Assume that m(p) = a+ bp for some constants a,b> 0. That is, the extinction rate increases with p because of competit dp ranish as p→0. The model for proportion of occupied sites is = cp(1- p) - (a + bp)p. Assuming that the subpopulat coefficients are a = 5, b= 2, but c is allowed to take any value, complete parts (a) through (c). a) Find the equilibrium values of p (your answer will depend on the unknown coefficient c). The equilibrium values are p= c-5 +2 (Simplify your answer. Use a comma to separate answers as needed.) b) What are the conditions on c for p to have a nontrivial equilibrium, that is, an equilibrium in which pE(0, 1]? c>5 (Type an inequality. Simplify your answer.) c) Show that if your condition from (b) is met, then the nontrivial equilibrium is also stable. At the nontrivial equilibrium, g' (p) O so it is stable.
Assume that m(p) = a+ bp for some constants a,b> 0. That is, the extinction rate increases with p because of competit dp ranish as p→0. The model for proportion of occupied sites is = cp(1- p) - (a + bp)p. Assuming that the subpopulat coefficients are a = 5, b= 2, but c is allowed to take any value, complete parts (a) through (c). a) Find the equilibrium values of p (your answer will depend on the unknown coefficient c). The equilibrium values are p= c-5 +2 (Simplify your answer. Use a comma to separate answers as needed.) b) What are the conditions on c for p to have a nontrivial equilibrium, that is, an equilibrium in which pE(0, 1]? c>5 (Type an inequality. Simplify your answer.) c) Show that if your condition from (b) is met, then the nontrivial equilibrium is also stable. At the nontrivial equilibrium, g' (p) O so it is stable.
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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![Assume that m(p) = a+ bp for some constants a,b>0. That is, the extinction rate increases with p because of competition between subpopulations, but m(p) does not
dp
= cp(1 - p) - (a + bp)p. Assuming that the subpopulations obey the differential equation and the
vanish as p+0. The model for proportion of occupied sites is
dt
coefficients are a = 5, b= 2, but c is allowed to take any value, complete parts (a) through (c).
(a) Find the equilibrium values of p (your answer will depend on the unknown coefficient c).
The equilibrium values are p= 0,
C-5
(Simplify your answer. Use a comma to separate answers as needed.)
(b) What are the conditions on c for p to have a nontrivial equilibrium, that is, an equilibrium in which pE(0, 1]?
c>5 (Type an inequality. Simplify your answer.)
(c) Show that if your condition from (b) is met, then the nontrivial equilibrium is also stable.
At the nontrivial equilibrium, g' (p)
VO so it is stable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72f0c3ea-f222-42fb-a1d0-90dbd040501b%2F86b3efa5-0092-469e-8b7c-0bfe04996dd5%2Fdc9zne_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Assume that m(p) = a+ bp for some constants a,b>0. That is, the extinction rate increases with p because of competition between subpopulations, but m(p) does not
dp
= cp(1 - p) - (a + bp)p. Assuming that the subpopulations obey the differential equation and the
vanish as p+0. The model for proportion of occupied sites is
dt
coefficients are a = 5, b= 2, but c is allowed to take any value, complete parts (a) through (c).
(a) Find the equilibrium values of p (your answer will depend on the unknown coefficient c).
The equilibrium values are p= 0,
C-5
(Simplify your answer. Use a comma to separate answers as needed.)
(b) What are the conditions on c for p to have a nontrivial equilibrium, that is, an equilibrium in which pE(0, 1]?
c>5 (Type an inequality. Simplify your answer.)
(c) Show that if your condition from (b) is met, then the nontrivial equilibrium is also stable.
At the nontrivial equilibrium, g' (p)
VO so it is stable.
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