Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms

Mihaela Negreanu

doi:10.3934/dcdsb.2020064

Article Contents

2020, Volume 25, Issue 9: 3335-3356. Doi: 10.3934/dcdsb.2020064

This issue Previous Article Geometric method for global stability of discrete population models Next Article Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations

Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms

Mihaela Negreanu

Depto. de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

Received: January 31, 2019

Revised: September 30, 2019

Early access: April 2020

Published: September 2020

The first author is supported by by project MTM2017- 83391-P MICINN (Spain)

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Abstract

This paper is concerned with a general asymptotic stabilization of arbitrary global positive bounded solutions for the Lotka Volterra reaction diffusion systems, with an additional chemotactic influence and constant coefficients. We consider the dynamics of a mathematical model involving two biological species, both of which move according to random diffusion and are attracted/ repulsed by chemical stimulus produced by the other. The biological species present the ability to orientate their movement towards the concentration of the chemical secreted by the other species. The nonlinear system consists of two parabolic equations with Lotka-Volterra-type kinetic terms coupled with chemotactic cross-diffusion, along with two elliptic equations describing the behavior of the chemicals. We prove that the solution to the corresponding Neumann initial boundary value problem is global and bounded for regular and positive initial data. Moreover, for different ranges of parameters, we show that any positive and bounded solution converges to a spatially constant homogeneous state.

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Mathematics Subject Classification: Primary: 92C17, 35K55, 35B35; Secondary: 35B51.

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