# Weak Equivalence Transformations for a Class of Models in Biomathematics

Authors: Igor Leite Freire; Mariano Torrisi
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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


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## Body

## 1. Introduction

In this paper we focus our attention on the following class of nonlinear advection-reaction-diffusion systems:(1)ut=(f(u)ux)x+g(u,v,ux),gux≠0,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 not only the reaction-diffusion effects but also some advection effects while the balance equation for density v takes into account only the so-called reaction terms. The advection effects are due to the presence in the function g of the gradient ux and appear when the individuals of population u feel external stimuli as, for instance, wind effects or water currents. Class (1) can be considered a generalization of the equations with the typical properties of the already knownAedes aegypti mathematical models [1–3]. We recall them shortly in the following.TheAedes aegypti mosquitos are the main vector of dengue, a viral disease that causes the so-called dengue hemorrhagic fever characterized by coagulation problems often leading the infected individual to death. Since the subtropical zone climate and environmental conditions are favorable to the development ofAedes aegypti, dengue is a serious public health problem in many countries around the world. However, due to the global warming, the interest in such considered mosquitoes is not restricted to those places affected by the disease, but it is also of interest for those countries whose weather, in the next decades, can become similar to the current environmental found in the subtropical zone. Therefore, the interest in modeling such a vector is not only a theoretical deal but also a way for finding methods and alternatives to overcome and control the problems arising from the dispersal dynamics of the mosquitos and, consequently, the propagation of the disease.The following system(2)ut=(upux)x-2νuqux+γkv(1-u)-m1u,vt=k(1-v)u-(m2+γ)v,
where p,q∈ℝ, belongs to class (1). It was introduced in [2] as a generalization of a model studied in [1].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, ν∈ℝ. Specifically k is the ratio between two constants k¯1 and k¯2, which are, respectively, the carrying capacity related to the amount offindable nutrients and the carrying capacity effect dependent on the occupation of the available breeder and γ denotes the specific rate of maturation of the aquatic form into winged female mosquitoes, while m1 and m2 are, respectively, the mortality of winged and the mortality of aquatic populations. Finally ν denotes a constant of velocity for flux due to wind currents that, in general, generate an advection motion of large masses of the winged population and consequently can facilitate a quick advance of the infestation. For further details, see [4] and references therein.Another system belonging to (1) is
(3)ut=(upux)x-2νuqux+γkv+(γk-m1)u,vt=ku+(k-m2-γ)v,
which was also introduced in [2] starting from (2) and, due to the weak interaction between the aquatic and winged populations, by modifying the source terms.The first equation of (3) gives the time rate of change of the mosquitoes density as a sum of the growth terms (γ/k)v, of theper capita death rate ((γ/k)-μ1)u and the diffusive-advective flux due to the movement of mosquito population.The second equation gives the corresponding time rate of change of the density of aquatic population as a sum of the growth termku of aquatic population, due to the new egg depositions of female mosquitoes, withper capita death rate (k-μ2)v of aquatic population. The term -γv represents the loss due to the change of the aquatic into winged form; see [4].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.Following [5], an equivalence transformation is a nondegenerated change of independent and dependent variables t, x, u, and v into t^, x^, u^, and v^:
(4)x=x(x^,t^,u^,v^),t=t(x^,t^,u^,v^),u=u(x^,t^,u^,v^),v=v(x^,t^,u^,v^),
which maps a system of class (1) in another one of the same class, that is, in an equation preserving differential structure but, in general, with
(5)f^(u^)≠f(u),g^(u^,v^,u^x^)≠g(u,v,ux),h^(u^,v^)≠h(u,v).Of course in the case(6)f^(u^)=f(u),g^(u^,v^,u^x^)=g(u,v,ux),h^(u^,v^)=h(u,v)
an equivalence transformation becomes a symmetry.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. Moreover we wish to stress that, as it is known, an equivalence transformation maps solutions of an equation in solutions of the transformed equation [6]. Then in order to find solutions for a certain equation one can look for the equivalence transformations that bring the equation in simpler other ones whose solutions are well studied; see, for example, [7, 8] and references inside.The plan of the paper is as follows. In the next section we provide some elements about equivalence transformations. In Section3 we apply these concepts in order to obtain a set of weak equivalence generators. In Section 4, after having introduced a projection theorem, we show how to apply it to find symmetries of (1). In Section 5, after having introduced a special structure of the advection-reaction function g that generalizes that one used in (3), we find extensions with respect to theprincipal Lie algebra. Conclusions and final remarks are given in Section 6.


