ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Equilibrium
CBT Bifurcation
Cusp-Bodanov-Takens Bifurcation
in a Predator-Prey Type of
Dynamical System
with Time-Periodic Perturbation
Periodic
Perturbation
Conclusion
References
Kus Prihantoso Krisnawan
Johan Matheus Tuwankotta
October 24th , 2011
ITB-Bandung Indonesia
Outline
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Predator-Prey System
Our goal
Scaling
Equilibrium
Cusp-Bogdanov-Takens Bifurcation
CBT Bifurcation
Periodic
Perturbation
System with Time-Periodic Perturbation
Conclusion
Conclusion
References
References
Predator-Prey System
ICREM 5
Krisnawan
Tuwankotta
Given Predator-Prey system
Outline
Pred-Prey
ẋ
= ax − λx 2 − yP(x)
ẏ
= −δy − µy 2 + yQ(x)
Our goal
Scaling
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
where a, λ, δ > 0 dan µ ≥ 0,
mx
αx 2 + βx + 1
Q(x) = cP(x)
P(x) =
√
where α ≥ 0 dan m, c > 0 dan β > −2 α.
(1)
Related works
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Broer, et al., Bifurcation of a predator-prey model with
non-monotonic response function, Comptes Rendus
Mathematique, V. 341, (2005) 601-604.
Broer, et al., A predator-prey model with non-monotonic response
function, Regular and Chaotic Dynamics 11(2) (2006),155-165.
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
Broer, et al., Dynamics of a predator-prey model with
non-monotonic response function, DCDS-A 18(2-3) (2007),
221-251.
Haryanto, E., Tuwankotta, J.M., Swallowtail in predator-prey type
of system with time-periodic perturbation, submitted to ICREM
2011.
Tuwankotta, J.M., Haryanto, E., On periodic solution of a
predator-prey type of dynamical system with time-periodic
perturbation, submitted to ICREM 2011.
Our goal:
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
Show that there is a codimension 3 bifurcation
(Cusp-Bogdanov-Takens) on the system.
Our goal:
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Show that there is a codimension 3 bifurcation
(Cusp-Bogdanov-Takens) on the system.
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
We give time perturbation on the carrying capacity to see its
influence to the bifurcation
the variation is λ = λ0 (1 + sin(ωt)).
Scaling
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Define:
t
=
1
τ,
a
x=
a
xe,
cm
Our goal
Scaling
Equilibrium
e
δ = aδ,
CBT Bifurcation
Periodic
Perturbation
Conclusion
µ = me
µ,
a
e
ye,
λ = cmλ,
m
c 2 m2
cm e
α=
α
e, β =
β
2
a
a
y=
substitute to the system (1) and discard the tilde on the new
variable:
References
ẋ
ẏ
y
)
+ βx + 1
y
).
= y (−δ − µy +
2
αx + βx + 1
= x(1 − λx −
αx 2
We fix the value λ = 0.01 and δ = 1.1.
(2)
Equilibrium points
ICREM 5
Krisnawan
Tuwankotta
Outline
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 +
3
(0, − µδ ), for µ 6= 0.
Pred-Prey
Our goal
Scaling
0.01
0.0001+0.01β+α .
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
4
for x 6= 0 and y 6= 0, the equilibrium point of the system
(2) is the solution of
y
αx 2 + βx + 1
x
−1.1 − µy +
αx 2 + βx + 1
1 − 0.01x −
=
0
=
0
Equilibrium points
ICREM 5
Krisnawan
Tuwankotta
Outline
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 +
3
(0, − µδ ), for µ 6= 0.
Pred-Prey
Our goal
Scaling
0.01
0.0001+0.01β+α .
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
4
for x 6= 0 and y 6= 0, the equilibrium point of the system
(2) is the solution of
y
αx 2 + βx + 1
x
−1.1 − µy +
αx 2 + βx + 1
1 − 0.01x −
=
0
=
0
The Cusp-Bogdanov-Takens bifurcation point is
(ᾱ, β̄, µ̄) = (0.0003986797, 0.8459684, 0.0013)
at (x̄, ȳ) = (49.38394909, 22.14429463).
Cusp-Bogdanov-Takens Bifurcation
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
Cusp-Bogdanov-Takens Bifurcation Diagram
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
Figure: Cusp-Bogdanov-Takens Bifurcation Diagram and its Phase Portrait
System with Periodic Perturbation
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Before the periodic perturbation, the system is:
y
)
ẋ = x(1 − λx −
2
αx + βx + 1
y
).
ẏ = y (−δ − µy +
2
αx + βx + 1
(3)
Scaling
Equilibrium
CBT Bifurcation
Parameter λ, is varied by
Periodic
Perturbation
λ = λ0 (1 + sin (ωt))
Conclusion
References
(4)
where λ0 , ω are constants and > 0 is the perturbation
parameter. Thus we get
ẋ
= x − λx 2 − λ sin(ωt)x 2 −
ẏ
= −δy − µy 2 +
αx 2
αx 2
xy
+ βx + 1
xy
+ βx + 1
(5)
the strategy
ICREM 5
Define
Krisnawan
Tuwankotta
·
u = ωv + u − u(u 2 + v 2 )
·
Outline
v
Pred-Prey
= −ωu + v − v (u 2 + v 2 ).
(6)
Our goal
Scaling
Equilibrium
System (6) has solutions u = sin ωt and v = cos ωt.
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
·
u = ωv + u − u(u 2 + v 2 )
v
·
= −ωu + v − v (u 2 + v 2 )
·
= x − λx 2 − λεux 2 −
x
·
y
xy
αx 2 + βx + 1
xy
= −δy − µy 2 +
2
αx + βx + 1
We investigate the system at ω = 1, λ = 0.01, and δ = 1.1
for the perturbations = 0.0003, = 0.001, and = 0.02.
Bifurcation Diagrams after the Perturbations
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
Figure: Bifurcation diagram of the system (5) for > 0
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Figure: Zoom in K (left), L (middle), dan M (right)
Conclusion
References
Figure: Zoom in K1 (left), L1 (middle), dan M1 (right)
Conclusion
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Equilibrium
CBT Bifurcation
Periodic
Perturbation
1. For the value λ = 0.01 and δ = 1.1 we get that the
Cusp-Bogdanov-Takens bifurcation happens at
(ᾱ, β̄, µ̄) = (0.0003986797, 0.8459684, 0.0013) for
(x̄, ȳ ) = (49.38394909, 22.14429463).
Conclusion
References
2. We found that the Cusp-Bogdanov-Takens bifurcation
is persist to the perturbation for > 0.
References
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
1. Broer, et al., Bifurcation of a predator-prey model with non-monotonic
response 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-monotonic
response function, DCDS-A 18(2-3) (2007), 221-251.
4. Doedel, E.J, et al. AUTO 2000: Continuation and Bifurcation Software for
Ordinary Differential Equations, User’s Guide, Concordia University,
Montreal, Canada 2000.
5. Freedman, H.I., Wolkowics, G.S., Predator-prey Systems with Group
Defense: The Paradox of Enrichment Revisited, Bull. Math. Biol. (1986), pp.
493-508.
6. Haryanto, E., Tuwankotta, J.M., Swallowtail in predator-prey type of system
with 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 type
of dynamical system with time-periodic perturbation, submitted to ICREM
2011.
ICREM 5
Krisnawan
Tuwankotta
Outline
Pred-Prey
Our goal
Scaling
Equilibrium
CBT Bifurcation
Periodic
Perturbation
Conclusion
References
THANK YOU
MATUR NUWUN