Question

Problem 3. [Predator-Prey with limit cycle] This problem is modified from 8.2.9 of [Stro- gatz]. Consider dx ху = x (b − x) = x (6 [0 — x) like 1 + x 1+2 dt dy dt xy 1+2 ay? = y ay 1 + x where x, y > 0 are the populations and a, b > 0) are parameters. (a) Sketch the nullclines and show that a positive fixed point æ*, y* > 0) exists for all a, b > 0. (Don't try to find the fixed point explicitly; use a graphical argument instead.) (b) Show that a Hopf bifurcation occurs at the positive fixed point if a = ac = 4 (b − 2) and b > 2. b2 (6+2) [Hint]: A necessary condition for a Hopf bifurcation to occur is T = 0 where t is the trace of the Jacobian matrix at the fixed point. Show that t = 0 if and only if 2x* = b – 2. Then use the fixed point conditions to express ac in terms of x*. Finally, substitute x* = (b − 2)/2 into the expression for ac. (c) Using a computer, check the validity of the expression in (b). Plot typical phase portraits above and below the Hopf bifurcation.

Answer 1

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(b) Jacobian matrix & A af 7 ду a 29. -an 99 ay 9=y ay f = aCbn) - ay colti of (b-x]f&(-1) At try ati x But 13 af ay at 29 ny an at дау. antly atl A = (-2) +9 -9AU et. Hati yaf)-ny (1418² _2072 nt A- (b-2x] - y - (hitlja nti ge (4132 — aqs- t1 at y* fixed point ** 6-Qu*) -, y (a* tu? A- att st 2a y* 2441 (au* 119² Trace of a -matrix is the sum of the diagonal elements.

Ti race T = (beat)_y* (soft19 & 7 am - 200k (**HJ T=0 24 Qout = 6-2 Ho* 62 co from yt it a* 90*+) put at درا Z a y* bio ab ☆ ī=0 (ها) . ( 62) + La Cbas ab acht ادا) 10 2 -266-2 ab3 b b t2 b 06-2) (bol 466-2) ab3 b 963 qe= a i net 2 Gb-2) 672 (1) B (172) if and only if a= ac = 466-2) 6 (642) and b72 and = 6-2 and gt bad ab can also confirm this from a (b-u* J Cakt f = a* q (b-Columns) (bad H 3* = we

by2 9_=9c = 465–2) (6 72 6² (c) phase portrait at a* = bod y*= bz ab y y* * t b V. o* Code

3 2. -2 1 0 1 2 3 4