Title: Predator-prey models with memory and kicks: Exact solution and discrete maps with memory Show Chinese title

Title: 带有记忆和踢动的捕食者-猎物模型：精确解和带记忆的离散映射

Abstract: In this paper, we proposed new predator-prey models that take into account memory and kicks. Memory is understood as the dependence of current behavior on the history of past behavior. The equations of these proposed models are generalizations of the Lotka-Volterra and Kolmogorov equations by using the Caputo fractional derivative of non-integer order and periodic kicks. This fractional derivative allows us to take into account memory with power-law fading. The periodic kicks, which are described by Dirac delta-functions, take into account short duration of interaction between predators and prey. For the proposed equations, which are fractional differential equations with kicks, we obtain exact solutions that describe behaviors of predator and prey with power-law fading memory. Using these exact solutions, we derive, without using any approximations, new discrete maps with memory that represent the proposed predator-prey models with memory. Show Chinese abstract

Abstract: 在本文中，我们提出了新的捕食者-猎物模型，这些模型考虑了记忆和脉冲。 记忆被理解为当前行为对过去行为历史的依赖。 这些提出模型的方程是通过使用非整数阶的Caputo分数阶导数和周期性脉冲对Lotka-Volterra和Kolmogorov方程的推广。 这种分数阶导数使我们能够考虑具有幂律衰减的记忆。 周期性脉冲由Dirac delta函数描述，考虑了捕食者和猎物之间相互作用的短持续时间。 对于这些提出的方程，即带有脉冲的分数阶微分方程，我们得到了精确解，这些解描述了具有幂律衰减记忆的捕食者和猎物的行为。 利用这些精确解，我们无需使用任何近似，推导出新的带有记忆的离散映射，这些映射代表了带有记忆的提出捕食者-猎物模型。