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The research identifies the conditions under which fixed points exist and remain stable. A key focus is placed on analyzing different types of bifurcation\u2014such as period doubling (PD), Neimark\u2013Sacker (NS), and strong resonances (1 : 2, 1 : 3, and\u20091 : 4)\u2014occurring at the positive fixed point to uncover their ecological significance. Bifurcation theory is applied to study these dynamics, and the theoretical findings are validated through numerical simulations performed with the MATLAB tool MatContM. In addition, a control mechanism is introduced to mitigate severe instabilities within the system. The results show that predation rate is key to ecological balance, while prey refuge has limited impact on stability. 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