Abstract
The local behavior with topological classifications, bifurcation analysis, chaos control, boundedness, and global attractivity of the discrete-time Kolmogorov model with piecewise-constant argument are investigated. It is explored that Kolmogorov model has trivial and two semitrival fixed points for all involved parameters, but it has an interior fixed point under definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrival, and interior fixed points. Further for the discrete Kolmogorov model, existence of periodic points is also investigated. It is also investigated the occurrence of bifurcations at interior fixed point and proved that at interior fixed point, there exists no bifurcation, except flip bifurcation by bifurcation theory. Next, feedback control method is utilized to stabilize chaos existing in discrete Kolmogorov model. Boundedness and global attractivity of the discrete Kolmogorov model are also investigated. Finally, obtained results are numerically verified.
Highlights
It is pointed out that in theoretical ecology, mutualist behavior of symbiosis or mutualism is very significant [1]. is field is not widely studied as the other fields of mathematical biology even for two species, its importance is equal to the other competitive interactions such as host-parasitoid and preypredator interactions
Paper Structure. e rest of the paper is structured as follows: Section 2 relates with the investigation of topological classifications of discrete Kolmogorov model (3) at fixed points
Work e work is about the local dynamical characteristics at fixed points, existence of periodic points, boundedness, global attractivity, chaos control, and bifurcations of a discrete Kolmogorov model with piecewise-constant argument
Summary
Motivated from aforementioned studies, the objective of the present work is to explore the global dynamics, bifurcations, and chaos in a discrete Kolmogorov model with piecewise-constant argument (3). Topological classifications at fixed points of the discrete Kolmogorov model (3) by linear stability theory. E rest of the paper is structured as follows: Section 2 relates with the investigation of topological classifications of discrete Kolmogorov model (3) at fixed points. Topological classifications at fixed points P, Q, R, and S are explored for the completion of this section. Based on eigenvalues λ1,2, we will summarize the topological classifications at Q of the discrete-time Kolmogorov model (3) as Table 3. We will summarize the topological classifications of discrete Kolmogorov model (3) at S as follows
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