## 2. Elements on Equivalence Transformations

In the past differential equation literature it is possible to find several examples of equivalence transformations. The direct search for the most general equivalence transformations through the finite form of the transformation is connected with considerable computational difficulties and quite often leads to partial solutions of the problem (e.g., [9, 10]).A systematic treatment to look for continuous equivalence transformations by using the Lie infinitesimal criterion was suggested by Ovsiannikov [11].In general, the equivalence transformations for class (1) can be considered as transformations acting on point of thebasic augmented space(7)A≡{t,x,u,v,ut,ux,vt,vx,f,g,h}.The previous elements allow us to consider, in the following, the one-parameter equivalence transformations as a group of transformations, acting on thebasic augmented space  A, of the type
(8)x=x(x^,t^,u^,v^,ε),t=t(x^,t^,u^,v^,ε),u=u(x^,t^,u^,v^,ε),v=v(x^,t^,u^,v^,ε),f=f(x^,t^,u^,v^,u^t^,u^x^,v^t^,v^x^,f^,g^,h^,ε),g=g(x^,t^,u^,v^,u^t^,u^x^,v^t^,v^x^,f^,g^,h^,ε),h=h(x^,t^,u^,v^,u^t^,u^x^,v^t^,v^x^,f^,g^,h^,ε),
which is locally a C∞-diffeomorphism, depending analytically on the parameter ε in a neighborhood of ε=0, and reduces to the identity transformation for ε=0.Following [6, 11–14] (see also, e.g., [5, 15–17] ) we consider the infinitesimal generator of the equivalence transformations (8) of the systems (1) that reads as follows:
(9)Y=ξ1∂x+ξ2∂t+η1∂u+η2∂v+μ1∂f+μ2∂g+μ3∂h,
where the infinitesimal components ξ1, ξ2, η1, and η2 are sought depending on x, t, u, and v, while the infinitesimal components μi(i=1,2,3) are sought, at least in principle, depending on x, t, u, v, ut, ux, ux, vt, vx, f, g, and h. In order to obtain thedetermining system which allows us to get the infinitesimal coordinates ξi, ηi, and μj (i=1,2 and j=1,2,3), we apply the Lie-Ovsiannikov infinitesimal criterion by requiring the invariance, with respect to suitable prolongations Y(1) and Y(2) of (9), of the following equations:
(10)ut-(fux)x-g=0,vt-h=0,
together with the invariance of theauxiliary conditions [13, 14, 18, 19]:
(11)ft=fx=fv=fux=fut=fvx=fvt=gt=gx=gut=gvt=gvx=0,ht=hx=hux=hut=hvx=hvt=0,
where u and v are (t,x) functions while f, g, and h are considered as functions depending,a priori, on (t,x,u,v,ut,ux,vt,vx). All of these functions are assumed to be analytical. The constraints, given by (11), characterize the functional dependence of f, g, and h.In this paper, instead, in view of further applications and following [20], we modify the previous classical procedure by looking for equivalence transformations whose generators are got by solving thedetermining system obtained from the following invariance conditions:
(12)Y(2)[ut-(f(u)ux)x-g(u,v,ux)]|ut-(f(u)ux)x-g(u,v,ux)=0,vt-h(u,v)=0=0,(13)Y(1)[vt-h(u,v)]|ut-(f(u)ux)x-g(u,v,ux)=0,vt-h(u,v)=0=0.
As the functional dependences of the parameters f, g, and h are knowna priori we do not require the invariance of theauxiliary conditions (11). In this way we work in abasic augmented space  A≡{t,x,u,v,ux,f,g,h}. Therefore the μi components must be sought, at least in principle, depending on x, t, u, v, ux, f, g, and h.The infinitesimal operators, obtained by following this shortening procedure, can generate transformations that map equations of our class into new equations of the same class where the transformed arbitrary functions may have new additional functional dependencies. Such transformationsare called weak equivalence transformations [13, 14].With respect to the application in biomathematical models, equivalence and weak equivalence transformations were applied not only to study of tumor models [21, 22] but also to the population dynamics in [20, 23, 24].


