A class of models with a nonlinear indirect chase-and-escape effect

In this manuscript, we examine a nonlinear indirect pursuit-evasion model represented by the following system $ \begin{equation*} \begin{split} \begin{cases} u_{1t} = \Delta u_{1}-\chi\nabla\cdot\bigl(u_{1}\nabla v_{1}\bigl)+u_{1}\bigl(\lambda_{1}-\mu_{1}u_{1}^{r_{1}-1}+au_{2}\bigl),&\quad x\in\Omega,t>0,\\ u_{2t} = \Delta u_{2}+\xi\nabla\cdot\bigl(u_{2}\nabla v_{2}\bigl)+u_{2}\bigl(\lambda_{2}-\mu_{2}u_{2}^{r_{2}-1}-bu_{1}\bigl),&\quad x\in\Omega,t>0,\\ v_{1t} = \Delta v_{1}-v_{1}+u_{2}^{\alpha},&\quad x\in\Omega,t>0,\\ 0 = \Delta v_{2}-v_{2}+u_{1}^{\beta},&\quad x\in\Omega,t>0 \end{cases} \end{split} \end{equation*} $ within a smoothly bounded domain $ \Omega \subset \mathbb{R}^N $($ N >2 $). We specifically consider this nonlinear indirect pursuit-evasion model under homogeneous Neumann boundary conditions. The model incorporates parameters $ \chi $, $ \xi $, $ a $, $ b $, $ \lambda_j $, $ \mu_j $, $ \alpha $, $ \beta > 0 $, and $ r_j > 1 $($ j = 1, 2 $). It has been established that if the parameters $ \alpha $, $ \beta $, $ r_1 $, and $ r_2 $ meet certain specific criteria: $ \begin{equation*} \beta\leq\frac{2}{N}, \mbox{for all}\; r_{1},r_{2}>1, \end{equation*} $ or $ \begin{equation*} \frac{2}{N}<\beta\leq1, \mbox{and}\; \frac{N\beta-2}{N(r_{1}-1)(r_{2}-1)}\max\{1,\alpha\}<1, \end{equation*} $ or$ \begin{equation*} \beta>1, \frac{\beta}{(r_{1}-1)(r_{2}-1)}\max\{1,\alpha\}<1,r_{1}>2,\; \mbox{and}\; r_{2}>2, \end{equation*} $then, for any appropriate nonnegative initial conditions $ u_{10} $, $ u_{20} $, and $ v_{10} $, the system possesses a globally and uniformly bounded classical solution. The present study proposes novel containment strategies that can be effectively implemented to tackle specific chemotactic models, thereby significantly enhancing or extending the findings previously reported in [38].