When population size tends to infinity

Angela Stevens

In 1952 Kac derived a telegrapher equation for the probability density of one particle moving on a lattice with a certain velocity and a certain reversal frequency. If the probability to change direction is 1/2, then a diffusion equation results from the same procedure. Similar derivations and comparisons can be done for more complex phenomena, e.g. single cell movement in a chemical gradient. Here the diffusion coefficient and the chemotactic sensitivity can be described in terms of the cells velocity and its reversal frequency.

On the population level the classical Keller-Segel model for chemotaxis can be derived as limit dynamics of many cells, each undergoing a biased random walk in dependence on the chemical gradient produced by itself and the other cells, when population size tends to infinity. The random walk for each single cell is similar to the one mentioned above. The only important assumption for the derivation of the limit dynamics is, that the cells interact moderately.

On the population level, again an ansatz like the one given by Kac for a single cell can be made, which means e.g. in 1D, that equations for the right and the left moving part of the cell population are given. Again the relevant parameters of this model can be explained in terms of the chemotaxis model, and vice versa, like on the individual level.

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