In this paper, we apply a new approach to a special class of discrete time evolution models and establish a solid mathematical foundation to analyse them. We propose new single and multi-species evolutionary competition models using the evolutionary game theory that require a more advanced mathematical theory to handle effectively. A key feature of this new approach is to consider the discrete models as non-autonomous difference equations. Using the powerful tools and results developed in our recent work [E. D'Aniello and S. Elaydi, The structure of ω-limit sets of asymptotically non-autonomous discrete dynamical systems, Discr. Contin. Dyn. Series B. 2019 (to appear).], we embed the non-autonomous difference equations in an autonomous discrete dynamical systems in a higher dimension space, which is the product space of the phase space and the space of the functions defining the non-autonomous system. Our current approach applies to two scenarios. In the first scenario, we assume that the trait equations are decoupled from the equations of the populations. This requires specialized biological and ecological assumptions which we clearly state. In the second scenario, we do not assume decoupling, but rather we assume that the dynamics of the trait is known, such as approaching a positive stable equilibrium point which may apply to a much broader evolutionary dynamics.

中文翻译：

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离散进化种群模型：一种新方法。

在本文中，我们将一种新方法应用于一类特殊的离散时间演化模型，并为分析它们建立了坚实的数学基础。我们使用进化博弈论提出了新的单物种和多物种进化竞争模型，该模型需要更先进的数学理论来有效处理。这种新方法的一个关键特征是将离散模型视为非自治差分方程。使用我们最近工作中开发的强大工具和结果[E. D'Aniello和S.Elaydi，渐近非自治离散动力系统的ω-极限集的结构，Discr。继续。达因 系列B. 2019（即将出现）。]，我们将非自治差分方程式嵌入到高维空间中的自治离散动力系统中，它是相空间的乘积空间和定义非自治系统的函数的空间。我们当前的方法适用于两种情况。在第一种情况下，我们假设特征方程与总体方程解耦。这需要我们明确指出的专门的生物学和生态学假设。在第二种情况下，我们不假设解耦，而是假设特征的动力学是已知的，例如接近正稳定平衡点，该平衡点可能适用于更广泛的进化动力学。这需要我们明确指出的专门的生物学和生态学假设。在第二种情况下，我们不假设解耦，而是假设特征的动力学是已知的，例如接近正稳定平衡点，该平衡点可能适用于更广泛的进化动力学。这需要我们明确指出的专门的生物学和生态学假设。在第二种情况下，我们不假设解耦，而是假设特征的动力学是已知的，例如接近正稳定平衡点，该平衡点可能适用于更广泛的进化动力学。