## 3. Calculation of Weak Equivalence Transformations

In order to avoid long formulas and writeY(1) and Y(2) in a compact way, we put
(14)x=x1,t=x2,u=u1,v=u2,f=h1,g=h2,h=h3.
For this reason system (1) is rewritten as
(15)ux21-h2-hu11(ux11)2-h1ux1x11=0,ux22-h3=0,
while the equivalence generator assumes the following form:
(16)Y=ξi∂xi+ηα∂uα+μA,∂hA,
where i=1,2, α=1,2, and A=1,2,3. Here the summation over the repeated indices is presupposed.After putting(17)(z1,z2,z3,z4,z5)=(x1,x2,u1,u2,u11),(ν1,ν2,ν3,ν4,ν5)=(ξ1,ξ2,η1,η2,ζ11),(18)uiα=uxiα,haA=hzaA,(19)D~ae=∂za+haA∂hA,(20)Dje=∂xj+ujα∂uα+uijα∂uiα+⋯,
the prolongations Y(1) and Y(2) assume the following form:
(21)Y(1)=Y+ζjα∂ujα+ωaA∂haA,Y(2)=Y(1)+ζ111∂u111=Y+ζjα∂ujα+ωaA∂haA+ζ111∂u111,
where
(22)ζjα=Djeηα-ukαDjeξk,ζijα=Djeζjα-uikαDjeξk,ωaA=D~aeμA-hbAD~aeνb.The invariant conditions read(23)Y(2)F1=ζ21-μ2-2ζ11hu11u11-(ω31)(u11)2-(μ1)u111-h1ζ111=0,(24)Y(1)F2=ζ22-μ3=0,
both under the constraints (15), which are after (18)
(25)u21=h2+hu11(u11)2+h1u111,(26)u22=h3.The coefficientsζ11, ζ21, ζ22, and ω31 are given, respectively, by
(27)ζ11=η11+(ηu11-ξ11)u11-ξ12u21+ηu21u12-ξu11(u11)2-ξu21u11u12-ξu12u11u21-ξu22u21u12,ζ21=η21-ξ21u11+(ηu11-ξ22)u21+ηu21u22-ξu11u11u21-ξu21u11u22-ξu12(u21)2-ξu22u21u22,ζ22=η22+ηu12u21+(ηu22-ξ22)u22-ξ21u12-ξu11u12u21-ξu21u12u22-ξu12u21u22-ξu22(u22)2,ω31=μu11+hu11μh11+hu12μh21+hu13μh31-hu11ηu11.Taking into account (24) and (27), we can write
(28)Y(1)F2=η22+ηu12u21+(ηu22-ξ22)u22-ξ21u12-ξu11u12u21-ξu21u12u22-ξu12u21u22-ξu22(u22)2-μ3=0,
with the constraints (25) and (26).Then, substituting (25) and (26) into (28), we get
(29)η22+ηu12(h2+hu11(u11)2+h1u111)+(ηu22-ξ22)h3-ξ21u12-ξu11u12(h2+hu11(u11)2+h1u111)-ξu21u12h3-ξu12(h2+hu11(u11)2+h1u111)h3-ξu22(h3)2-μ3=0.From this condition we obtain the following determining equations:(30)η22+(ηu22-ξ22)h3-ξu22(h3)2-μ3=0,ξ21=0,ξu11=0,ξu21=0,ηu12=0,ξu12=0.
Then it follows that
(31)ξ1=ξ1(x1),ξ2=ξ2(x1,x2,u2),η2=η2(x1,x2,u2),η1=η1(x1,x2,u1,u2),μ3=η22+(ηu22-ξ22)h3-ξu22(h3)2
and, consequently, μu113=0.Following the same procedure we can write the invariance condition (23) as
(32)Y(2)F1=η21+(ηu11-ξ22)(h2+hu11(u11)2+h1u111)+ηu21h3-ξu22(h2+hu11(u11)2+h1u111)h3-μ2-2{η11+(ηu11-ξ11)u11-ξ12(h2+hu11(u11)2+h1u111)+ηu21u12+-ξu22(h2+hu11(u11)2+h1u111)u12}hu11u11-(μu11+hu11μh11+hu12μh21+hu13μh31-hu11ηu11)×(u11)2-(μ1)u111-h1(D1eη11+D1e(ηu11-ξ11)u11+(ηu11-ξ11)u111+(D1eηu21)u12+ηu21u112)-h1(-D1e(ξ12)(h2+hu11(u11)2+h1u111)-ξ12u211-D1eξu22(h2+hu11(u11)2+h1u111)u12-ξu22u211u12-ξu22(h2+hu11(u11)2+h1u111)u112-u111ξ11)=0.Collecting the terms with(hu11)2 and with hu11 and equating their respective coefficients to zero it is obtained that,
(33)ξ12=0,ξu22=0,η11=0,μh11=2ξ11-ξ22,ηu21=0η11=0.Thus, from the coefficients ofhu12 and hu13(34)μh21=μh31=0.
Considering the coefficients of u111 we get
(35)μ1=(2ξ11-ξ22)h1=0⟹μu11=0.
The remaining terms of (32) give the following form for the infinitesimal component of h2:
(36)μ2=η21+(ηu11-ξ22)h2+(ξ111-ηu1u11u11)h1.
Therefore, once having taken into account all restrictions obtained we are finally able to write the following infinitesimal components for theweak equivalence generators:
(37)ξ1=α(x1),ξ2=β(x2),η1=δ(x2,u1),η2=λ(x1,x2,u2),μ1=(2α′-β′)h1,μ3=(λu2-β′)h3+λx2,
while, from (36), we get
(38)μ2=δx2+(δu1-βx2)h2+(αx1x1-δu1u1ux11)h1,
where α(x1), β(x2), δ(x2,u1), and λ(x1,x2,u2) are arbitrary real functions of their arguments. Then, going back to the original variables, the most general operator of these continuous weak equivalence transformations reads
(39)Y=α(x)∂x+β(t)∂t+δ(t,u)∂u+λ(x,t,v)∂v+(2α′-β′)f∂f+(δt+(δu-βt)g+(α′′-δuuux)f)∂g+((λv-β′)h+λt)∂h.


