1. Ecological consequences of global bifurcations
George van Voorn
Vrije Universiteit, Amsterdam
Edinburgh, ECMTB 2008
With: Bob Kooi, Lia Hemerik, Yuri Kuznetsov, Martin Boer & Eusebius Doedel

2. Overview Allee model (2D) Rozenzweig-MacArthur model (3D) Conclusion
What is an Allee model?
Analysis of the model, including bifurcation analysis
Conflicting results  global bifurcation
Develop new analysis techniques
Results
Rozenzweig-MacArthur model (3D)
Model equations
Existing connecting orbits
Conclusion

3. Allee model Density-dependency affects prey population
Below threshold  extinction (Allee, 1931)
x1 = prey population
x2 = predator population
l = extinction threshold, no fixed value
k = carrying capacity, by default 1
c = conversion ratio, by default 1
m = predator mortality rate, no fixed value
Note: dimensionless
LEG DEZE SLIDE UIT, a.u.b.!

4. Equilibria System has the following equilibria: E0 = (0,0) E1 = (l,0)
E2 = (k,0), with k ≥ l
E3 = (m,(m-l)(k-m))

5. Analysis
Two-parameter bifurcation diagram: Predator mortality m vs Allee threshold l
Note that Kent et al. have a similar picture, but without the bifurcation names. They do not mention these, just the “regions of behaviour” (B and C are not qualitatively different; they are just the transition between node and focus).

6. Analysis
Two-parameter bifurcation diagram: Predator mortality m vs Allee threshold l
Equilibrium:
Only prey
m > 1
Equilibrium:
Predator-prey
Note that Kent et al. have a similar picture, but without the bifurcation names. They do not mention these, just the “regions of behaviour” (B and C are not qualitatively different; they are just the transition between node and focus).
Transcritical bifurcation TC2: transition to a positive equilibrium

7. Analysis
Two-parameter bifurcation diagram: Predator mortality m vs Allee threshold l
Predator-prey
Equilibrium
Predator-prey
Cycles
Note that Kent et al. have a similar picture, but without the bifurcation names. They do not mention these, just the “regions of behaviour” (B and C are not qualitatively different; they are just the transition between node and focus).
Hopf bifurcation H3: transition from equilibrium to stable cycle

8. Problem…
Running time-integrated simulations result in extinction of both populations
Predator-prey
Cycles
Note that Kent et al. have a similar picture, but without the bifurcation names. They do not mention these, just the “regions of behaviour” (B and C are not qualitatively different; they are just the transition between node and focus).
Extinction
Prey AND predator !!
Kent et al. (2003): “Allee effect does not support limit cycles.”

9. Problem… What do we miss?
Running time-integrated simulations result in extinction of both populations
What do we miss?
Predator-prey
Cycles
Note that Kent et al. have a similar picture, but without the bifurcation names. They do not mention these, just the “regions of behaviour” (B and C are not qualitatively different; they are just the transition between node and focus).
Extinction
Prey AND predator !!
Kent et al. (2003): “Allee effect does not support limit cycles.”

10. Let’s take a look at the manifolds…
Orbits starting here go to (0,0)
 Allee effect
Attracting region
Bistability:
Depending on initial conditions
to E0 or E3/Cycle
l = 0.5, m =

11. Connecting orbit
Manifolds of two equilibria connect:
Limit cycle “touches” E1/E2
l = 0.5, m =

12. Extinction l = 0.5, m = 0.735 All orbits go to extinction! “Tunnel”
Bistability lost
l = 0.5, m = 0.735

13. Homotopy method How can we find connecting orbit?
Take some value of m (l fixed)
Starting point at equilibrium E2
Step ε in direction of unstable eigenvalue v
End point: x-value equal to x-value E1 – ζ w
Difference in y-value
Difference in vectors end point and E1

