Stability analysis of a certain class of difference equations by using KAM theory

Kalabušić, Senada; Bešo, Emin; Mujić, Naida; Pilav, Esmir

doi:10.1186/s13662-019-2148-7

Cited by 2 publications

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“…We can see that on a logarithmic scale, map ( 26) is area-preserving. So, it is possible to use KAM theory to investigate the stability property of E p [15]. Computer simulations suggest that when m < 1, all orbits converge to the interior (26) equilibrium E p .…”

Section: Example 1: F (Y) = 1 1+y Mmentioning

confidence: 99%

Dynamics of a class of host–parasitoid models with external stocking upon parasitoids

et al. 2021

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This paper is motivated by the series of research papers that consider parasitoids’ external input upon the host–parasitoid interactions. We explore a class of host–parasitoid models with variable release and constant release of parasitoids. We assume that the host population has a constant rate of increase, but we do not assume any density dependence regulation other than parasitism acting on the host population. We compare the obtained results for constant stocking with the results for proportional stocking. We observe that under a specific condition, the release of a constant number of parasitoids can eventually drive the host population (pests) to extinction. There is always a boundary equilibrium where the host population extinct occurs, and the parasitoid population is stabilized at the constant stocking level. The constant and variable stocking can decrease the host population level in the unique interior equilibrium point; on the other hand, the parasitoid population level stays constant and does not depend on stocking. We prove the existence of Neimark–Sacker bifurcation and compute the approximation of the closed invariant curve. Then we consider a few host–parasitoid models with proportional and constant stocking, where we choose well-known probability functions of parasitism. By using the software package Mathematica we provide numerical simulations to support our study.

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Section: Example 1: F (Y) = 1 1+y Mmentioning

confidence: 99%

Dynamics of a class of host–parasitoid models with external stocking upon parasitoids

et al. 2021

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Stability of May’s Host–Parasitoid model with variable stocking upon parasitoids

Kalabušić

Pilav

2021

Int. J. Biomath.

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Using the Kolmogorov–Arnold–Mozer (KAM) theory, we investigate the stability of May’s host–parasitoid model’s solutions with proportional stocking upon the parasitoid population. We show the existence of the extinction, boundary, and interior equilibrium points. When the host population’s intrinsic growth rate and the releasement coefficient are less than one, both populations are extinct. There are an infinite number of boundary equilibrium points, which are nonhyperbolic and stable. Under certain conditions, there appear 1:1 nonisolated resonance fixed points for which we thoroughly described dynamics. Regarding the interior equilibrium point, we use the KAM theory to prove its stability. We give a biological meaning of obtained results. Using the software package Mathematica, we produce numerical simulations to support our findings.

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Stability analysis of a certain class of difference equations by using KAM theory

Cited by 2 publications

References 25 publications

Dynamics of a class of host–parasitoid models with external stocking upon parasitoids

Dynamics of a class of host–parasitoid models with external stocking upon parasitoids

Stability of May’s Host–Parasitoid model with variable stocking upon parasitoids

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