500Intrinsic growth rate…

Q: For a Pacific halibut population, the value of r is 0.71 per year, and the environmental carrying…

A: a) We have to find time in logistic growth: Rate per year,r=0.71 Population growth,N=77,430 Value of…

Concept explainers

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.

Question

(2) A common growth model for a population P(t) is so-called logistic growth:
dP
— k. Р. (М- Р)
dt
(2)
For small times the growth dP/dtwill be roughly proportional to P, but for later times, the growth will
slow down because M – P will approach zero. In the logistic equation the constant k is called the growth
factor and M is called the carrying capacity. This differential equation can be solved as
M
P(t)
(3)
(* -1)
M
1+
Ро
kMt
e
where P(0) = Po-
(a) Show that when we insert t = 0 in the general solution (3), we indeed find P(0) = Po.
(b) Show that for large times, the general solution (3) yields lim;→∞ P(t) = M
(c) Now solve a special case of the logistic equation with k = 3, M = 2 and Po = 1. Use separation by
variables and for the P-integral use partial fractions.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

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, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Exercise 0.2.12: Is there a solution to y' = y, such that y(0) = y(1)? Exercise 0.2.13: The population of city X was 100 thousand 20 years ago, and the population of city X was 120 thousand 10 years ago. Assuming constant growth, you can use the exponential population model (like for the bacteria). What do you estimate the population is now?
The number N(t) of people in a community who are exposed to a particular advertisement is governed by the logistic equation. Initially, N(0) = 200, and it is observed that N(1) = 400. Solve for N(t) if it is predicted that the limiting number of people in the community who will see the advertisement is 20,000. (Round all coefficients to four decimal places.) N(t) =
Suppose the population of a species of animals on an island is governed by the logistic model with relative rate of growth k = 0.05 and carrying capacity M= 15000. I.e., the population function P(t) satisfies the equation P' = bP(15000 - P), where b = k/M. If the current population is P(0) = 20000, which one of the following is closest to P(1)? O a) 19700 O b) 19890 Oc) 19680 Od) 19690 Oe) 19805 Of) 19742 Motion
At time t = 0, a bacterial culture weighs 2 grams. Three hours later, the culture weighs 5 grams. The maximum weight of the culture is 20 grams. (a) Write a logistic equation that models the weight of the bacterial culture. (Round your coefficients to four decimal places.) 20 y = dy dt = 1 +9e 8 g (c) When will the culture's weight reach 16 grams? (Round your answer to two decimal places.) 9.79 hr (d) Write a logistic differential equation that models the growth rate of the culture's weight. Then repeat part (b) using Euler's Method with a step size of h = 1. (Round your answer to the nearest whole number.) y(5) = 6 -0.3662t = 1.37 0.366y| 1 (1-20) (b) Find the culture's weight after 5 hours. (Round your answer to the nearest whole number.) X g (e) At what time is the culture's weight increasing most rapidly? (Round your answer to two decimal places.) hr
7. A commercial fishery is estimated to have carrying capacity 40 tons of salmon. Suppose the annual growth rate of total salmon population P (measured in tons) is governed by the logistic equation P' = P(1 – P/40) and initially there are 30 tons of fish. (i) What is the fish population after 1 year? (You may use e 2 2.72.) Suppose that the owner of the fishery decides to harvest h tons of salmon annually at a constant rate. Then (ii) ( What is the differential equation governing the fish population now? (iii) What is the maximun sustainable harvesting rate? (Justification is required.)
During the period from 1790 to 1920, a country's population P(t) (t in years) grew from 3.6 million to 101.6 million. Throughout this period, P(t) remained close to the solution of the initial value problem -0.0001494P² P(0) = 3.6. dP dt = 0.03136P-( (a) What 1920 population does this logistic equation predict? (b) What limiting population does it predict? (c) The country's population in 2000 was 259 million. Has this logistic equation continued since 1920 to accurately model the country's population? (a) The logistic equation predicts the population in 1920 to be million. (Do not round until the final answer. Then round to the nearest thousandth as needed.) K AI C H TU U →> C ¢ ➡ H i O + 2.4 e В → نه Y + - 8 .... : *** 5 Ø Co Less . N 0 .. H A E X U i 9 A C ש Σ VI 8 FR > in X