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\begin{document}

\title{A diffusive predator-prey system with hunting cooperation in predators
and prey-taxis: II stationary pattern formation}



\author[1]{Kimun Ryu}%
\author[1]{Wonlyul Ko}%
\affil[1]{Cheongju University}%


\vspace{-1em}



  
  \date{May 18, 2024}


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\begin{abstract}
This paper is a continuation of the study conducted by Ko and Ryu (2024)
{[}5{]}, which introduces and analyzes a generalized predator-prey
reaction-diffusion system incorporating (repulsive) prey-taxis and a
hunting cooperation effect in predators, under homogeneous Neumann
boundary conditions. In the study, the existence and uniqueness of
global and classical solutions for the time- and space-dependent system
are analytically examined. Furthermore, the study examines the local and
global stability and convergence rate at the constant
predator-extinction and coexistence states. In our paper, we analyze the
stationary system corresponding to the system in {[}5{]}, with a
specific focus on examining the existence and nonexistence of positive
and nonconstant solutions. The nonexistence occurs when the diffusion
rate of prey is sufficiently high. On the other hand, the existence
occurs when the prey-tactic rate is sufficiently high, indicating a
strong repulsive prey-taxis, and the diffusion rate of prey is
sufficiently low. For this investigation, we separately employ the
energy method and the Leray-Schauder degree theory.%
\end{abstract}%



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