Dynamic complexities in a discrete predator-prey system with lower critical point for the prey
Dynamic complexities in a discrete predator-prey system with lower critical point for the prey
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DOI：
10.1016/j.matcom.2014.04.010
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Limin Zhang;Chaofeng Zhang;Min Zhao
中科院分区：
文献类型：
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作者：
Limin Zhang;Chaofeng Zhang;Min Zhao
本文提出并分析了一类离散的捕食-食饵系统。假设食饵种群有一个较低的临界点，也被称为灭绝阈值。这种行为已经在许多开花植物、许多鱼类、流行病学等方面得到了报道。利用流形定理和分支理论，得到了系统存在翻转分支和Hopf分支的条件。数值模拟，包括分岔图，相图和最大李雅普诺夫指数，不仅与理论分析的一致性，但也显示了其他复杂的动力学和某些生物现象.复杂动力学包括准周期性、导致混沌的倍周期分岔、周期窗口混沌带、间歇性、超瞬态等，模拟结果表明，适当的增长率可以使系统稳定，但过高的增长率可能使稳定的系统失稳为更复杂的动力学.同时，模拟表明，当下临界点参数较小时，系统是稳定的，但当下临界点参数增大到超过临界值时，系统从准周期转变为崩溃。此外，根据生物学的观点解释了模拟结果。
In this paper, a discrete predator–prey system is proposed and analyzed. It is assumed that the prey population has a lower critical point, which is also referred to as extinction threshold. Such behavior has been reported for many flowering plants, many fishes, epidemiology, and so on. The existence and stability of nonnegative fixed points are explored. The conditions for the existence of flip bifurcation and Hopf bifurcation are obtained by using manifold theorem and bifurcation theory. Numerical simulations, including bifurcation diagrams, phase portraits and Maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit other complex dynamics and certain biological phenomena. Complex dynamics include quasi-periodicity, period-doubling bifurcations leading to chaos, chaotic bands with periodic windows, intermittent, supertransient, and so on. Simulations suggest that appropriate growth rate can stabilize the system, but the high growth rate may destabilize the stable system into more complex dynamics. As well, simulations suggest that the system is stable when the lower critical point parametercis small, but whencincreases beyond the critical values, the system changes from quasi-period to collapses. Furthermore, the simulated results are explained according to a biological point of view.