(b) Suppose that we want to model the evolution of the population of a cer- tain type of organisms. Observations indicate that if the population drops below a survival level of 10² individuals, it goes extinct. Moreover, the population growth is limited: the available resources of space and food can sustain at most 10 individuals. We treat the population size P(t) as a continuous function of time. (i) Explain briefly how the following model incorporates the above ob- servations: dP dt = k(A — P)(P – B), k>0, where P(t) denotes the population size at time t and B < A. Using the informations given in the text of the question, determine the val- ues of the constants A and B. Can you also determine the value of k? [2] (ii) Suppose that A < P < B. At which value of the population is the growth fastest? [2] (c) Perform a linear stability analysis for the model below: dP dt Find the equilibrium values and determine their stability. - P(1 - P)e-P², P≥ 0. [6]
(b) Suppose that we want to model the evolution of the population of a cer- tain type of organisms. Observations indicate that if the population drops below a survival level of 10² individuals, it goes extinct. Moreover, the population growth is limited: the available resources of space and food can sustain at most 10 individuals. We treat the population size P(t) as a continuous function of time. (i) Explain briefly how the following model incorporates the above ob- servations: dP dt = k(A — P)(P – B), k>0, where P(t) denotes the population size at time t and B < A. Using the informations given in the text of the question, determine the val- ues of the constants A and B. Can you also determine the value of k? [2] (ii) Suppose that A < P < B. At which value of the population is the growth fastest? [2] (c) Perform a linear stability analysis for the model below: dP dt Find the equilibrium values and determine their stability. - P(1 - P)e-P², P≥ 0. [6]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(b) Suppose that we want to model the evolution of the population of a cer-
tain type of organisms. Observations indicate that if the population drops
below a survival level of 10° individuals, it goes extinct. Moreover, the
population growth is limited: the available resources of space and food
can sustain at most 106 individuals. We treat the population size P(t) as
a continuous function of time.
(i) Explain briefly how the following model incorporates the above ob-
servations:
dP
— К(А- Р)(Р — В), k>0,
dt
where P(t) denotes the population size at time t and B < A. Using
the informations given in the text of the question, determine the val-
ues of the constants A and B. Can you also determine the value of
[2]
(ii) Suppose that A < P < B. At which value of the population is the
[2]
k?
growth fastest?
(c) Perform a linear stability analysis for the model below:
dP
P(1 – P)e-P*,P > 0.
dt
Find the equilibrium values and determine their stability.
[6]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b379c3f-af09-4b4d-b202-fe506c36fa98%2F2465b259-c3ae-4e6d-88b0-62b4b1c67b32%2F7n1xt4m_processed.png&w=3840&q=75)
Transcribed Image Text:(b) Suppose that we want to model the evolution of the population of a cer-
tain type of organisms. Observations indicate that if the population drops
below a survival level of 10° individuals, it goes extinct. Moreover, the
population growth is limited: the available resources of space and food
can sustain at most 106 individuals. We treat the population size P(t) as
a continuous function of time.
(i) Explain briefly how the following model incorporates the above ob-
servations:
dP
— К(А- Р)(Р — В), k>0,
dt
where P(t) denotes the population size at time t and B < A. Using
the informations given in the text of the question, determine the val-
ues of the constants A and B. Can you also determine the value of
[2]
(ii) Suppose that A < P < B. At which value of the population is the
[2]
k?
growth fastest?
(c) Perform a linear stability analysis for the model below:
dP
P(1 – P)e-P*,P > 0.
dt
Find the equilibrium values and determine their stability.
[6]
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