This paper considers the stationary problem of density-suppressed motility models proposed in Fu et al. (2012) and Liu et al. (2011) in one dimension with Neumman boundary conditions. The models consist of parabolic equations with cross-diffusion and degeneracy. We employ the global bifurcation theory and Helly compactness theorem to explore the conditions under which non-constant stationary (pattern) solutions exist and asymptotic profiles of solutions as some parameter value is small. When the cell growth is not considered, we are able to show the monotonicity of solutions and hence achieve a global bifurcation diagram by treating the chemical diffusion rate as a bifurcation parameter. Furthermore, we show that the solutions have boundary spikes as the chemical diffusion rate tends to zero and identify the conditions for the non-existence of non-constant solutions. When transformed to specific motility functions, our results indeed give sharp conditions on the existence of non-constant stationary solutions. While with the cell growth, the structure of global bifurcation diagram is much more complicated and in particular the solution loses the monotonicity property. By treating the growth rate as a bifurcation parameter, we identify a minimum range of growth rate in which non-constant stationary solutions are warranted, while a global bifurcation diagram can still be attained in a special situation. We use numerical simulations to test our analytical results and illustrate that patterns can be very intricate and stable stationary solutions may not exist when the parameter value is outside the minimal range identified in our paper.

中文翻译：

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密度抑制运动模型的稳态和模式形成

本文考虑了 Fu 等人提出的密度抑制运动模型的平稳问题。（2012）和刘等人。（2011）在一维与纽曼边界条件。这些模型由具有交叉扩散和退化的抛物线方程组成。我们采用全局分岔理论和 Helly 紧致性定理来探索存在非常量平稳（模式）解的条件以及由于某些参数值较小的解的渐近分布。当不考虑细胞生长时，我们能够显示溶液的单调性，从而通过将化学扩散速率作为分岔参数来实现全局分岔图。此外，我们表明，当化学扩散率趋于零时，解决方案具有边界尖峰，并确定了不存在非常量解决方案的条件。当转换为特定的运动函数时，我们的结果确实给出了非常量静止解存在的尖锐条件。而随着细胞的增长，全局分岔图的结构更加复杂，尤其是解失去了单调性。通过将增长率视为分岔参数，我们确定了一个最小的增长率范围，在该范围内保证了非常数的平稳解，而在特殊情况下仍然可以获得全局分岔图。