The amount of fish in a lake is modeled by the logistic equation: = 2y(4 – y). If a certain amount of fish k is caught per unit of time, the equation becomes y = 2y(4 – 3) – k, k > 0. (a) Determine for which h the equation has two, one, and no equilibriums respectively. (b) Let h be such that the equation has one equilibrium. Find out for which initial amount of fish y(0) the population will eventually die out, and for which it will not. Sketch solution curves. (C) For k = 6, find equilibriums and study their stability. Sketch solution curves. Find the solution with initial condition y(0) = 2.

The amount of fish in a lake is modeled by the logistic equation: = 2y(4 – y). If a certain amount of fish k is caught per unit of time, the equation becomes y = 2y(4 – 3) – k, k > 0. (a) Determine for which h the equation has two, one, and no equilibriums respectively. (b) Let h be such that the equation has one equilibrium. Find out for which initial amount of fish y(0) the population will eventually die out, and for which it will not. Sketch solution curves. (C) For k = 6, find equilibriums and study their stability. Sketch solution curves. Find the solution with initial condition y(0) = 2.

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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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

The amount of fish in a lake is modeled by the logistic equation: y' = 2y(4 – y). If a certain amount of fish k is caught per unit of time, the
equation becomes y' = 2y(4 – 3) – k, k > 0.
(a) Determine for which h the equation has two, one, and no equilibriums respectively.
(b) Let h be such that the equation has one equilibrium. Find out for which initial amount of fish y(0) the population will eventually die out, and
for which it will not. Sketch solution curves.
(C) For k = 6, find equilibriums and study their stability. Sketch solution curves. Find the solution with initial condition y(0) = 2.

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