Modeling of Allee effect in biofilm formation via the stochastic bistable Allen–Cahn partial differential equation

Abstract

The evolution of biofilms on surfaces of medical implants, food, machinery, etc. may be modeled via reaction-diffusion partial differential equations. According to Allee effect, the microbes growth rate is positively correlated with the microbes density. Thus, instead of using Fisher’s equation, we employ the bistable Allen–Cahn equation for the modeling of biofilm formation. We work in space dimension two. Accounting for incomplete knowledge on the microbes movements and error measurements, we consider the diffusion, the growth rate and the Allee effect parameters as random quantities that depend on the outcome of the experiment. Their probability distributions may be defined by means of the maximum entropy principle. The solution to the model becomes a random field in time-space. Uncertainty quantification is conducted via non-intrusive (interpolation and pseudospectral) generalized polynomial chaos expansions, which give accurate approximations to the statistics of the response.