In recent decades, studying the behavior of biological species has become one of the most fascinating areas of applied mathematics. The high importance of conservation of rare species in nature has prompted researchers in various fields to pay particular attention to this issue. Therefore, it is essential to develop mathematical models that examine the dynamics of their behavior. On the other hand, the development of new concepts in numerical analysis has enabled us to preserve more information on the evolutionary behavior history of a dynamic system and to use it in predicting the new features of the system. Fractional derivatives have provided such a valuable tool. This paper studies a dynamic system that models the interactions between two densities of immature and mature prey and predator populations. In the model, prey population is divided into two populations, including mature prey and immature prey. Another feature of the model is that predator depends on mature prey only and it followed by Crowley-Martin type functional response. Moreover, the fractional operator used in this model as derivative is of the Atangana-Baleanu $\mathsf{AB}$ type. Using this kind of fractional derivative causes the results to depend on the fractional order of the derivative. The addition of the concept of memory to the model is another highlight of using this type of derivative for the biological model. This helps the model to apply all the essential information of the phenomenon from the beginning to the desired time in the calculations. Existence and uniqueness of solutions to the fractional model are also investigated in this manuscript. The numerical method used in the article is also one of the most efficient patterns in solving problems with fractional derivatives. Using this effective method makes the results very consistent with what we actually expect to happen. Many simulations have been carried out to investigate the effect of parameters in the model on its overall behavior. Numerical results show the impressive performance of the fractional operator on the dynamic behavior of the considered predator-prey model. This efficient fractional operator can also be tested in the structure of other existing biological models.

在最近的几十年中，研究生物物种的行为已成为应用数学中最引人入胜的领域之一。保护自然界中稀有物种的高度重要性促使各个领域的研究人员特别关注此问题。因此，开发检查其行为动态的数学模型至关重要。另一方面，数值分析中新概念的发展使我们能够保留有关动态系统演化行为历史的更多信息，并将其用于预测系统的新功能。小数导数提供了这样一种有价值的工具。本文研究了一个动态系统，该系统模拟了两个密度的未成熟和成熟的猎物和捕食者种群之间的相互作用。在模型中 猎物种群分为两个种群，包括成熟猎物和未成熟猎物。该模型的另一个特点是，捕食者仅依赖成熟的猎物，其后是Crowley-Martin类型的功能响应。此外，在此模型中用作导数的分数算子是Atangana-Baleanu的$\mathsf{AB}$类型。使用这种分数导数会导致结果取决于导数的分数阶。在模型中增加记忆概念是将这种类型的导数用于生物学模型的另一个亮点。这有助于模型在计算中从开始到期望的时间应用现象的所有基本信息。该手稿中还研究了分数模型解的存在性和唯一性。本文中使用的数值方法也是解决分数导数问题的最有效模式之一。使用这种有效的方法可使结果与我们实际预期发生的结果非常一致。已经进行了许多模拟以研究模型中的参数对其整体行为的影响。数值结果表明分数算子对所考虑的捕食者－猎物模型的动力学行为具有令人印象深刻的性能。这种有效的分数运算符也可以在其他现有生物学模型的结构中进行测试。