Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2020-08-27 , DOI: 10.1007/s00033-020-01376-8 Yuanyuan Liu , Yuehong Zhuang
We investigate a forager–exploiter model in a high-dimensional smooth bounded domain with zero-flux Neumann boundary condition:
$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{cc} {\displaystyle u_{t}=\Delta u-\chi _{1}\nabla \cdot (u\nabla w),}&{}{}\quad \,\,\,\,x\in \Omega ,\,t>0,\\ {\displaystyle v_{t}=\Delta v-\chi _{2}\nabla \cdot (v\nabla u),}&{}{}\quad \,\,\,\,x\in \Omega ,\,t>0,\\ w_{t}=\mathrm {d}\Delta w-\frac{u+v}{(1+u+v)^{\gamma }}w-\mu w+r(x,t),&{}{}\quad \,\,\,\,x\in \Omega ,\,t>0.\end{array}\right. \end{aligned} \end{aligned}$$This model characterizes the social interactions between the two species, foragers and exploiters, denoted by u and v, searching for the same food resource w. The positive taxis effects \(\chi _{1}\) and \(\chi _{2}\) reflect doubly tactic modelling hypothesis that the foragers chase food resource directly, while the exploiters follow after them. The spatio-temporal dynamics of food resource include its reaction-diffusion at rate d, natural reduction at rate \(\mu \), renewed production at rate r and especially its nonlinear consumption by the two species. For a positive constant \(\gamma \) weighing the nonlinear sensitivity of resource consumption rate, we find a sufficient condition such that the system possesses a unique nonnegative global bounded classical solution.
中文翻译：
具有两个物种非线性资源消耗的高维觅食动物探索模型的有界性
我们在零通量诺伊曼边界条件下研究高维光滑有界域中的觅食者-探索者模型：
$$ \ begin {aligned} \ begin {aligned \\ left \ {\ begin {array} {cc} {\ displaystyle u_ {t} = \ Delta u- \ chi _ {1} \ nabla \ cdot（u \ nabla w），}＆{} {} \ quad \，\，\，\，x \ in \ Omega，\，t> 0，\\ {\ displaystyle v_ {t} = \ Delta v- \ chi _ {2 } \ nabla \ cdot（v \ nabla u），}＆{} {} \ quad \，\，\，\，x \ in \ Omega，\，t> 0，\\ w_ {t} = \ mathrm { d} \ Delta w- \ frac {u + v} {（1 + u + v）^ {\ gamma}} w- \ mu w + r（x，t），＆{} {} \ quad \，\ ，\，\，x \ in \ Omega，\，t>0。\ end {array} \ right。\ end {aligned} \ end {aligned} $$该模型描述了用u和v表示的两种物种，觅食者和剥削者之间的社会互动，以寻找相同的食物资源w。正面的滑行效应\（\ chi _ {1} \）和\（\ chi {2} \）反映了双重策略建模假设，即觅食者直接追逐食物资源，而剥削者则追随它们。粮食资源的时空动态包括其在速率d下的反应扩散，在速率\（\ mu \）下的自然还原，在速率r下的更新生产，尤其是两种物种的非线性消耗。对于正常数\（\ gamma \） 通过权衡资源消耗率的非线性敏感性，我们找到了一个充分条件，使得系统拥有唯一的非负全局有界经典解。
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