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modelling of reaction term of the adr equation

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MODELLING OF REACTION TERM OF THE ADVECTION-DIFFUSION-REACTION EQUATION FOR ENVIRONMENTAL PROBLEMS. Introduction The Advection Diffusion Reaction Equation (ADRE), φ φ φ φ Γ + ⎟ ⎟  ⎠  ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = ∂ ∂ + ∂ ∂ i i T i i  x  x K  x U t , (1) where K T is the transference coefficient and Г φ is the reaction term, is not only used to evaluate the hydrodynamics (velocity vectors) of a system, it’s also used to calculate scalars, such as Temperature, Salinity, Concentration of a Pollutant, Population Dynamics, Equilibrium Variables of the system (BOD, OD), etc. The main difference in the treatment of the reaction term in the ADRE is that it’s represented by an Ordinary Differential Equation (ODE). Since the methods to solve the ODE’s are different from those to solve the former parts of the ADRE, a new choose of method should be made. Problems of interest 1. Population Dynamics. The choose model for this dynamics is the Lotka-Volterra model, which consists in differential equations that are used to model predator-prey interactions. The system considers of two entities. The Lotka-Volterra equations for the two dimensional system with exponential growth is defined by the following differential equations: ( ) ( ) ( ) ⎩ ⎨ ⎧ + − = + − = − = − = = ) ( ) ( ,  Dx C  y  Dxy Cy  y  By  A  x  Bxy  Ax  x t  y t  x F (2) Where x is the prey population, y is the predator population, A, B, C, and D are positive constants representing the growth and decay rate, and the interspecies interactions given in day -1 . In order to extend the two entities system above into a three entities system, a third specie is added to the system. In this case the third specie feeds exclusively on the first predator population. The three species Lotka-Volterra system is defined by: ( ) ( ) ( ) ( ) ⎪ ⎩ ⎪ ⎨ ⎧ + − = + − = − + − = − + − = − = − = = Gy F  z Gzy Fz  z  Ez  Dx C  y  Eyz  Dxy Cy  y  By  A  x  Bxy  Ax  x t  y t  x F ) ( ) ( , (3) Where z is the new predator population, E, F and G are positive constants representing decay rate and the new interspecies interactions given in day -1 .

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