There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the fac (1 - m/P). Thus the modified logistic model is given by the differential equation (a) Use the differential equation to show that any solution is increasing if m
There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the fac (1 - m/P). Thus the modified logistic model is given by the differential equation (a) Use the differential equation to show that any solution is increasing if 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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Transcribed Image Text:There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the factor
(1 – m/P). Thus the modified logistic model is given by the differential equation
-(1 -(1-)
dP
= kPI 1
dt
P
M
(a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P < m.
(Ko(1 - 4)(1 - ).
dP
P
m
If --Select--- V
then dP/dt = (+)(+)(+) = +
P is ---Select--- ♥
. If |---Select--- ♥
then dP/dt = (+)(+)(-) =
P is ---Select--- ♥
dt
M
(b) For the case where k = 0.08, M =
1000, and m =
150, draw a direction field and use it to sketch several solution curves.
P
P
3000)
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P
3000 t /
1400
2500
1200
2000.
1000,
800
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1000
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![Describe what happens to the population for various initial populations. (Enter your answers using interval notation.)
For Po E
the population dies out. For Po E
the population increases and approaches 1000. For Po
E
the population decreases and approaches 1000.
What are the equilibrium solutions? (Enter your answers as a comma-separated list.)
P =
(c) Solve the differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial population Po.
P =
(d) Use the solution in part (c) to show that if Po < m, then the species will become extinct. [Hint: Show that the numerator in your expression for P(t) is 0 for some value of t.]
If Po < m, then Po - m < 0. Let N(t) be the numerator of the expression for P(t) in part (c). Then N(0) = Po(M – m) ?v 0, and Po - m ? v 0
lim N(t)
?v 0. Since N is ---Select---
?v 00
there is a number t
lim M(Po - m)e(M – m)(k/M)t
such that N(t)
O and thus P(t) = 0. So the species will become extinct.
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Transcribed Image Text:Describe what happens to the population for various initial populations. (Enter your answers using interval notation.)
For Po E
the population dies out. For Po E
the population increases and approaches 1000. For Po
E
the population decreases and approaches 1000.
What are the equilibrium solutions? (Enter your answers as a comma-separated list.)
P =
(c) Solve the differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial population Po.
P =
(d) Use the solution in part (c) to show that if Po < m, then the species will become extinct. [Hint: Show that the numerator in your expression for P(t) is 0 for some value of t.]
If Po < m, then Po - m < 0. Let N(t) be the numerator of the expression for P(t) in part (c). Then N(0) = Po(M – m) ?v 0, and Po - m ? v 0
lim N(t)
?v 0. Since N is ---Select---
?v 00
there is a number t
lim M(Po - m)e(M – m)(k/M)t
such that N(t)
O and thus P(t) = 0. So the species will become extinct.
%3D
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