| 自全国各地100多位理事Road Machinery Branch of China Highway and Transportation Society He |
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| 作者单位： | Zhao Zhong（Dept. of Appl. Math., Dalian Univ. of Tech., Dalian 116024, China;Dept. of Math., Huanghuai College, Zhumadian 463000, China） ;
Song Xinyu（Dept. of Math., Xinyan Normal Univ., Henan 464000, China） ; |
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| 基金项目： | 国家自然科学基金
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河南省高校杰出科研人才创新工程项目
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the Scientific Research Foundation of Education Ministry for the Returned Overseas Chinese Scholars |
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| 摘 要： | In this paper, a three dimensional ratio-dependent chemostat model with periodically pulsed input is considered. By using the discrete dynamical system determined by the stroboscopic map and Floquet theorem, an exact periodic solution with positive concentrations of substrate and predator in the absence of prey is obtained. When β is less than some critical value the boundary periodic solution (xs(t), 0, zs(t)) is locally stable, and when β is larger than the critical value there are periodic oscillations in substrate, prey and predator. Increasing the impulsive period τ, the system undergoes a series of period-doubling bifurcation leading to chaos, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex.
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| 收稿时间： | 28 March 2007 |
BIFURCATION AND COMPLEXITY IN A RATIO-DEPENDENT PREDATOR-PREY CHEMOSTAT WITH PULSED INPUT |
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| Authors: | Zhao Zhong Song Xinyu |
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| Institution: | (1) Dept. of Appl. Math., Dalian Univ. of Tech., Dalian, 116024, China;(2) Dept. of Math., Huanghuai College, Zhumadian, 463000, China;(3) Dept. of Math., Xinyan Normal Univ., Henan, 464000, China |
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| Abstract: | In this paper, a three dimensional ratio-dependent chemostat model with periodically pulsed input is considered. By using the discrete dynamical system determined by the stroboscopic map and Floquet theorem, an exact periodic solution with positive concentrations of substrate and predator in the absence of prey is obtained. When β is less than some critical value the boundary periodic solution (xs(t), 0, zs(t)) is locally stable, and when β is larger than the critical value there are periodic oscillations in substrate, prey and predator. Increasing the impulsive period τ, the system undergoes a series of period-doubling bifurcation leading to chaos, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex. |
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| Keywords: | chemostat model periodical solution stability bifurcation |
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