AI 中文总结
研究与双抛物型Keller-Segel方程相关的两物种粒子系统，在细胞对化学吸引剂敏感度足够小等条件下，当\(N\to\infty\)时近似抛物-抛物型Keller-Segel方程，\(N\)固定且产生速率趋于无穷时近似单物种系统。
AI 中文摘要
我们考虑由Stevens（2000）引入的与双抛物型Keller-Segel方程相关的两物种粒子系统。它由\(N\)个细胞和数量可变的化学吸引剂粒子组成。细胞在平面上扩散并遵循化学吸引剂浓度的（平滑化）经验梯度。化学吸引剂粒子以一定恒定速率由细胞产生，以一定恒定速率扩散和消失。我们表明，当细胞对化学吸引剂的敏感度足够小时，在关于平滑函数族的一些相当弱的条件下，当\(N\to\infty\)时，该系统近似抛物-抛物型Keller-Segel方程。我们还证明，当\(N\)固定且化学吸引剂粒子的产生速率趋于无穷大时，该系统近似Talay-Tomašević（2020）引入并由作者（2023）进一步研究的（非马尔可夫）单物种系统。
英文摘要
We consider the two-species particle system introduced by Stevens (2000) related to the doubly parabolic Keller-Segel equation. It consists of $N$ cells and of a varying number of chemoattractant particles. Cells diffuse in the plane and follow the (mollified) empirical gradient of concentration of chemoattractant. Chemoattractant particles are produced by cells at some constant rate, diffuse and disappear at some constant rate. We show that when the sensitivity of cells to the chemoattractant is small enough, under some rather weak condition on the family of mollifiers, this system approximates the parabolic-parabolic Keller-Segel equation as $N\to \infty$. We also prove that when $N$ is fixed and when the production rate of chemoattractant particles tends to infinity, this system approximates the (non-Markovian) one-species system introduced Talay-Tomašević (2020) and further studied by the authors (2023).
