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\begin{document}

\title{Stationary and oscillatory patterns of a food chain model with diffusion
and predator-taxis}



\author[1]{Renji Han}%
\author[2]{Gergely Rost}%
\affil[1]{Zhejiang University of Science and Technology}%
\affil[2]{University of Szeged}%


\vspace{-1em}



  
  \date{June 17, 2024}


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\begin{abstract}
In this paper, we investigate pattern dynamics in a
reaction-diffusion-chemotaxis food chain model with predator-taxis,
which enriches previous studies about diffusive food chain models. By
virtue of diffusion semigroup theory, we first show the global classical
solvability and uniform boundedness of the considered model over a
bounded domain  [?] R N ( N [?] 1 ) with smooth boundary. Then the linear
stability analysis for the considered model shows that chemotaxis can
induce the unique positive spatially homogeneous steady state loses its
stability via Turing bifurcation and Turing-spatiotemporal Hopf
bifurcation, which results in the formation of two kinds of important
spatiotemporal patterns: stationary Turing pattern and oscillatory
pattern. Simultaneously, the threshold values for Turing bifurcation and
Turing-spatiotemporal Hopf bifurcation are given explicitly. In
addition, the existence and stability of non-constant positive steady
state that bifurcates from the positive constant steady state is
investigated by the abstract bifurcation theory of Crandall-Rabinowitz
and eigenvalue perturbation theory. Finally, numerical simulations are
performed to verify our theoretical results, and some interesting
non-Turing pattern are found in temporal Hopf parameter space by
numerical simulation.%
\end{abstract}%



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