ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Cusp-Bodanov-Takens Bifurcationin a Predator-Prey Type of

Dynamical Systemwith Time-Periodic Perturbation

Kus Prihantoso KrisnawanJohan Matheus Tuwankotta

October 24th, 2011

ITB-Bandung Indonesia

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Outline

Predator-Prey System

Cusp-Bogdanov-Takens Bifurcation

System with Time-Periodic Perturbation

Conclusion

References

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Predator-Prey System

Given Predator-Prey system

x = ax − λx2 − yP(x)

y = −δy − µy2 + yQ(x) (1)

where a, λ, δ > 0 dan µ ≥ 0,

P(x) =mx

αx2 + βx + 1Q(x) = cP(x)

where α ≥ 0 dan m, c > 0 dan β > −2√α.

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Related works

Broer, et al., Bifurcation of a predator-prey model withnon-monotonic response function, Comptes RendusMathematique, V. 341, (2005) 601-604.

Broer, et al., A predator-prey model with non-monotonic responsefunction, Regular and Chaotic Dynamics 11(2) (2006),155-165.

Broer, et al., Dynamics of a predator-prey model withnon-monotonic response function, DCDS-A 18(2-3) (2007),221-251.

Haryanto, E., Tuwankotta, J.M., Swallowtail in predator-prey typeof system with time-periodic perturbation, submitted to ICREM2011.

Tuwankotta, J.M., Haryanto, E., On periodic solution of apredator-prey type of dynamical system with time-periodicperturbation, submitted to ICREM 2011.

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Our goal:

Show that there is a codimension 3 bifurcation(Cusp-Bogdanov-Takens) on the system.

We give time perturbation on the carrying capacity to see itsinfluence to the bifurcationthe variation is λ = λ0(1 + ε sin(ωt)).

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Our goal:

Show that there is a codimension 3 bifurcation(Cusp-Bogdanov-Takens) on the system.

We give time perturbation on the carrying capacity to see itsinfluence to the bifurcationthe variation is λ = λ0(1 + ε sin(ωt)).

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Scaling

Define:

t =1aτ, x =

acm

x , y =am

y , λ = cmλ,

δ = aδ, µ = mµ, α =c2m2

a2 α, β =cmaβ

substitute to the system (1) and discard the tilde on the newvariable:

x = x(1− λx − yαx2 + βx + 1

)

y = y(−δ − µy +y

αx2 + βx + 1). (2)

We fix the value λ = 0.01 and δ = 1.1.

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Outline

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Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Equilibrium points

The system (2) has some equilibrium points:

1 O(0,0), where λ1 = 1 and λ2 = −1.1.2 (100,0), λ1 = −1 and λ2 = −1.1 + 0.01

0.0001+0.01β+α .

3 (0,− δµ), for µ 6= 0.

4 for x 6= 0 and y 6= 0, the equilibrium point of the system(2) is the solution of

1 − 0.01x −y

αx2 + βx + 1= 0

−1.1 − µy +x

αx2 + βx + 1= 0

The Cusp-Bogdanov-Takens bifurcation point is(α, β, µ) = (0.0003986797,0.8459684,0.0013)at (x , y) = (49.38394909,22.14429463).

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Equilibrium points

The system (2) has some equilibrium points:

1 O(0,0), where λ1 = 1 and λ2 = −1.1.2 (100,0), λ1 = −1 and λ2 = −1.1 + 0.01

0.0001+0.01β+α .

3 (0,− δµ), for µ 6= 0.

4 for x 6= 0 and y 6= 0, the equilibrium point of the system(2) is the solution of

1 − 0.01x −y

αx2 + βx + 1= 0

−1.1 − µy +x

αx2 + βx + 1= 0

The Cusp-Bogdanov-Takens bifurcation point is(α, β, µ) = (0.0003986797,0.8459684,0.0013)at (x , y) = (49.38394909,22.14429463).

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Cusp-Bogdanov-Takens Bifurcation

ICREM 5

KrisnawanTuwankotta

Outline

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Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Cusp-Bogdanov-Takens Bifurcation Diagram

Figure: Cusp-Bogdanov-Takens Bifurcation Diagram and its Phase Portrait

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

System with Periodic Perturbation

Before the periodic perturbation, the system is:

x = x(1− λx − yαx2 + βx + 1

)

y = y(−δ − µy +y

αx2 + βx + 1). (3)

Parameter λ, is varied by

λ = λ0 (1 + ε sin (ωt)) (4)

where λ0, ω are constants and ε > 0 is the perturbationparameter. Thus we get

x = x − λx2 − λε sin(ωt)x2 − xyαx2 + βx + 1

y = −δy − µy2 +xy

αx2 + βx + 1(5)

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Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

the strategy

Define·u = ωv + u − u(u2 + v2)·v = −ωu + v − v(u2 + v2). (6)

System (6) has solutions u = sinωt and v = cosωt .

·u = ωv + u − u(u2 + v2)·v = −ωu + v − v(u2 + v2)·x = x − λx2 − λεux2 − xy

αx2 + βx + 1·y = −δy − µy2 +

xyαx2 + βx + 1

We investigate the system at ω = 1, λ = 0.01, and δ = 1.1for the perturbations ε = 0.0003, ε = 0.001, and ε = 0.02.

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KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Bifurcation Diagrams after the Perturbations

Figure: Bifurcation diagram of the system (5) for ε > 0

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CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Figure: Zoom in K (left), L (middle), dan M (right)

Figure: Zoom in K1 (left), L1 (middle), dan M1 (right)

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

Conclusion

1. For the value λ = 0.01 and δ = 1.1 we get that theCusp-Bogdanov-Takens bifurcation happens at(α, β, µ) = (0.0003986797,0.8459684,0.0013) for(x , y) = (49.38394909,22.14429463).

2. We found that the Cusp-Bogdanov-Takens bifurcationis persist to the perturbation for ε > 0.

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

References

1. Broer, et al., Bifurcation of a predator-prey model with non-monotonicresponse function, Comptes Rendus Mathematique, V. 341, (2005) 601-604.

2. Broer, et al., A predator-prey model with non-monotonic response function,Regular and Chaotic Dynamics 11(2) (2006),155-165.

3. Broer, et al., Dynamics of a predator-prey model with non-monotonicresponse function, DCDS-A 18(2-3) (2007), 221-251.

4. Doedel, E.J, et al. AUTO 2000: Continuation and Bifurcation Software forOrdinary Differential Equations, User’s Guide, Concordia University,Montreal, Canada 2000.

5. Freedman, H.I., Wolkowics, G.S., Predator-prey Systems with GroupDefense: The Paradox of Enrichment Revisited, Bull. Math. Biol. (1986), pp.493-508.

6. Haryanto, E., Tuwankotta, J.M., Swallowtail in predator-prey type of systemwith time-periodic perturbation, submitted to ICREM 2011.

7. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, second edition,Springer-Verlag New York, Inc., 1998.

8. Tuwankotta, J.M., Haryanto, E., On periodic solution of a predator-prey typeof dynamical system with time-periodic perturbation, submitted to ICREM2011.

ICREM 5

KrisnawanTuwankotta

Outline

Pred-PreyOur goal

Scaling

Equilibrium

CBT Bifurcation

PeriodicPerturbation

Conclusion

References

THANK YOUMATUR NUWUN