scholarly journals Large time behavior in a predator-prey system with indirect pursuit-evasion interaction

2017 ◽  
Vol 22 (11) ◽  
pp. 0-0
Author(s):  
Genglin Li ◽  
◽  
Youshan Tao ◽  
Michael Winkler ◽  
◽  
...  
Keyword(s):  
Large Time ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Predator Prey ◽  
Pursuit Evasion ◽  
Prey System

scholarly journals Large time behavior in a predator-prey system with pursuit-evasion interaction

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Dayong Qi ◽  
Yuanyuan Ke
Keyword(s):  
Large Time ◽  
Smooth Boundary ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Predator Prey ◽  
Pursuit Evasion ◽  
Prey System ◽  
Spatially Homogeneous ◽  
Coexistence State ◽  
Large Time Limit

<p style='text-indent:20px;'>This work considers a pursuit-evasion model</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1000">\begin{document}$\begin{equation} \left\{ \begin{split} &amp;u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+u(\mu-u+av),\\ &amp;v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda-v-bu),\\ &amp;w_t = \Delta w-w+v,\\ &amp;z_t = \Delta z-z+u\\ \end{split} \right. \ \ \ \ \ (1) \end{equation}$\end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>with positive parameters <inline-formula><tex-math id="M1">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> in a bounded domain <inline-formula><tex-math id="M7">\begin{document}$ \Omega\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M8">\begin{document}$ N $\end{document}</tex-math></inline-formula> is the dimension of the space) with smooth boundary. We prove that if <inline-formula><tex-math id="M9">\begin{document}$ a&lt;2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \frac{N(2-a)}{2(C_{\frac{N}{2}+1})^{\frac{1}{\frac{N}{2}+1}}(N-2)_+}&gt;\max\{\chi,\xi\} $\end{document}</tex-math></inline-formula>, (1) possesses a global bounded classical solution with a positive constant <inline-formula><tex-math id="M11">\begin{document}$ C_{\frac{N}{2}+1} $\end{document}</tex-math></inline-formula> corresponding to the maximal Sobolev regularity. Moreover, it is shown that if <inline-formula><tex-math id="M12">\begin{document}$ b\mu&lt;\lambda $\end{document}</tex-math></inline-formula>, the solution (<inline-formula><tex-math id="M13">\begin{document}$ u,v,w,z $\end{document}</tex-math></inline-formula>) converges to a spatially homogeneous coexistence state with respect to the norm in <inline-formula><tex-math id="M14">\begin{document}$ L^\infty(\Omega) $\end{document}</tex-math></inline-formula> in the large time limit under some exact smallness conditions on <inline-formula><tex-math id="M15">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>. If <inline-formula><tex-math id="M17">\begin{document}$ b\mu&gt;\lambda $\end{document}</tex-math></inline-formula>, the solution converges to (<inline-formula><tex-math id="M18">\begin{document}$ \mu,0,0,\mu $\end{document}</tex-math></inline-formula>) with respect to the norm in <inline-formula><tex-math id="M19">\begin{document}$ L^\infty(\Omega) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M20">\begin{document}$ t\rightarrow \infty $\end{document}</tex-math></inline-formula> under some smallness assumption on <inline-formula><tex-math id="M21">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> with arbitrary <inline-formula><tex-math id="M22">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Boundedness and asymptotic behavior in a predator-prey model with indirect pursuit-evasion interaction

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chao Liu ◽  
Bin Liu
Keyword(s):  
Asymptotic Behavior ◽  
Large Time ◽  
Neumann Boundary ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Smooth Bounded Domain ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pursuit Evasion ◽  
Predator Model

<p style='text-indent:20px;'>In this paper, we study the prey-predator model with indirect pursuit-evasion interaction defined on a smooth bounded domain with homogeneous Neumann boundary conditions. We obtain the globa existence and boundedness of the classical solution of the model by estimating <inline-formula><tex-math id="M1">\begin{document}$ L^{p} $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ v $\end{document}</tex-math></inline-formula>, and we also show the large time behavior and convergence rate of the solution.</p>

A nonlocal swarm model for predators–prey interactions

2015 ◽  
Vol 26 (02) ◽  
pp. 319-355 ◽  
Author(s):  
Marco Di Francesco ◽  
Simone Fagioli
Keyword(s):  
Large Time ◽  
Space Dimension ◽  
Large Time Behavior ◽  
Nonlocal Interaction ◽  
Time Behavior ◽  
Predator Prey ◽  
Prey Interaction ◽  
Gradient Type ◽  
Repulsive Forces ◽  
Nonlinear Continuum

We consider a two-species system of nonlocal interaction PDEs modeling the swarming dynamics of predators and prey, in which all agents interact through attractive/repulsive forces of gradient type. In order to model the predator–prey interaction, we prescribed proportional potentials (with opposite signs) for the cross-interaction part. The model has a particle-based discrete (ODE) version and a continuum PDE version. We investigate the structure of particle stationary solution and their stability in the ODE system in a systematic form, and then consider simple examples. We then prove that the stable particle steady states are locally stable for the fully nonlinear continuum model, provided a slight reinforcement of the particle condition is required. The latter result holds in one space dimension. We complement all the particle examples with simple numerical simulations, and we provide some two-dimensional examples to highlight the complexity in the large time behavior of the system.

scholarly journals Global well-posedness of the full compressible Hall-MHD equations

2021 ◽  
Vol 10 (1) ◽  
pp. 1235-1254
Author(s):  
Qiang Tao ◽  
Canze Zhu
Keyword(s):  
Global Solution ◽  
Large Time ◽  
Initial Temperature ◽  
Large Time Behavior ◽  
Initial Energy ◽  
Mhd Equations ◽  
Well Posedness ◽  
Time Behavior ◽  
Magnetohydrodynamic Flows ◽  
Hall Mhd

Abstract This paper deals with a Cauchy problem of the full compressible Hall-magnetohydrodynamic flows. We establish the existence and uniqueness of global solution, provided that the initial energy is suitably small but the initial temperature allows large oscillations. In addition, the large time behavior of the global solution is obtained.

Large Time Behavior of Solutions to One Nonlinear Integro-Differential Equation

2008 ◽  
Vol 15 (3) ◽  
pp. 531-539
Author(s):  
Temur Jangveladze ◽  
Zurab Kiguradze
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Boundary Conditions ◽  
Large Time ◽  
Dirichlet Boundary ◽  
Large Time Behavior ◽  
Dirichlet Boundary Conditions ◽  
Time Behavior ◽  
Integro Differential Equation ◽  
Homogeneous Data

Abstract Large time behavior of solutions to the nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. The rate of convergence is given, too. Dirichlet boundary conditions with homogeneous data are considered.

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