We propose and study a class of parabolic-ordinary differential equation models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumour microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature constituting a core difference to both classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

中文翻译：

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信号产生的间接性是否会降低趋化性 - 趋化性系统中的爆炸支持潜力？一类癌症侵袭（及更多）模型中的全局经典可解性

我们提出并研究了一类抛物线普通微分方程模型，该模型涉及一个物种的趋化性和趋触性，这些信号是由另一个非运动物种间接产生的信号。该设置的动机是通过与肿瘤微环境的相互作用介导的癌症侵​​袭，但具有更广泛的适用性，能够描述生物学上完全不同的问题。作为构成经典 Keller-Segel 趋化系统和 Chaplain-Lolas 型趋化-趋化系统的核心差异的主要数学特征，所考虑的模型解释了某些类型的间接信号产生机制。主要结果在相关的空间二维初始边界值问题中的系统参数函数的适当温和假设下断言了独特的全局经典可解性。特别是，这严格证实，至少在二维环境中，信号产生中所考虑的间接性在出租车系统中也引起了显着的爆炸抑制趋势，这比最近观察到相应影响的一些特定示例更普遍。