Abstract
Based on Hastings and Powell’s research, this paper extends a three-species food-chain system to fractional-order form, whose dynamics are analyzed and explored. The necessary conditions for generating chaos are confirmed by the stability theory of fractional-order systems, chaos is characterized by its phase diagrams, and bifurcation diagrams prove that the dynamic behaviors of the fractional-order food-chain system are affected by the order. Next, the chaotic control of the fractional-order system is realized by the feedback control method with a good effect in a relative short period. The stability margin of the controlled system is revealed by the theory and numerical analysis. Finally, the results of theory analysis are verified by numerical simulations.
Highlights
Chaos is the common phenomenon in non-linear science, and it is a special motion of non-linear systems
This paper extends an integer-order three-species food-chain system to fractional-order form, and determines the range of order with chaos in the fractional-order ecosystem
Toexplore explorethe theeffect effect environmental factors on the ecosystems, the dynamics of system the varying parameters are analyzed by Lyapunov exponents and bifurcation diagrams
Summary
Chaos is the common phenomenon in non-linear science, and it is a special motion of non-linear systems. Due to the complexity and existence of non-linear effects from natural or unnatural factors, a fractional model of species systems provides a new feasibility to precisely describe dynamic behaviors of the multi-species food-chain ecosystems. The dynamic behaviors of the fractional-order two-species cooperative systems with harvesting were studied, which provided several sufficient conditions to stabilize the system [25]. To further explore the dynamics of the three-species food chain, the corresponding fractional-order system is presented in this paper. The stability control of the three-species food-chain system with the Holling I-type functional response was realized, which suppressed chaos of the system successfully [33]. To balance and develop the ecosystem better, the chaos control of the fractional-order three-species food-chain system is explored and realized in this paper.
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