(4) (a) Derive the solution MPO P(t) = Po+(M-Po)ek Mt of the extinction-explosion initial value problem P' = kP(P - M), P(0) = Po- (b) How does the behavior of P(t) as t increases depend on whether 0 < Po M?
(4) (a) Derive the solution MPO P(t) = Po+(M-Po)ek Mt of the extinction-explosion initial value problem P' = kP(P - M), P(0) = Po- (b) How does the behavior of P(t) as t increases depend on whether 0 < Po M?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**(4) (a) Derive the solution**
\[
P(t) = \frac{M P_0}{P_0 + (M - P_0)e^{kMt}}
\]
of the extinction-explosion initial value problem \( P' = kP(P - M) \), \( P(0) = P_0 \).
**(b) How does the behavior of \( P(t) \) as \( t \) increases depend on whether \( 0 < P_0 < M \) or \( P_0 > M \)?**
---
### Explanation:
The expression given, \( P(t) = \frac{M P_0}{P_0 + (M - P_0)e^{kMt}} \), represents a solution to a differential equation describing a process often referred to as "extinction-explosion."
- **Initial Value Problem:**
- The differential equation is given by \( P' = kP(P - M) \).
- \( P(0) = P_0 \) specifies the initial condition.
- **Parameters:**
- \( k \) is a rate constant.
- \( M \) is a threshold value.
- \( P_0 \) is the initial population or quantity.
- **Behavior Analysis:**
- **If \( 0 < P_0 < M \):** The solution \( P(t) \) will approach zero as \( t \) increases, indicating extinction, since the initial population is below the threshold \( M \).
- **If \( P_0 > M \):** The solution \( P(t) \) will increase and potentially approach infinity or a stable state above \( M \), indicating explosion or rapid growth, as the initial population exceeds the threshold.
This mathematical model can be applied in contexts like population dynamics, chemical reactions, or ecological studies where thresholds determine system behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3edbe93-d39c-43f4-8aa9-2c034a88e626%2Fd9f7f9dc-bf1c-4f87-bc29-3ade3f6d2d68%2Fie8f5cv_processed.png&w=3840&q=75)
Transcribed Image Text:**(4) (a) Derive the solution**
\[
P(t) = \frac{M P_0}{P_0 + (M - P_0)e^{kMt}}
\]
of the extinction-explosion initial value problem \( P' = kP(P - M) \), \( P(0) = P_0 \).
**(b) How does the behavior of \( P(t) \) as \( t \) increases depend on whether \( 0 < P_0 < M \) or \( P_0 > M \)?**
---
### Explanation:
The expression given, \( P(t) = \frac{M P_0}{P_0 + (M - P_0)e^{kMt}} \), represents a solution to a differential equation describing a process often referred to as "extinction-explosion."
- **Initial Value Problem:**
- The differential equation is given by \( P' = kP(P - M) \).
- \( P(0) = P_0 \) specifies the initial condition.
- **Parameters:**
- \( k \) is a rate constant.
- \( M \) is a threshold value.
- \( P_0 \) is the initial population or quantity.
- **Behavior Analysis:**
- **If \( 0 < P_0 < M \):** The solution \( P(t) \) will approach zero as \( t \) increases, indicating extinction, since the initial population is below the threshold \( M \).
- **If \( P_0 > M \):** The solution \( P(t) \) will increase and potentially approach infinity or a stable state above \( M \), indicating explosion or rapid growth, as the initial population exceeds the threshold.
This mathematical model can be applied in contexts like population dynamics, chemical reactions, or ecological studies where thresholds determine system behavior.
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