Team:Peking/ProjectApplication

Introduction

In order to the estimate the effect of killing and improve the application method, other than varied tests in laboratory, we also develop a macro-level model to examine whether our E. coli will work well considering spatial distribution and diffusion. We described interaction and diffusion of elements in the fresh water system by using Partial Differential Equations (PDEs). Even though we don’t know the analytical expression of interaction of elements in water, we can propose a possible interaction function, and analyze its spatial properties, in order to find an optimized way to pour bacterium solution.

Method

Finite-difference methods are applied in solving PDEs in this model.

Diffuision

Differentials to space coordinates are given by Fick Law. The length of a time step is approximated to Δt = 1min, and the discretization of space coordinates is approximated to Δs = 1dm(s = x, y, z), considering the scale of boundary is 10²m and the time scale is 10¹ days.

Interaction

A, E, L indicate the concentration of algae, our programming E. coli, lysozym which kills algae. We also introduce a variable which describes the organic nutrition released by cracked algae(indicated by N), which contributes to respiration of E. coli. Interacitons between them are given below.

The first term in Equ(1) describes the growth of algae by logistic model, in which K1 indicate the growth rate of algae and k2 indicate the steady value. The second term means the Killing effect by lysozyme. Equ(2) describes the accumulation and consumption of organic nutrition. Equ(3) describes the reproduction and natural mortality of our E. coli and Equ(4) describes secretion and reduction of lysozyme

There are several factors which are not considered in equations above such as nutrition accumulation by algae which dies naturally, degradation of lysozyme, evaluation by binding part and so on, because they are either too small compared with other items, or they are irrelevant to the issue that we care. As this model is focused on the application in macro-scale water system, accurate parameters are not required meanwhile not available, considering the form of equations could not be proved. So some parameters are given by speculation to satisfy our expactation

Spatial Analysis

We control the total volume of bacteriuma solution, and we tried several basic mode of pouring: at one point, at several points, along a line or evenly on the whole surface. Then we measured the time scale of algae elimination. From this, we can find out if mode of pouring

Result

Parameter manipulation

Ignoring the diffusion, the PDEs were converted to Ordinary Differential Equations (ODE).

Speculatively the equations have two typical form of solution, indicating the success and failure of the project. For there is a positive feedback mechanic by introducing organic nutrition, oscillatory solution could not exist.

Through the test, we noticed that the reasonable type of the solution is the only “success” (Fig. 1) regardless of reasonable variation of k9 and k5. The other paremeters are not the crucial factor of the type of the solution as the magnitude of order are estimable, for example ,K₆, the growth rate of E. coli.