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A competition between two species is modelled using a pair of coupled dıfferential quations dP 2P(t) – 2Q(t) %3D dt OP -2P(t) – Q(t) dt where t is the time, 2000 + P(t) is the number of the original species, and Q(t) is the number of the competitor species. a) Find the general solution P(t) and Q(t) of the system of differential equations above. b) Find the particular solution when P(0) = -20 and Q(0) = 90. %3D c) Show that for the particular solution above, the original species goes extinct. In other words, show that 2000 + P(t) : 0 has a positive solution for t.

A competition between two species is modelled using a pair of coupled dıfferential quations dP 2P(t) – 2Q(t) %3D dt OP -2P(t) – Q(t) dt where t is the time, 2000 + P(t) is the number of the original species, and Q(t) is the number of the competitor species. a) Find the general solution P(t) and Q(t) of the system of differential equations above. b) Find the particular solution when P(0) = -20 and Q(0) = 90. %3D c) Show that for the particular solution above, the original species goes extinct. In other words, show that 2000 + P(t) : 0 has a positive solution for t.

Advanced Engineering Mathematics
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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A competition between two species is modelled using a pair of coupled differential equations dP 2P(t) - 2Q(t) dt OP dt 2P(t)- Q(t) where t is the time, 2000 + P(t) is the number of the original species, and Q(t) is the number of the competitor species. a) Find the general solution P(t) and Q(t) of the system of differential equations above. b) Find the particular solution when P(0) = -20 and Q(0) = 90. %3D c) Show that for the particular solution above, the original species goes extinct. In other words, show that 2000 + P(t) : 0 has a positive solution for t. You do not need to find the value of time t at which it ddes so. d) The original species will not necessarily go extinct when the competitor is introduced. Assuming that Q(0) = 90 remains the same, what is the smallest value of P(0) such that the original species survives?
Transcribed Image Text:A competition between two species is modelled using a pair of coupled differential equations dP 2P(t) - 2Q(t) dt OP dt 2P(t)- Q(t) where t is the time, 2000 + P(t) is the number of the original species, and Q(t) is the number of the competitor species. a) Find the general solution P(t) and Q(t) of the system of differential equations above. b) Find the particular solution when P(0) = -20 and Q(0) = 90. %3D c) Show that for the particular solution above, the original species goes extinct. In other words, show that 2000 + P(t) : 0 has a positive solution for t. You do not need to find the value of time t at which it ddes so. d) The original species will not necessarily go extinct when the competitor is introduced. Assuming that Q(0) = 90 remains the same, what is the smallest value of P(0) such that the original species survives?
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