## 4. Symmetries for the System (1)

In the next sections in order to carry out symmetries for the system (1) we do not use the classical Lie approach. Instead of the mentioned method we apply theprojection theorem, introduced in [25] and eventually reconsidered in [13, 14, 18, 19]. In agreement with these references, we can affirm the following.Theorem 1.
Let(40)Y=α(x)∂x+β(t)∂t+δ(t,u)∂u+λ(x,t,v)∂v+(2α′-β′)f∂f+(δt+(δu-βt)g+(α′′-δuuux)f)∂g+((λv-β′)h+λt)∂h
be an infinitesimal equivalence generator for the system (1); then the operator
(41)X=α(x)∂x+β(t)∂t+δ(t,u)∂u+λ(x,t,v)∂v,
which corresponds to the projection of  Y on the space (x,t,u,v), is an infinitesimal symmetry generator of the system (1) if and only if the constitutive equations, specifying the forms of f, h, and g, are invariant with respect to Y.For the system under consideration, in general, the constitutive equations whose invariance must be requested are(42)f=D(u),g=G(u,v,ux),h=F(u,v).
The request of invariance
(43)Y(f-D(u))|(42)=0,Y(1)(g-G(u,v,ux))|(42)=0,Y(h-F(u,v))|(42)=0
brings us to the following equations:
(44)μ1-η1Du=0,μ2-ζ11Gux-η1Gu-η2Gv=0,μ3-η1Fu-η2Fv=0,
under the restrictions (42).Then substituting(45)ζ11=(ηu11-ξ11)u11=(δu-α′)ux
and taking into account the constraints we can write (44) as
(46)(2α′-β′)D(u)-δ(t,u)Du=0,(47)δt+(δu-βt)G-(δuuux2-αxxux)D-(δu-α′)uxGux-δGu-λ(t,x,v)Gv=0,(48)((λv-β′)F+λt)-δFu-λ(t,x,v)Fv=0.
We recall here that the principal Lie algebra L𝒫 [5, 12] is the Lie algebra of the principal Lie group, that is, the group of the all Lie point symmetries
(49)X=ξ(x,t,u,v)∂∂x+τ(x,t,u,v)∂∂t+η1(x,t,u,v)∂∂u+η2(x,t,u,v)∂∂v
that leave the system (1) invariant for any form of the functions D(u), G(u,v,ux), and F(u,v). In other words we can remark that the principal Lie algebra is the subalgebra of the equivalence algebra such that any operator Y of this subalgebra leaves the equations f=D(u), g=G(u,v,ux), and h=F(u,v) invariant for any form of the functions D(u), G(u,v,ux), and F(u,v). Then we can say [5] the following.Corollary 2.
An equivalence operator for the system (1) belongs to theprincipal Lie algebra  L𝒫 if and only if ηi=0, μj=0, i=1,2, and j=1,2,3.Taking Corollary2 into account from the previous equations (46)-(48) it is a simple matter to ascertain that the L𝒫 [5, 12] is spanned by the following translation generators:
(50)X0=∂t,X1=∂x.


