Stability, delay, and chaotic behavior in a LotkaVolterra predatorprey system

1.
Department of Systems Engineering, Faculty of Engineering, Shizuoka University, Johoku 351, Hamamatsu, Shizuoka 4328561

Received:
01 February 2005
Accepted:
29 June 2018
Published:
01 November 2005


MSC :
34D35.


We consider the following LotkaVolterra predatorprey system with two delays:
$x'(t) = x(t) [r_1  ax(t \tau_1)  by(t)]$
$y'(t) = y(t) [r_2 + cx(t)  dy(t \tau_2)]$ (E)
We show that a positive equilibrium of system (E) is globally asymptotically stable for small delays. Critical values of time delay through which system (E) undergoes a Hopf bifurcation are analytically determined. Some numerical simulations suggest an existence of subcritical Hopf bifurcation near the critical values of time delay. Further system (E) exhibits some chaotic behavior when $tau_2$ becomes large.
Citation: S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a LotkaVolterra predatorprey system[J]. Mathematical Biosciences and Engineering, 2006, 3(1): 173187. doi: 10.3934/mbe.2006.3.173

Abstract
We consider the following LotkaVolterra predatorprey system with two delays:
$x'(t) = x(t) [r_1  ax(t \tau_1)  by(t)]$
$y'(t) = y(t) [r_2 + cx(t)  dy(t \tau_2)]$ (E)
We show that a positive equilibrium of system (E) is globally asymptotically stable for small delays. Critical values of time delay through which system (E) undergoes a Hopf bifurcation are analytically determined. Some numerical simulations suggest an existence of subcritical Hopf bifurcation near the critical values of time delay. Further system (E) exhibits some chaotic behavior when $tau_2$ becomes large.
References

