Abstract
This study examines the complexity of a discrete-time predator-prey system with ratio-dependent functional response. We establish algebraically the conditions for existence of fixed points and their stability. We show that under some parametric conditions the system passes through a bifurcation (flip or Neimark-Sacker). Numerical simulations are presented not only to justify theoretical results but also to exhibit new complex behaviors which include phase portraits, orbits of periods 9, 19, and 26, invariant closed circle, and attracting chaotic sets. Moreover, we measure numerically the Lyapunov exponents and fractal dimension to confirm the chaotic dynamics of the system. Finally, a state feedback control method is applied to control chaos which exists in the system.
Highlights
The interaction between predator and prey is one of the most studied topics in ecology and mathematical biology
Many authors qualitatively analysed other types of predator-prey models in which response function of predator depends on prey densities only
A number of respectable researchers [3,4,5] have claimed that in some environments, the response function of predator may depend on the ratio of prey to predator abundance
Summary
The interaction between predator and prey is one of the most studied topics in ecology and mathematical biology. For detailed results in predator-prey systems with ratio-dependent response function, we refer to [6,7,8,9]. When the size of populations is small, the discretization of predator-prey systems is more suitable compared to continuous ones to understand unpredictable dynamic behaviors which exist in the system. There are a few number of articles discussing the dynamics of ratio-dependent discrete-time predator-prey systems [21,22,23]. For discrete ratio-dependent predator-prey systems, it is shown that positive equilibrium is globally asymptotically stable [21], the strong and the weak Allee effects are investigated in [22], and it is discussed that periodic solutions exist in [23].
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