## 5. Some Extensions ofL𝒫

In order to show some extensions of the principal algebra, which could be of interest in biomathematics, we assume that the advection-reaction function is of the form(51)G=ρuruxs+Γ1ua+Γ2vb,
where the parameters ρ, Γ1, Γ2, r, s, a, and b are constitutive parameters of the considered phenomena.This form ofG is a generalization of
(52)G=-2νuqux+γkv+(γk-m1)u,
appearing in (3), where
(53)ρ=-2ν,Γ1=(γk-m1),Γ2=γk
Consequently in (51) we must consider Γ2>0 and as limit cases Γ1=0 and a=0. Moreover, in this section, we assume that the value s=0 will not be considered because in this case the advective effects disappear. We also assume that b≠0. This last restriction implies that the balance equation of the density u depends on the density v. Finally for the sake of simplicity we omit the limit case Γ1=0 and assume that the diffusion is only nonlinear; that is, Du≠0.In the following we continue the discussion of invariance conditions written in the previous section.From (46), by deriving with respect to x, we get
(54)α′′=0.
Then
(55)α=α1x+α0
with α1 and α0 arbitrary constants, so (46) becomes
(56)(2α1-β′)D(u)-δ(t,u)Du=0
while, after having taken (51) into account, (47) reads
(57)δt+(δu-βt)(ρuruxs+Γ1ua+Γ2vb)-Dδuuux2-(δu-α1)ρsuruxs+-δ(ρrur-1uxs+Γ1aua-1)-λ(t,x,v)Γ2bvb-1=0.
From (57) we get immediately
(58)λ=λ(t,v).In the following we analyze separately the cases≠2 and the case s=2.


### 5.1.s≠2

Fromux2 coefficient we get δuu=0; that is,
(59)δ=δ1(t)u+δ0(t).Therefore, from the remaining terms we have(60)ρur-1uxs[u((1-s-r)δ1+sα1-β′)-rδ0]+δ0′+δ1′u+Γ1((1-a)δ1-β′)+-aΓ1δ0ua-1+Γ2(δ1-β′)vb-Γ2λbvb-1=0.As we assumeds≠0, from the coefficient of uxs in (60) we conclude that δ0(t)=0 and
(61)β′=(1-s-r)δ1+sα1.
Then, still from (60) we have the following constraints to consider:
(62)δ1′u+Γ1ua((1-a)δ1-β′)=0,(63)(δ1-β′)vb-λbvb-1.
From (62) two cases are obtained.(i) Case
a
≠
1
. Then, from (62) we conclude that δ1=const and it follows that
(64)β′=(1-a)δ1.
From (64) and (61) we obtain
(65)α1=s+r-asδ1,β=(1-a)δ1t+β0,
with β0 and δ1 arbitrary constants.The analysis of (63) leads to the following two subcases.(
1
) Consider λ(v)=λ0v, with
(66)λ0=abδ1.
Taking into account the previous results and going back to (56) and (48) we get that the system (3) with G of the form (51) admits the 3-dimensional Lie algebra spanned by the translations in space and time and by the following additional generator:
(67)X3=(1-a)∂t+1s(s+r-a)x∂x+u∂u+abv∂v,
provided that D and F are solutions of the following differential equations:
(68)uDu=((1+a)+2r-as)D,buFu+avFv=(a-b(1-a))F.(
2
) Consider λ(v)=δ1=0. In this case the only symmetries admitted are translations of the independent variables and the form of D and F is arbitrary, so there is not an extension of the principal Lie algebra.(ii) Case
a
=
1
. In this case from (62) it follows that
(69)δ1′-Γ1β′=0.
After having substituted (69) into (61) we obtain
(70)β(t)=-β0+c1e(1-s-r)Γ1t,δ1(t)=ss+r-1α1+c1Γ1e(1-s-r)Γ1t,
with α1, β0, and c1 arbitrary constants and once assumed s+r-1≠0. Finally we analyze the contribution of (63) from where the following two subcases arise.(
1
)  Consider λ(t,v)=λ0(t)v, with
(71)λ0(t)=1bδ1(t).
Then taking into account the previous results it reads
(72)λ0(t)=1b[ss+r-1α1+c1Γ1e(1-s-r)Γ1t].
Going to put the previous result in the condition (46) and by separating the variable we get
(73)β(t)=-β0,δ1(t)=ss+r-1α1,λ0(t)=1b[ss+r-1α1],
with α1 and β0 arbitrary constants and provided that the diffusion coefficient D is solution of
(74)uDu=2(1+r-1s)D.
From condition (48) we do not get further restrictions on the infinitesimal components of the symmetry generator but only the following constraint on the reaction function F:
(75)buFu+vFv=F.
Taking into account the arbitrariness of α0 in this case we have got a 3-dimensional Lie algebra. The additional generator is
(76)X3=x∂x+ss+r-1u∂u+1bss+r-1v∂v.
In particular, from (75), setting F=ku+(k-m2-γ)v, we obtain b=1, while from (75) we obtain the power function D=D0u2((r+s-1)/s).Remark 3.
By puttingr=1, the corresponding system of class (1) admits still a 3-dimensional symmetry Lie algebra generated by translations in time and in space and by X3. These results are in agreement with the ones obtained in [2].By considerings+r-1=0 we do not get extension of L𝒫 for D and F arbitrary. We get, instead, the following extension:
(77)X3=st∂t+x∂x+Γ1stu∂u+Γ1stbv∂v,
provided that D=D0=const. and F=(Γ1/b)v.(
2
) Consider λ(t,v)=δ1(t)=0. In this case the only symmetries admitted are the translations in space and time and the form of D and F is arbitrary.


