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>

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 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

Dynamics of a Discrete Prey–Predator Model with Mixed Functional Response

2019 ◽  
Vol 29 (14) ◽  
pp. 1950199
Author(s):  
Mohammed Fathy Elettreby ◽  
Aisha Khawagi ◽  
Tamer Nabil
Keyword(s):  
Type I ◽  
Time Behavior ◽  
Stable Fixed Point ◽  
Functional Responses ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Long Time ◽  
Predator Model ◽  
The Stability

In this paper, we propose a discrete Lotka–Volterra predator–prey model with Holling type-I and -II functional responses. We investigate the stability of the fixed points of this model. Also, we study the effects of changing each control parameter on the long-time behavior of the model. This model contains a lot of complex dynamical behaviors ranging from a stable fixed point to chaotic attractors. Finally, we illustrate the analytical results by some numerical simulations.

scholarly journals Asymptotic Behavior of Solutions to Fast Diffusive Non-Newtonian Filtration Equations Coupled by Nonlinear Boundary Sources

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Wang Zejia ◽  
Wang Shunti ◽  
Zhang Chengbin
Keyword(s):  
Asymptotic Behavior ◽  
Global Existence ◽  
Nonlinear Boundary ◽  
Large Time ◽  
Large Time Behavior ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Multidimensional Problem ◽  
Critical Curves

This paper is concerning the asymptotic behavior of solutions to the fast diffusive non-Newtonian filtration equations coupled by the nonlinear boundary sources. We are interested in the critical global existence curve and the critical Fujita curve, which are used to describe the large-time behavior of solutions. It is shown that the above two critical curves are both the same for the multidimensional problem we considered.

scholarly journals Boundedness and Large-Time Behavior Results for a Diffusive Epidemic Model

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Lamine Melkemi ◽  
Ahmed Zerrouk Mokrane ◽  
Amar Youkana
Keyword(s):  
Asymptotic Behavior ◽  
Large Time ◽  
System Modeling ◽  
Uniform Boundedness ◽  
Reaction Diffusion ◽  
Large Time Behavior ◽  
Positive Answer ◽  
Time Behavior ◽  
Epidemic Disease ◽  
Infection Disease

We consider a reaction-diffusion system modeling the spread of an epidemic disease within a population divided into the susceptible and infective classes. We first consider the question of the uniform boundedness of the solutions for which we give a positive answer. Then we deal with the asymptotic behavior of the solutions where in particular we are interested in reasonable conditions leading to the extinction of the infection disease as the time goes to infinity.

scholarly journals Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Rong Zhang ◽  
Liangchen Wang
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Large Time ◽  
Neumann Boundary ◽  
Rates Of Convergence ◽  
Large Time Behavior ◽  
Global Dynamics ◽  
Time Behavior ◽  
Chemical Signalling ◽  
Chemotaxis System

<p style='text-indent:20px;'>This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla v)+\mu_1 u(1-u-a_1w),\, x\in \Omega,\, t&gt;0,\\ 0 = \Delta v-v+w,\,x\in\Omega,\, t&gt;0,\\ w_t = \Delta w-\chi_2\nabla\cdot(w\nabla z)-\chi_3\nabla\cdot(w\nabla v)+\mu_2 w(1-w-a_2u), \,x\in \Omega,\,t&gt;0,\\ 0 = \Delta z-z+u, \,x\in\Omega,\, t&gt;0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ n\geq1 $\end{document}</tex-math></inline-formula>, where the parameters <inline-formula><tex-math id="M3">\begin{document}$ a_1,a_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \chi_1, \chi_2, \chi_3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu_1, \mu_2 $\end{document}</tex-math></inline-formula> are positive constants. We first showed some conditions between <inline-formula><tex-math id="M6">\begin{document}$ \frac{\chi_1}{\mu_1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \frac{\chi_2}{\mu_2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \frac{\chi_3}{\mu_2} $\end{document}</tex-math></inline-formula> and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.</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.

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