The dynamical behaviors of a Ivlev-type two-prey two-predator system with impulsive effect.(English)Zbl 1273.34053
Summary: Considering the strategy of integrated pest management (IPM), a class of two-prey two-predator system with the Ivlev-type functional response and impulsive effect at different fixed times is established. By using impulsive comparison theorem, Floquent theory and small amplitude perturbation, the sufficient conditions for the system to be extinct of prey and permanence are proved. Moreover, we give two sufficient conditions for the extinction of one of two prey species and the permanence of the remaining three species. A numerical simulation shows that there exist complex dynamics in the system, such as symmetry-breaking pitchfork bifurcation, periodic doubling bifurcation, chaos and periodic halving cascade. Finally, a brief discussion is given.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 92D25 | Population dynamics (general) |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
Keywords:
prey-predator;impulsive differential system;impulsive comparison theorem;chaos;integrated pest managementCite
Full Text:DOI
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