### 5.2.s=2

We analyze this case by beginning with the discussion of (56) from which two cases arise.(
1
)  D(u)Is Arbitrary.  It follows, of course, that δ=0 and β′=2α1 so
(78)β=2α1t+β0
with β0 arbitrary constant.Moreover (46) and (48) become
(79)-2α1G+α1uxGux-λ(v)Gv=0,(80)(λv-2α1)F-λ(v)Fv=0.
But taking into account that
(81)G=ρurux2+Γ1ua+Γ2vb
(79) becomes
(82)-2α1(ρurux2+Γ1ua+Γ2vb)+2α1ρurux2-λ(v)bΓ2vb-1=0
and gives us the following conditions:
(83)2α1Γ1=0,2α1v+λ(v)b=0
from where, taking into account the work hypotheses at the beginning of this section, we get α1=λ=0 with F arbitrary function of u and v.In this case the only admitted symmetries are translations in time and space.(
2
)
Consider
D
u
/
D
=
(
2
α
1
-
β
′
)
/
δ
(
t
,
u
)
. Then by requiring
(84)∂∂t(2α1-β′δ(t,u))=0,
we get
(85)-δ(t,u)β′′-(2α1-β′)δt=0
from where we derive(a)δ=δ(t,u) arbitrary function and 2α1-β′=0 and then
(86)β=2α1t+β0
and Du=0, so we omit this case,(b)(87)δ(t,u)=(2α1-β′)A(u).
In this case
(88)DuD=1A(u),
which implies
(89)D(u)=D0e∫‍du/A(u)≡D0ea(u),
where
(90)a′(u)=1A(u).
Equation (47) after (54), (58), and (81) becomes
(91)-β′′A(u)+((2α1-β′)A′(u)-βt)(ρurux2+Γ1ua+Γ2vb)-(δuuux2)D0e∫‍du/A(u)+-((2α1-β′)A′(u)-α1)ux2ρurux-(2α1-β′)A(u)(rρur-1ux2+aΓ1ua-1)-λ(t,v)bΓ2vb-1=0.
From terms in v we get the following condition:
(92)((2α1-β′)A′(u)-β′)v-λ(t,v)b=0,
which gives us A′′(u)=0 and then A=A1u+A0. Therefore, from (87)
(93)δ(t,u)=(2α1-β′)(A1u+A0),
which implies that
(94)δuu=0.From (92) arise two cases.(i) Considerλ(t,v)=0 and (2α1-β′)A1-β′=0. In this case after having derived from (91) the following additional conditions
(95)A0=0,β′′A1=0,2A1(2α1-β′)(2-r)-2α1=0,A1(2α1-β′)aΓ1=0,
it is a simple matter to ascertain that there does not exist extension of L𝒫.(ii) Considerλ(t,v)=λ0(t)v with
(96)λ0(t)=1b((2α1-β′)A1-β′).Then (91) assumes the following form:
(97)-β′′(A1u+A0)+((2α1-β′)A1-βt)(ρurux2+Γ1ua)+-((2α1-β′)A1-α1)ux2ρurux-(2α1-β′)(A1u+A0)(rρur-1ux2+aΓ1ua-1)=0.Forr≠1, from (97) we obtain the following:
(98)A0=0,β=β1t+β0,(2α1-β1)(1-A1(1+r))=0,Γ1[(2α1-β1)A1(1-a)-β1]=0.From the previous conditions we consider the following subclasses(A) Consider2α1-β1=0 and (2α1-β1)A1(1-a)-β1=0. As a consequence we get
(99)β1=α1=δ=λ=0,
and then there is no extension of L𝒫.(B) For1-A1(1+r)=0 and (2α1-β1)A1(1-a)-β1=0 we get, for a≠1,
(100)A1=11+r,α=β12(2-a-r)x+α0,β=β1t+β0,δ=β11-au,λ=β11ba1-av
and then in this subcase we got an extension by one of L𝒫 given by
(101)X3=t∂t+2-a-r2x∂x+u1-a∂tu+av(1-a)b∂v
provided that D(v) and F(u,v) are solutions of the following equations:
(102)u1-aDu=(1-a-r)D,(a-b(1-a))F-buFu-avFv=0.
For a=1, instead, we conclude that
(103)β1=0,δ=2α11+r,λ=1b2α11+rv.
Therefore, the extension is given by
(104)X3=x∂x+21+ru∂u+1b21+rv
provided that D(u) and F(u,v) are solutions of the following differential equations:
(105)u1+rDu=D,F-buFu-vFv=0.


