Weak Equivalence Transformations for a Class of Models in Biomathematics

Igor Leite Freire,Mariano Torrisi

doi:10.1155/2014/546083

Igor Leite Freire, Mariano Torrisi

Open Access

https://doi.org/10.1155/2014/546083

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Abstract

A class of reaction-diffusion systems unifying severalAedes aegyptipopulation dynamics models is considered. Equivalence transformations are found. Extensions of the principal Lie algebra are derived for some particular cases.

Highlights

In this paper we focus our attention on the following class of nonlinear advection-reaction-diffusion systems: ut = (f (u) ux)x + g (u, V, ux), gux ≠ 0, (1) Vt = h (u, V) .These systems can describe the evolution of the densities u and V of two interacting populations where the balance equation for u takes into account the reaction-diffusion effects and some advection effects while the balance equation for density V takes into account only the so-called reaction terms
The studied class includes, as particular cases, all partial differential equation models concerned with the Aedes aegypti mosquito that have been proposed until now
We have investigated such a system from the point of view of equivalence transformations in the spirit of the Lie-Ovsiannikov algorithm based on the Lie infinitesimal criterion

Summary

Introduction

We recall that in (2), as well as in [1], u and V are, respectively, nondimensional densities of winged and aquatic populations of mosquitoes and k, γ, m1, and m2 are nondimensional, in general, positive parameters, ] ∈ R. The family of system (1) contains arbitrary functions or numerical parameters, which specifies the individual characteristics of phenomena belonging to large subclasses In this sense, the knowledge of equivalence transformations can provide us with certain relations between the solutions of different phenomena of the same class and allows us to get symmetries in a quite direct way. In this paper we look for certain equivalence transformations for the class of systems (1) in order to find symmetries for special systems belonging to (1) and to get information about constitutive parameters f, g, and h appearing there.

Elements on Equivalence Transformations

Calculation of Weak Equivalence Transformations

Some Extensions of LP

Conclusions