Journal of Evolution Equations ( IF 1.4 ) Pub Date : 2023-11-28 , DOI: 10.1007/s00028-023-00931-w Chuanjia Wan , Pan Zheng , Wenhai Shan
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In this paper, we study the quasilinear fully parabolic predator–prey model with indirect pursuit-evasion interaction
$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot \left( D_{1}(u)\nabla u\right) -\chi \nabla \cdot \left( S_{1}(u)\nabla z\right) +u\left( \alpha v-a_{1} -b_{1}u\right) ,&x \in \varOmega , t>0, \\&v_t=\nabla \cdot \left( D_{2}(v)\nabla v\right) +\xi \nabla \cdot \left( S_{2}(v)\nabla {w}\right) +v\left( a_{2} -b_{2} v-u\right) ,&x \in \varOmega , t>0, \\&{w_t}=\Delta w+\beta {u}-\gamma {w},&x \in \varOmega , t>0,\\&{z_t}=\Delta z+\delta {v}-\rho z,&x \in \varOmega , t>0,\\ \end{aligned} \right. \end{aligned} \end{aligned}$$under homogeneous Neumann boundary conditions in a smoothly bounded domain \(\varOmega \subset \mathbb {R}^{n}(n\ge 1)\), where \( \chi , \xi , \alpha , \beta , \gamma , \delta , \rho , a_{1},a_{2},\) \(b_{1},b_{2}\) are positive parameters, the functions \(D_{i} \in C^{2}([0,\infty ))\) and \(S_{i}\in C^{2}([0,\infty ))\) with \(S_{i}(0)=0(i=1,2)\). Firstly, under certain suitable conditions, we prove that the system admits a unique globally bounded classical solution when \(n\le 4\). Moreover, we investigate the asymptotic stability and precise convergence rates of globally bounded solutions by constructing appropriate Lyapunov functionals. Finally, we present numerical simulations that not only support our theoretical results, but also involve new and interesting phenomena.
中文翻译：
具有间接追逐-逃避交互作用的拟线性全抛物线捕食者-被捕食者模型
在本文中，我们研究了具有间接追逐-逃避相互作用的拟线性全抛物线捕食者-被捕食者模型
$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot \left( D_{1}(u)\nabla u\right) -\chi \nabla \cdot \left( S_{1}(u)\nabla z\right) +u\left( \alpha v-a_{1} -b_{1}u\right) ,&x \in \varOmega , t>0, \ \&v_t=\nabla \cdot \left( D_{2}(v)\nabla v\right) +\xi \nabla \cdot \left( S_{2}(v)\nabla {w}\right) +v \left( a_{2} -b_{2} vu\right) ,&x \in \varOmega , t>0, \\&{w_t}=\Delta w+\beta {u}-\gamma {w},&x \in \varOmega , t>0,\\&{z_t}=\Delta z+\delta {v}-\rho z,&x \in \varOmega , t>0,\\ \end{aligned} \right。\end{对齐} \end{对齐}$$在平滑有界域中的齐次诺伊曼边界条件下\(\varOmega \subset \mathbb {R}^{n}(n\ge 1)\)，其中\( \chi , \xi , \alpha , \beta , \ gamma , \delta , \rho , a_{1},a_{2},\) \(b_{1},b_{2}\)为正参数，函数\(D_{i} \in C^{ 2}([0,\infty ))\)和\(S_{i}\in C^{2}([0,\infty ))\)且\(S_{i}(0)=0(i =1,2)\)。首先，在某些合适的条件下，我们证明当\(n\le 4\)时系统承认唯一的全局有界经典解。此外，我们通过构造适当的李雅普诺夫泛函来研究全局有界解的渐近稳定性和精确收敛率。最后，我们提出的数值模拟不仅支持我们的理论结果，而且还涉及新的有趣的现象。
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