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.

中文翻译：

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具有化学物质和捕食者项的趋化系统解的整体存在性和渐近行为

本文涉及Lotka Volterra反应扩散系统的任意全局正定解的一般渐近稳定，具有附加的趋化影响和常数。我们考虑了涉及两个生物物种的数学模型的动力学，这两个生物物种均根据随机扩散而运动，并被另一生物物种产生的化学刺激所吸引/排斥。生物物种具有使它们的运动朝着其他物种分泌的化学物质的浓度定向的能力。非线性系统由两个具有Lotka-Volterra型动力学项的抛物线方程与化学趋化交叉扩散，以及两个描述化学药品行为的椭圆方程组成。我们证明，相应的Neumann初始边值问题的解决方案是全局的，并且对于常规和正初始数据都是有界的。此外，对于参数的不同范围，我们表明任何正解和有界解都收敛到空间恒定的均匀状态。