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.

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Chapter2: Second-order Linear Odes
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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?
Transcribed Image Text: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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