We introduce two models of biological aggregation, based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighborhood. In the first-order model the location of each individual is subject to a density-dependent random walk, while in the second-order model the density-dependent random walk acts on the velocity variable, together with a density-dependent damping term. The main novelty of our models is that we do not assume any explicit aggregative force acting on the individuals; instead, aggregation is obtained exclusively by reducing the individual stochasticity in response to higher perceived density. We formally derive the corresponding mean-field limits, leading to nonlocal degenerate diffusions. Then, we carry out the mathematical analysis of the first-order model, in particular, we prove the existence of weak solutions and show that it allows for measure-valued steady states. We also perform linear stability analysis and identify conditions for pattern formation. Moreover, we discuss the role of the nonlocality for well-posedness of the first-order model. Finally, we present results of numerical simulations for both the first- and second-order model on the individual-based and continuum levels of description.

我们介绍了两种生物聚集模型，它们基于随机移动的粒子，具有个体随机性，具体取决于其邻域中感知的平均人口密度。在一阶模型中，每个个体的位置都受密度相关的随机游走的影响，而在二阶模型中，密度相关的随机游走作用于速度变量，以及与密度相关的阻尼项。我们模型的主要新颖之处在于我们不假设任何明确的集体力量作用于个人；相反，聚合是通过减少个体随机性以响应更高的感知密度来获得的。我们正式推导出相应的平均场限制，导致非局部退化扩散。然后，我们进行一阶模型的数学分析，特别是，我们证明了弱解的存在，并表明它允许测量值稳定状态。我们还进行线性稳定性分析并确定模式形成的条件。此外，我们讨论了非定域性对一阶模型适定性的作用。最后，我们展示了基于个体和连续描述层次的一阶和二阶模型的数值模拟结果。