The existence and nonexistence of semi-trivial or coexistence steady-state solutions of one-dimensional competition models in an unstirred chemostat are studied by establishing new results on systems of Hammerstein integral equations via the classical fixed point index theory. We provide three ranges for the two parameters involved in the competition models under which the models have no semi-trivial and coexistence steady-state solutions or have semi-trivial steady-state solutions but no coexistence steady-state solutions or have semi-trivial or coexistence steady-state solutions. It remains open to find the largest range for the two parameters under which the models have only coexistence steady-state solutions. We apply the new results on systems of Hammerstein integral equations to obtain results on steady-state solutions of systems of reaction-diffusion equations with general separated boundary conditions. Such type of results have not been studied in the literature. However, these results are very useful for studying the competition models in an unstirred chemostat. Our results on Hammerstein integral equations and differential equations generalize and improve some previous results.

中文翻译：

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基于不动点指数理论的非搅拌恒化器中一维竞争模型的稳态解

通过经典不动点指数理论建立Hammerstein积分方程组的新结果，研究了非搅拌恒化器中一维竞争模型的半平凡或共存稳态解的存在与不存在。我们为竞争模型中涉及的两个参数提供了三个范围，其中模型没有半平凡和共存的稳态解或具有半平凡的稳态解但没有共存的稳态解或具有半平凡或共存稳态解决方案。找到模型只有共存稳态解的两个参数的最大范围仍然是开放的。我们将新结果应用于 Hammerstein 积分方程组，以获得具有一般分离边界条件的反应扩散方程组的稳态解的结果。此类结果尚未在文献中进行研究。然而，这些结果对于研究未搅拌恒化器中的竞争模型非常有用。我们在 Hammerstein 积分方程和微分方程上的结果推广和改进了以前的一些结果。