14. Homotopy method Differences  homotopy (“dummy”) parameter
Define boundary conditions on starting and end points orbit (a.o. Beyn, 1990)
Implement in AUTO
Stepwise continuation, including m and homotopy parameter, until parameter = 0

15. Method
Δx1 = 0
ζ*w
ε*v E1 E2
l = 0.5, m = 0.7 (shot in direction unstable eigenvector)
l = 0.5, m = (connecting orbit)

16. Global bifurcation in Allee
Curve G≠
Continue with two bifurcation parameters m and l
Extremely close to Hopf  limit cycles are immediately destroyed
Van Voorn, Hemerik, Boer, Kooi, Math. Biosci. 209 (2007), 451

17. Global bifurcation in Allee
Regions:
Only prey
Predator –prey
0. Extinct
Van Voorn, Hemerik, Boer, Kooi, Math. Biosci. 209 (2007), 451

18. Global bifurcation in Allee
Regions:
Only prey
Predator –prey
0. Extinct
Decrease in predator mortality m  crossing global bifurcation
Van Voorn, Hemerik, Boer, Kooi, Math. Biosci. 209 (2007), 451

19. Conclusions Allee Global bifurcation is the interesting bifurcation
Kent et al. “No limit cycles” explained
Decrease in m  extinction both populations
Overexploitation or ecological suicide
Observe: simple 2D system
How about 3D?
G.A.K. van Voorn, L. Hemerik, M.P. Boer, and B.W. Kooi,
Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect,
Math. Biosci. 209 (2007) 451.

20. RM model
Three-dimensional example: Rozenzweig-MacArthur food chain model
where (Holling type II)
x = variable
d = death rate
note: dimensionless

21. Connections There exist (at least) 2 types of connections:
Homoclinic connection of a limit cycle to itself
Heteroclinic connection of an equilibrium to a limit cycle
Connections are of codimension 0, meaning they exist in a parameter range (rather than just at one specific parameter value, as with Allee)

22. Approach Homotopy method, in AUTO:
Collect data involved equilibria/limit cycles, cycle manifolds and approximate connecting orbits
Define as Boundary Value Problem (BVP)
Please, find details in:
E.J. Doedel, B.W. Kooi, Y.A. Kuznetsov and G.A.K. van Voorn
Continuation of connecting orbits in 3D-ODEs:
(I) Point-to-cycle connections, Int. J. Bif. Chaos, in press (2008a)
(II) Cycle-to-cycle connections, Int. J. Bif. Chaos, in press (2008b)

23. Cycle-to-cycle
3D representation of a connection at a1 = 5, a2 = 0.1, b1 = 3, b2 = 2, d1 = 0.25, d2 =

24. Cycle-to-cycle One pair of connections  Primary branch
Region where connection exists

25. Primary branch One-parameter diagram (Boer et al.) Stable limit cycle
minimum
Saddle limit cycle minimum

26. Primary branch Boundary of chaos: homoclinic orbit disappears Chaos
Global bifurcation
Stable limit cycle
minimum
Saddle limit cycle minimum

27. Point-to-cycle
2D representation of the connection at a1 = 5, a2 = 0.1, b1 = 3, b2 = 2, d1 = 0.25, d2 =

28. Point-to-cycle
2D bifurcation diagram: region where connection (actually two connections) exists, bounded by Tc and Thet

29. Point-to-cycle Example of bistability structure
Region of attraction to attractor with predator, d1 = 0.22, d2 =

30. Conclusions
In Allee (2D): bifurcation of heteroclinic connection boundary of region where bistability exists
In RM (3D): bifurcation of heteroclinic connection also boundary of region where bistability exists, but:
Very complicated structure (Boer et al., 1999)
Depending on initial conditions convergence to attractor (x3 > 0) or extinction predator (x3 = 0)

31. Conclusions Homoclinic cycle connection  boundary of chaos
Global bifurcation analysis vital to understanding of ODE model dynamics
e.g. overexploitation and regions of chaos

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