(a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P < m. (b) For the case where k = 1, M = 100,000 and m it to sketch several solutions for various initial populations. What are the equilibrium 10, 000, draw a direction field and use solutions? (c) One can show that k(M-m)t - m(Po – M) M(Po – m)eM P(t) k(М-т) (Ро — т)е м - (Ро — М) is a solution with initial population P(0) = Po. Use this to show that, if P(0) < m, then there is a time t at which P(t) = 0 (and so the population will be extinct).
(a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P < m. (b) For the case where k = 1, M = 100,000 and m it to sketch several solutions for various initial populations. What are the equilibrium 10, 000, draw a direction field and use solutions? (c) One can show that k(M-m)t - m(Po – M) M(Po – m)eM P(t) k(М-т) (Ро — т)е м - (Ро — М) is a solution with initial population P(0) = Po. Use this to show that, if P(0) < m, then there is a time t at which P(t) = 0 (and so the population will be extinct).
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:(a) Use the differential equation to show that any solution is increasing if m < P < M and
decreasing if 0 < P < m.
(b) For the case where k = 1, M = 100, 000 and m = 10, 000, draw a direction field and use
it to sketch several solutions for various initial populations. What are the equilibrium
solutions?
(c) One can show that
k(M–m)+
M(Po – m)eª M
т(Ро — М)
P(t)
k(М-т)
(Ро — т)е м
- (Ро — М)
M
-
is a solution with initial population P(0) =
there is a time t at which P(t) = 0 (and so the population will be extinct).
Po. Use this to show that, if P(0) < m, then
Transcribed Image Text:6. A population of fish is living in an environment with limited resources. This environment can
only support the population if it contains no more than M fish (otherwise some fish would
starve due to an inadequate supply of food, etc.). There is considerable evidence to support
the theory that, for some fish species, there is a minimum population m such that the species
will become extinct if the size of the population falls below m. Such a population can be
modelled using a modified logistic equation:
dP
= kP (1
dt
m
1
M
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