Qualitative behaviour of a ratio-dependent predator-prey system. (English) Zbl 1159.93352
Summary: This paper deals with the qualitative properties of an autonomous system of differential equations, modeling ratio-dependent predator-prey interactions.
This model differs from traditional ratio dependent models essentially in the predator mortality term, the death rate of the predator is not constant but instead increases when there is overcrowding.
We incorporate delay(s) into the system. The most important observation is that as the delay(s) is (are) increased the originally asymptotic stable interior equilibrium loses its stability. Furthermore at a certain critical value a Hopf bifurcation takes place: small amplitude periodic solutions arise.
This model differs from traditional ratio dependent models essentially in the predator mortality term, the death rate of the predator is not constant but instead increases when there is overcrowding.
We incorporate delay(s) into the system. The most important observation is that as the delay(s) is (are) increased the originally asymptotic stable interior equilibrium loses its stability. Furthermore at a certain critical value a Hopf bifurcation takes place: small amplitude periodic solutions arise.
MSC:
| 93D20 | Asymptotic stability in control theory |
| 49N75 | Pursuit and evasion games |
| 93C15 | Control/observation systems governed by ordinary differential equations |
Keywords:
ratio-dependence; steady states; global asymptotic stability; time delay; characteristic equation; Hopf bifurcationReferences:
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