Cusp-Bodanov-Takens Bifurcation in a Predator-Prey Type of Dynamical System with Time-Periodic Perturbation

Transcription

1 Cusp-Bodanov-Takens Bifurcation in a Predator-Prey Type of Dynamical System with Time- Kus Prihantoso Johan Matheus October 24 th, 2011 ITB-Bandung Indonesia

2 Predator-Prey System Cusp-Bogdanov-Takens Bifurcation System with Time-

3 Predator-Prey System Given Predator-Prey system where a, λ, δ > 0 dan µ 0, ẋ = ax λx 2 yp(x) ẏ = δy µy 2 + yq(x) (1) P(x) = mx αx 2 + βx + 1 Q(x) = cp(x) where α 0 dan m, c > 0 dan β > 2 α.

4 Related works Broer, et al., Bifurcation of a predator-prey model with non-monotonic response function, Comptes Rendus Mathematique, V. 341, (2005) Broer, et al., A predator-prey model with non-monotonic response function, Regular and Chaotic Dynamics 11(2) (2006), Broer, et al., Dynamics of a predator-prey model with non-monotonic response function, DCDS-A 18(2-3) (2007), Haryanto, E.,, J.M., Swallowtail in predator-prey type of system with time-periodic perturbation, submitted to ICREM 2011., J.M., Haryanto, E., On periodic solution of a predator-prey type of dynamical system with time-periodic perturbation, submitted to ICREM 2011.

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

6 : Show that there is a codimension 3 bifurcation (Cusp-Bogdanov-Takens) on the system. We give time perturbation on the carrying capacity to see its influence to the bifurcation the variation is λ = λ 0 (1 + ɛ sin(ωt)).

7 Define: t = 1 a τ, x = a cm x, y = a ỹ, λ = cm λ, m δ = a δ, µ = m µ, α = c2 m 2 a 2 α, β = cm a β substitute to the system (1) and discard the tilde on the new variable: ẋ = y x(1 λx αx 2 + βx + 1 ) ẏ = y y( δ µy + αx 2 ). + βx + 1 (2) We fix the value λ = 0.01 and δ = 1.1.

8 points The system (2) has some equilibrium points: 1 O(0, 0), where λ 1 = 1 and λ 2 = (100, 0), λ 1 = 1 and λ 2 = β+α. 3 (0, δ µ ), for µ 0. 4 for x 0 and y 0, the equilibrium point of the system (2) is the solution of y x αx 2 + βx + 1 x 1.1 µy + αx 2 + βx + 1 = 0 = 0

9 points The system (2) has some equilibrium points: 1 O(0, 0), where λ 1 = 1 and λ 2 = (100, 0), λ 1 = 1 and λ 2 = β+α. 3 (0, δ µ ), for µ 0. 4 for x 0 and y 0, the equilibrium point of the system (2) is the solution of y x αx 2 + βx + 1 x 1.1 µy + αx 2 + βx + 1 = 0 = 0 The Cusp-Bogdanov-Takens bifurcation point is (ᾱ, β, µ) = ( , , ) at ( x, ȳ) = ( , ).

10 Cusp-Bogdanov-Takens Bifurcation

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

12 System with Before the periodic perturbation, the system is: y ẋ = x(1 λx αx 2 + βx + 1 ) ẏ = y( δ µy + Parameter λ, is varied by y αx 2 ). (3) + βx + 1 λ = λ 0 (1 + ɛ sin (ωt)) (4) where λ 0, ω are constants and ɛ > 0 is the perturbation parameter. Thus we get ẋ = x λx 2 λɛ sin(ωt)x 2 xy αx 2 + βx + 1 ẏ = δy µy 2 xy + αx 2 + βx + 1 (5)

13 the strategy Define u = ωv + u u(u 2 + v 2 ) v = ωu + v v(u 2 + v 2 ). (6) System (6) has solutions u = sin ωt and v = cos ωt. u = ωv + u u(u 2 + v 2 ) v = ωu + v v(u 2 + v 2 ) x = x λx 2 λεux 2 y = δy µy 2 + xy αx 2 + βx + 1 xy αx 2 + βx + 1 We investigate the system at ω = 1, λ = 0.01, and δ = 1.1 for the perturbations ɛ = , ɛ = 0.001, and ɛ = 0.02.

14 Bifurcation Diagrams after the s Figure: Bifurcation diagram of the system (5) for ɛ > 0

15 Figure: Zoom in K (left), L (middle), dan M (right) Figure: Zoom in K1 (left), L1 (middle), dan M1 (right)

16 1. For the value λ = 0.01 and δ = 1.1 we get that the Cusp-Bogdanov-Takens bifurcation happens at (ᾱ, β, µ) = ( , , ) for ( x, ȳ) = ( , ). 2. We found that the Cusp-Bogdanov-Takens bifurcation is persist to the perturbation for ɛ > 0.

17 1. Broer, et al., Bifurcation of a predator-prey model with non-monotonic response function, Comptes Rendus Mathematique, V. 341, (2005) Broer, et al., A predator-prey model with non-monotonic response function, Regular and Chaotic Dynamics 11(2) (2006), Broer, et al., Dynamics of a predator-prey model with non-monotonic response function, DCDS-A 18(2-3) (2007), Doedel, E.J, et al. AUTO 2000: Continuation and Bifurcation Software for Ordinary Differential Equations, User s Guide, Concordia University, Montreal, Canada Freedman, H.I., Wolkowics, G.S., Predator-prey Systems with Group Defense: The Paradox of Enrichment Revisited, Bull. Math. Biol. (1986), pp Haryanto, E.,, J.M., Swallowtail in predator-prey type of system with time-periodic perturbation, submitted to ICREM Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, second edition, Springer-Verlag New York, Inc., , J.M., Haryanto, E., On periodic solution of a predator-prey type of dynamical system with time-periodic perturbation, submitted to ICREM 2011.

18 THANK YOU MATUR NUWUN