Cells in a living body are like little travelers. They can walk, sense where to go, and reproduce. Sometimes they move as a wide crowd, filling all the space. Other times they gather into a dense group and crawl together, like a drop on glass. Why the difference?
Scientists built a mathematical model where each cell moves at a constant speed, occasionally turns toward a chemical signal, and divides from time to time. It turns out that one thing decides everything: how fast cells divide compared to how fast they walk. If division and walking go at the same speed, cells have time to spread out, and you get a flow. If division is slow, cells don't have time to disperse, they stick together, and you get a clump.
When division is slow, the clump doesn't stay still: it slowly grows and crawls. To describe this, mathematicians had to look at time with two eyes: one fast, on which the clump seems motionless, and one slow, on which you see it growing. This approach helped connect the behavior of individual cells to the behavior of the whole group.
A clump of cells behaves like a droplet crawling along a gradient: its speed depends on mass but does not grow indefinitely, because cells at the leading edge move slower, limiting the speed of the whole group.