Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals

Cruz, Elena de la; Negreanu, Mihaela; Tello, J. Ignacio

doi:10.1007/s00033-018-1002-1

Cited by 20 publications

(22 citation statements)

References 25 publications

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“…It shows that in the mutually repulsive case χ 1 , χ 2 > 0, regardless of how large χ 1 , χ 2 are, the system (1.6)-(1.7) is globally well-posed in all dimensions n ≥ 1, and any solution is globally bounded in time. Recall that global well-posedness has been previously established by [17,65] under some smallness assumptions of |χ 1 | and |χ 2 |. Also, for any χ 1 , χ 2 > 0, global well-posedness for the system (1.6)-(1.7) without the competition terms was obtained in [56] in dimensions n ≤ 3.…”

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confidence: 84%

“…Next we move on to question (b) that concerns the weak competition regime a 1 , a 2 ∈ (0, 1). In this regime, for small |χ 1 |, |χ 2 |, [17] showed that the constant steady state (1.2) is the global attractor for the system (1.6)-(1.7). Clearly we do not expect this property to hold for large χ 1 , χ 2 : a simple linear stability criterion in Section 3.1 shows that (1.2) is linearly unstable for large χ 1 , χ 2 .…”

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confidence: 96%

“…χ 1 , χ 2 < 0), and showed that when −χ 1 , −χ 2 are below certain thresholds, the system is globally well-posed and converges towards the steady state (1.2). [17] showed that when |χ 1 |, |χ 2 | are below certain thresholds, (1.6)-(1.7) has a global solution, and the solution converges towards (1.2) in the weak competition regime, and converges towards (1, 0) if 0 < a 1 < 1 < a 2 . [59] studied the mutually attractive case, and showed that solutions converge towards constant steady states when b 1 , b 2 are sufficiently large.…”

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confidence: 99%

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Two-species competition model with chemotaxis: well-posedness, stability and dynamics

Li,

Yao

2021

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We study a system of PDEs modeling the population dynamics of two competitive species whose spatial movements are governed by both diffusion and mutually repulsive chemotaxis effects. We prove that solutions to this system are globally well-posed, without any smallness assumptions on the chemotactic coefficients. Moreover, in the weak competition regime, we prove that neither species can be driven to extinction as the time goes to infinity, regardless of how strong the chemotaxis coefficients are. Finally, long-time behaviors of the system are studied both analytically in the weakly nonlinear regime, and numerically in the fully nonlinear regime.

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confidence: 84%

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confidence: 96%

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confidence: 99%

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Two-species competition model with chemotaxis: well-posedness, stability and dynamics

Li,

Yao

2021

Preprint

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“…Recently, this results were improved by Wang et al [38]. There are some another results on various the system (1.3) ( [3,5,10,18,23,26,29,27,43,47,46]).…”

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confidence: 94%

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

Zhang

Wang

2021

era

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This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop<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>0,\\ 0 = \Delta v-v+w,\,x\in\Omega,\, t>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>0,\\ 0 = \Delta z-z+u, \,x\in\Omega,\, t>0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \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.

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“…There exist many studies dedicated to the analysis of PDE-ODE systems in the literature, see for instance [4,6,7,11,14,15,20,24] and the references therein. Some of them such as [14,24], use results of classical solutions given in [1,2].…”

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confidence: 99%

Theoretical and numerical analysis for a hybrid tumor model with diffusion depending on vasculature

Fernández-Romero,

Guillén-González,

Suárez

2020

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In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of global in time weak-strong solutions using a regularization technique via an artificial diffusion in the ODE-system and a fixed point argument. In addition, stability results of the critical points are given under some constraints on parameters. Finally, we design a fully discrete finite element scheme for the model which preserves the pointwise and energy estimates of the continuous problem.

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Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals

Cited by 20 publications

References 25 publications

Two-species competition model with chemotaxis: well-posedness, stability and dynamics

Two-species competition model with chemotaxis: well-posedness, stability and dynamics

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

Theoretical and numerical analysis for a hybrid tumor model with diffusion depending on vasculature

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