Abstract
We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations.
Highlights
Predator-prey interaction shows widespread existence in nature and can take many forms, such as resource-consumer, plant-herbivore, and phytoplankton-zooplankton forms [1]
We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations
Due to the ubiquity and importance of the predator-prey interaction between populations, the dynamics of predatorprey systems have attracted the attention of many scholars, and the research on predator-prey systems has become one of the dominant themes in ecology [2,3,4]
Summary
Predator-prey interaction shows widespread existence in nature and can take many forms, such as resource-consumer, plant-herbivore, and phytoplankton-zooplankton forms [1]. It is employed to demonstrate the predator-prey stability It shows ecological significance for researching the predator-prey models with Crowley-Martin functional response. Ren et al have investigated a Leslie-Gower type discrete predator-prey model with Crowley-Martin functional response and found the existence of Marotto chaos [32]. They controlled chaotic orbits to be a fixed point by a feedback control method. The bifurcations and complex dynamics of the discrete predator-prey system with Crowley-Martin functional response are explored through map (3).
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