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In this article, we determine all fixed points of quadratic stochastic operators with invariant coordinates, and we conduct a comprehensive analysis of the local and global dynamics of the given operator. In particular, we investigate the stability of the fixed points and classify their types using appropriate dynamical system techniques. Furthermore, we study the asymptotic behavior of trajectories generated by the operator and establish conditions under which convergence occurs. Several illustrative examples are provided to demonstrate the theoretical results and to highlight different dynamical regimes, including stability, periodicity, and convergence to boundary points.
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