12. [Kaplan & Glass(1995)] Limpets and seaweed live in a tide pool. The dynamics of this system are given by the differential equations ds s – s² – sl, dt dl = sl dt 1², 1>0,s>0, 2 where the densities of seaweed and limpets are given by s and l, respectively. (i) Determine all equilibria of this system. (ii) For each nonzero equilibrium determined in part (a), evaluate the stability and classify it as a node, focus, or saddle point. (iii) Sketch the flows in the phase plane.

12. [Kaplan & Glass(1995)] Limpets and seaweed live in a tide pool. The dynamics of this system are given by the differential equations ds s – s² – sl, dt dl = sl dt 1², 1>0,s>0, 2 where the densities of seaweed and limpets are given by s and l, respectively. (i) Determine all equilibria of this system. (ii) For each nonzero equilibrium determined in part (a), evaluate the stability and classify it as a node, focus, or saddle point. (iii) Sketch the flows in the phase plane.

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

12. [Kaplan & Glass(1995)] Limpets and seaweed live in a tide pool. The dynamics
of this system are given by the differential equations
ds
s² – sl,
= S
dt
dl
sl
--2, 1>0,s > 0,
dt
2
where the densities of seaweed and limpets are given by s and l, respectively.
(i) Determine all equilibria of this system.
(ii) For each nonzero equilibrium determined in part (a), evaluate the stability
and classify it as a node, focus, or saddle point.
(iii) Sketch the flows in the phase plane.
(iv) What will the dynamics be in the limit as t → o for initial conditions
(i) s(0) = 0, 1(0) = 0?
(iї) s(0) — 0, 1(0) — 15?
(iii) s(0) = 2, 1(0) = 0?
(iv) s(0) = 2, 1(0) = 15?

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