# American Institute of Mathematical Sciences

September  2020, 25(9): 3335-3356. doi: 10.3934/dcdsb.2020064

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

Received  February 2019 Revised  October 2019 Published  April 2020

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

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.

Citation: Mihaela Negreanu. Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3335-3356. doi: 10.3934/dcdsb.2020064
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