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 dt P 1- т 1- (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)t М (Ро — т)е м -m(Po- M) E(M=m), (Po- M) P(t) = (Po – m)eM 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 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 dt P 1- т 1- (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)t М (Ро — т)е м -m(Po- M) E(M=m), (Po- M) P(t) = (Po – m)eM 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: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
m
kP 1
dt
%D
-
M
(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)t
M
M(Po – m)e
(Ро — т)е м
-m(Po – M)
t- (Po – M)
P(t) =
k(М-т)
M
|
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).
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