NUMERICAL SIMULATION FOR A MODEL OF BIOREMEDIATION WITH CHEMOTAXIS

The aim of this paper is to present some numerical simulations for different values of parameters and to compare their influence for a reaction-diffusion model which describes chemotaxis phenomena in a bioremediation process of a polluted environment (soil or water). We set a model of Keller-Segel type, and we consider for this model a small influence of the degradation rate of pollutant ( ) and its diffusion coefficient ( ). For the numerical simulations, we used the package COMSOL Multiphysics v3.5a, working with the finite element method.


INTRODUCTION
Chemotaxis is a biological phenomenon describing the change of motion when a population of individuals (in particular, bacteria that have a density denoted by b ) reacts in response to an external stimulus spread in the environment by another population or substance (chemoattractant c ).As a consequence, the population b directs its movement towards (positive chemotaxis) a higher concentration of the chemical substance.For a brief history of the chemotaxis phenomenon see [1][2][3].
Starting from this definition, several transport phenomena in biological systems have been labeled with the term of chemotaxis.We recall here the systems for the bacteria Escherichia Coli, amoebae Discoideum Distiostelium or for the endothelial cells of the human body responding to angiogenic factors secreted by a tumour [4].Apart from applications in biology, this model is also used in cases of environment bioremediation, when specific bacteria b cleans a medium (soil or water) polluted with a certain chemical c , as for instance oil [5].The aim of soil bioremediation is to reduce the concentration of pollutants to undetectable levels using microorganisms (bacteria).In the case of soil bioremediation it must be satisfied some conditions: the bacteria must exist in this environment and the soil must have suitable nutrients and a convenient temperature to sustain the growth of bacteria and its activity.More, the soil must not contain any chemicals that could slow down the chemotaxis.We suppose that the process is not influenced by any other fluid flow.Other physical models, in which besides the pollutant, flow other fluids influencing pollutant and bacteria concentrations and the porosity of soil can be found in [6][7][8].
During the bioremediation process the pollutant c and the bacteria b are moving by diffusion, too.Diffusion is the movement of molecules in the sense of decreasing concentration (from an area with higher concentration to an area with lower concentration).In order to make the model be as real, we will consider that the kinetic term and the diffusion coefficient of the pollutant c have a weak influence on the flow.This means that the rate of degradation of the chemoattractant is slow and it diffuses very little (as in the case of oil) [9].
From mathematical point of view, processes involving the chemotaxis and the diffusion are represented by a system of partial differential equations, written for the concentration of bacteria population b and for the concentration of pollutant c respectively, with initial and boundary conditions.
The main aim of this work is to make some numerical simulations for this model of reaction-diffusion with chemotaxis and to compare the influence of different parameter values on the systems.

MATHEMATICAL MODEL
A mathematical reaction-diffusion model of chemotaxis is expressed by a system of partial differential equations for the movement of bacteria whose density is denoted by b in response to chemical gradients emitted by the pollutant c [10].In the system, we denote the diffusion coefficient of bacteria with D and the diffusion coefficient of pollutant with  .The chemotactic sensitivity of bacteria is denoted by   c b K , [1] To write the equations system, we consider the cylindrical domain given by (1): where 2  is an open bounded subset of 2  R .
We consider that the domain  is composed of n parallel layers along the Ox axis, denoted by i  .We suppose that these layers i  have the boundaries given by (2): , ,..., 1 , Each layer of the soil represented in the Figure 1 is characterized by the following parameters and functions that do not explicitly depend on 0 , , , , , , : . We assume that the values 0 , , , are constant in each layer, but different from one layer to another.Functions as


have different expressions from one layer to another.We assume that the system is closed for both bacteria and pollutant, namely the fluxes across the exterior frontiers are zero.
We have as mathematical model the following dimensionless system given by the equations ( 4)-( 7): ), ( , 0 0 , (7) with the conditions ( 8) and ( 9) at the interface between two layers: where  is the unit outer normal to .
lat i


This mathematical model with a small influence of the degradation rate of the pollutant and of its diffusion coefficient (for example, oil), in relation to other parameters was mathematical studied in [9] and [11].This means      were considered as small parameters.For studying this model was made an asymptotic analysis.To this end all the functions were developed with respect to the powers of the small parameter and were treated two systems for the 0-order and 1-order approximations.The existence and uniqueness of a global in time solution for the asymptotic model were studied within the framework of the evolution equations with maccretive operators in Hilbert spaces, under certain assumptions for the functions  , f and K .
We will use the small parameters for numerical purposes and we will compare the influence of different parameter values on the solution behaviour.

NUMERICAL SIMULATIONS. RESULTS AND DISCUSSION
In order to achieve the numerical simulations we used the package COMSOL Multiphysics v3.5a [12], working with the finite element method.We consider a 1D domain, We make the simulations for a time interval   shows that the bacteria diffuses uniformly.
For the first simulation we still consider the chemotactic sensitivity of bacteria with the form (19) The values for b and c are evaluated from the system (4)- (7).
We consider the same value for the chemotactic constant  If we compare the Figures 4 and 5, we observe that in both situations the pollutant behavior is not significantly influenced by changes of chemotactic sensitivity.For the case with chemotactic sensitivity depending only on pollutant concentration (Figure 5.a), the density of bacteria has higher values than in the other case.Moreover, we observe that in each layer, the concentration of bacteria b remains around on the same values level, but it has steep variations from one layer to another, especially at the beginning of the process. .

CONCLUSIONS
The study of such a model is justified by the application of bioremediation processes occurring in stratified . The function denoted by   c b f , describes the rates of growth or death of bacteria b and   c b,  is the function describing the degradation of the chemoattractant c .
boundaries between layers, 0  and n  are the external boundaries and lat i  are the lateral boundaries of i  .This model is represented in the Figure 1.The model is studied in a finite time interval   T , 0 and we denote i Q by (3): conditions (10)-(15) for the horizontal and lateral boundaries: given by (18).The initial concentration of the pollutant   x c 0 has the maximum value on the first layer values, next is halved on the second layer and then completely destroyed.The diffusion coefficient   x  shows that the pollutant spread quickly in the middle layer and the diffusion coefficient   x D

.
So the Figures 2.a. and 3.a represent the graph of the solution b according to x , at fixed time moments b. and 3.b.represent the solution c according to x , at the same time moments.If we compare Figures 2. and Fig. 2. Solutions   x t b , (a) and   x t c , (b) for chemotactic coefficient 5  