Abstract
A discrete-time parasite-host system with bifurcation is investigated in detail in this paper. The existence and stability of nonnegative fixed points are explored and the conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. And we also prove the chaos in the sense of Marotto. The numerical simulations not only illustrate the consistence with the theoretical analysis, but also exhibit other complex dynamical behaviors, such as bifurcation diagrams, Maximum Lyapunov exponents, and phase portraits. More specifically, when the integral step size is chosen as a bifurcation parameter, this paper presents the finding of period orbits, attracting invariant cycles and chaotic attractors of the discrete-time parasite-host system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point by using the feedback control method.
Highlights
In the theory of ecology, population dynamics are generally governed by continuous-time and discrete-time systems
We prove the chaos in the sense of Marotto
We study the stability of these fixed points
Summary
In the theory of ecology, population dynamics are generally governed by continuous-time and discrete-time systems. We apply the forward Euler Scheme to discrete the parasite-past model and mainly focus on the existence and stability of nonnegative fixed points and flip bifurcation, Neimark-Sacker bifurcation, and possible chaos in the sense of Marotto’s definition [32] in the discrete-time parasite-host system by using the center manifold theorem [33] and the bifurcation theory [17,18,19,20, 33,34,35,36]. We conclude this paper with comments and discuss the future work
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