Abstract
Starting from a particle model we derive a macroscopic aggregation-diffusion equation for the evolution of slime mold under the assumption of propagation of chaos in the large particle limit. We analyze properties of the macroscopic model in the stationary case and study the behavior of the slime mold between food sources. The efficient numerical simulation of the aggregation-diffusion equation allows for a detailed analysis of the interplay between the different regimes drift, interaction and diffusion.
Highlights
Introduction and backgroundP. polycephalum, or a slime mold, is an amoeboid organism that is notable for its ability to perform complex tasks despite its relatively simple biological structure and lack of a brain
A=10, B=5, C=1 time = 0 time = 2 time = 4 time = 6. In this manuscript we presented a model hierarchy aimed at modeling the food seeking behavior of the slime mold P. polycephalum
We first presented a particle based model which includes three main features - a drift term to model a gradient of chemoattractant produced by food sources, an interaction term to model the slime mold’s propensity to maintain a connected mass, and a diffusion term to model slime mold foraging behavior
Summary
P. polycephalum , or a slime mold, is an amoeboid organism that is notable for its ability to perform complex tasks despite its relatively simple biological structure and lack of a brain. The maze is described as a network topology and P. polycephalum is characterized with one dimensional partial differential equations for chemotaxis (in particular the Keller-Segel model) defined on the edges of the network with coupling conditions at the junctions of the maze [23, 24] Both of these macroscopic descriptions attempt to model the path that P. polycephalum builds between food sources once they are located. We demonstrate under the assumption of propagation of chaos that as the number of agents approaches infinity, the marginal distribution of each agent converges to a deterministic density based model in the form of an aggregation-diffusion equation This macroscopic model includes terms that correspond to the drift, interaction and noise terms present in the microscopic model. Aggregation diffusion equations without a drift term are well known to exhibit "blow up" below a critical mass - we do not observe such a phenomena in our simulations ( we do not vary the mass) - it could be an interesting area of future research to investigate the effect of the drift term on the occurrence of blow up
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