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dr kr(M- x)-h (5) models a fish population which is growing according to the logistic law, and harvested at a constant rate h(h > 0). (a) Show that the critical points of (5) can be obtained by solving the quadratic equation -ka? + kMx - h = 0. (b) Give the conditions under which the roots of the quadratic equations are real. (c) Denote the roots by H, N where 0 < H < N < M. Show that Equation (5) can be written as dr = k(N – x)(x – H) (6) dt (d) Using partial fractions, show that Equation (6) has the solution N(ro – H) – H(xo – N)e-^(N-H)t (ro - H) – (xo – N)e-k(N-H)t r(t) = (7) Deduce from Equation (7) that r(t) →N as t+o0

dr kr(M- x)-h (5) models a fish population which is growing according to the logistic law, and harvested at a constant rate h(h > 0). (a) Show that the critical points of (5) can be obtained by solving the quadratic equation -ka? + kMx - h = 0. (b) Give the conditions under which the roots of the quadratic equations are real. (c) Denote the roots by H, N where 0 < H < N < M. Show that Equation (5) can be written as dr = k(N – x)(x – H) (6) dt (d) Using partial fractions, show that Equation (6) has the solution N(ro – H) – H(xo – N)e-^(N-H)t (ro - H) – (xo – N)e-k(N-H)t r(t) = (7) Deduce from Equation (7) that r(t) →N as t+o0

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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Given that the equation dr = kr(M – x) – h (5) dt models a fish population which is growing according to the logistic law, and harvested at a constant rate h(h > 0). (a) Show that the critical points of (5) can be obtained by solving the quadratic equation -kr? + kMx – h = 0. (b) Give the conditions under which the roots of the quadratic equations are real. (c) Denote the roots by H, N where 0 < H < N < M. Show that Equation (5) can be written as dr = k(N – x)(r – H) (6) dt (d) Using partial fractions, show that Equation (6) has the solution N(ro – H) – H(xo – N)e-k(N-H)t r(t) = (7) %3! (xo – H) – (19 – N)e-k(N-H)t Deduce from Equation (7) that r(t) → N as t → +∞
Transcribed Image Text:Given that the equation dr = kr(M – x) – h (5) dt models a fish population which is growing according to the logistic law, and harvested at a constant rate h(h > 0). (a) Show that the critical points of (5) can be obtained by solving the quadratic equation -kr? + kMx – h = 0. (b) Give the conditions under which the roots of the quadratic equations are real. (c) Denote the roots by H, N where 0 < H < N < M. Show that Equation (5) can be written as dr = k(N – x)(r – H) (6) dt (d) Using partial fractions, show that Equation (6) has the solution N(ro – H) – H(xo – N)e-k(N-H)t r(t) = (7) %3! (xo – H) – (19 – N)e-k(N-H)t Deduce from Equation (7) that r(t) → N as t → +∞
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