Neimark–Sacker Bifurcation in a Discrete Host–Parasitoid System with Host Refuge Mechanism

Ardianto Ramadhan, Abadi Abadi

Abstract


This study investigates the local dynamics of a discrete-time host–parasitoid model incorporating a proportional host refuge mechanism. The objective is to clarify how host growth and refuge intensity govern the transitions among extinction, stable coexistence, and oscillatory
dynamics. Linearization, stability conditions, discriminant analysis, and local normal-form
1calculations are used to determine the existence and stability of the extinction and coexistence
fixed points and to characterize their local behavior. The boundary between extinction and
coexistence is identified as a nonhyperbolic threshold supporting a continuum of boundary
fixed points. Within the stable coexistence region, the discriminant separates node-type
convergence from damped spiral convergence. Variation in the refuge fraction is shown to
induce a supercritical Neimark–Sacker bifurcation, producing a stable invariant closed curve
and bounded oscillations on the weak-refuge side near the critical threshold. Numerical phase
portraits, time series, bifurcation diagrams, and a two-parameter map support the analytical
findings. These results provide a unified description of extinction, stable coexistence, and
oscillatory behavior in the refuge model.


Keywords


Host–parasitoid Model; Discrete Dynamical System; Invariant closed curve; Neimark– Sacker bifurcation; proportional refuge

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DOI: https://doi.org/10.18860/cauchy.v11i2.45034

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