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<p>In this work, we identify the regions of the parameter space in which a predator–prey system-derived from the classical Gause model with a generalized Holling response function and logistic prey growth in the absence of predators-fails to be Liouvillian integrable. Although the model parameters have biological meaning only when restricted to appropriate real domains, our analysis is carried out in the complex setting, which provides a unified algebraic framework; the resulting nonintegrability conditions remain valid in the biologically relevant regime. As a consequence, we establish the nonintegrability of an Abel differential equation of the second kind with polynomial coefficients obtained from the system. Finally, we analyze the existence of a local analytic first integral in neighborhoods of the equilibrium points.</p>

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Liouvillian and Analytic Integrability of a Generalized Gause System

  • 期刊论文

  • 发表日期：

    2026年4月29日

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  • Jorge A. Borrego-Morell

摘要

In this work, we identify the regions of the parameter space in which a predator–prey system-derived from the classical Gause model with a generalized Holling response function and logistic prey growth in the absence of predators-fails to be Liouvillian integrable. Although the model parameters have biological meaning only when restricted to appropriate real domains, our analysis is carried out in the complex setting, which provides a unified algebraic framework; the resulting nonintegrability conditions remain valid in the biologically relevant regime. As a consequence, we establish the nonintegrability of an Abel differential equation of the second kind with polynomial coefficients obtained from the system. Finally, we analyze the existence of a local analytic first integral in neighborhoods of the equilibrium points.