Team:ULB-Brussels/Modelling/Population-Dynamics/Lotka

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

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- Université Libre de Bruxelles - Population Dynamics by Logistic & Lotka Equations 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 these are less consistent with our global and partial systems.
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