Team:ULB-Brussels/Modelling/Population-Dynamics

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<h3>$1.2)$ $By$ $Logistic$ $Equation$</h3>
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<h3>$1.2)$ $By$ $Logistic$ $\small\&\normalsize$ $Lotka$ $Equation$</h3>
The Logistic Equation was initially introduced during the beginning of the XIXth Century, by the belgian mathematician P.F. Verhulst. Now, this equation is mainly used in Population Dynamics Models, especially in Biological Sciences. Mathematicians currently finish Ph.D thesis using this, and the analytical Lotka-Volterra model is directly associated with the Verhulst theory.</p>
The Logistic Equation was initially introduced during the beginning of the XIXth Century, by the belgian mathematician P.F. Verhulst. Now, this equation is mainly used in Population Dynamics Models, especially in Biological Sciences. Mathematicians currently finish Ph.D thesis using this, and the analytical Lotka-Volterra model is directly associated with the Verhulst theory.</p>
Another interesting model is obtained from Euler-Lotka equation to the Leslie matrix coefficients. By identifying the caracteristic eigenvalues and eigenvectors analyse, we estimate the asymptotic stability (stable steady state) and the profile of the growth rate, as in [Fig.m3]. An advantage is that this model is yet discrete, so by computing it, we don't loose info of the background theory.  
Another interesting model is obtained from Euler-Lotka equation to the Leslie matrix coefficients. By identifying the caracteristic eigenvalues and eigenvectors analyse, we estimate the asymptotic stability (stable steady state) and the profile of the growth rate, as in [Fig.m3]. An advantage is that this model is yet discrete, so by computing it, we don't loose info of the background theory.  

Revision as of 10:23, 5 September 2014

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newcommand{\MyColi}{{\small Mighty\hspace{0.12cm}Coli}} \newcommand{\Stabi}{\small Stabi}$ $\newcommand{\EColi}{\small E.coli} \newcommand{\SCere}{\small S.cerevisae}\\[0cm] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newcommand{\PI}{\small PI}$ $\newcommand{\Igo}{\Large\mathcal{I}} \newcommand{\Tgo}{\Large\mathcal{T}} \newcommand{\Ogo}{\Large\mathcal{O}} ~$ Example of a hierarchical menu in CSS

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- Université Libre de Bruxelles -



Population Dynamics Model

A Population Dynamics Model can be fitted in our system. Theoretically, two approaches have been planned:

$1.1)$ $By$ $Probabilities$

When some new plasmids are genetically introduced into the cytoplasm of E.Coli bacteria, this doesn't garantee that the future copies will contain it. Indeed, these plasmids can be lost after cell division or replication, so it's interesting to study a model based on the different possibilities of plasmid combinations in bacteria, like in the studies of mutations in animals. A typical example of a similar way is found if we study the mutations of the eyes or hair color in a family, by vertical genes transfer. In this case, there's a horizontal genes transfer too, originated by the plasmids.

Figure m2 : A first example of bacteria generations at same acceptation plasmid rate for a Toxin (T), an Antitoxin (A), the two (T&A) or no plasmid (-). The number on the right of the letter G indicates the generation number.

We remark in [Fig m2] that we'll converge to finally obtain bacteria without Toxin or Antitoxin. In this case, four kinds of plasmids are ingered at same rate by bacteria. A Probabilistic Model is useful because easily undertsood, but necessits some assumptions to make sense.

$1.2)$ $By$ $Logistic$ $\small\&\normalsize$ $Lotka$ $Equation$

The Logistic Equation was initially introduced during the beginning of the XIXth Century, by the belgian mathematician P.F. Verhulst. Now, this equation is mainly used in Population Dynamics Models, especially in Biological Sciences. Mathematicians currently finish Ph.D thesis using this, and the analytical Lotka-Volterra model is directly associated with the Verhulst theory.

Another interesting model is obtained from Euler-Lotka equation to the Leslie matrix coefficients. By identifying the caracteristic eigenvalues and eigenvectors analyse, we estimate the asymptotic stability (stable steady state) and the profile of the growth rate, as in [Fig.m3]. An advantage is that this model is yet discrete, so by computing it, we don't loose info of the background theory.

Figure m3 : Quantity of bacteria normalized by the maximal value in function of time, with the second bacterial growth evolution theory purposed. The population goes to an asymptotic stable state.

The idea is to compare these two models, applicated in our experimental growth conditions and to fit the better corresponding with our colonies, balanced with explicative biological arguments to interprete the choice. Other models exist, f.e. by Monod equation, but this idea is probably not consistent with our global system.
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