## 5.1.s≠2

Fromux2 coefficient we get δuu=0; that is,
(59)δ=δ1(t)u+δ0(t).Therefore, from the remaining terms we have(60)ρur-1uxs[u((1-s-r)δ1+sα1-β′)-rδ0]+δ0′+δ1′u+Γ1((1-a)δ1-β′)+-aΓ1δ0ua-1+Γ2(δ1-β′)vb-Γ2λbvb-1=0.As we assumeds≠0, from the coefficient of uxs in (60) we conclude that δ0(t)=0 and
(61)β′=(1-s-r)δ1+sα1.
Then, still from (60) we have the following constraints to consider:
(62)δ1′u+Γ1ua((1-a)δ1-β′)=0,(63)(δ1-β′)vb-λbvb-1.
From (62) two cases are obtained.(i) Case
a
≠
1
. Then, from (62) we conclude that δ1=const and it follows that
(64)β′=(1-a)δ1.
From (64) and (61) we obtain
(65)α1=s+r-asδ1,β=(1-a)δ1t+β0,
with β0 and δ1 arbitrary constants.The analysis of (63) leads to the following two subcases.(
1
) Consider λ(v)=λ0v, with
(66)λ0=abδ1.
Taking into account the previous results and going back to (56) and (48) we get that the system (3) with G of the form (51) admits the 3-dimensional Lie algebra spanned by the translations in space and time and by the following additional generator:
(67)X3=(1-a)∂t+1s(s+r-a)x∂x+u∂u+abv∂v,
provided that D and F are solutions of the following differential equations:
(68)uDu=((1+a)+2r-as)D,buFu+avFv=(a-b(1-a))F.(
2
) Consider λ(v)=δ1=0. In this case the only symmetries admitted are translations of the independent variables and the form of D and F is arbitrary, so there is not an extension of the principal Lie algebra.(ii) Case
a
=
1
. In this case from (62) it follows that
(69)δ1′-Γ1β′=0.
After having substituted (69) into (61) we obtain
(70)β(t)=-β0+c1e(1-s-r)Γ1t,δ1(t)=ss+r-1α1+c1Γ1e(1-s-r)Γ1t,
with α1, β0, and c1 arbitrary constants and once assumed s+r-1≠0. Finally we analyze the contribution of (63) from where the following two subcases arise.(
1
)  Consider λ(t,v)=λ0(t)v, with
(71)λ0(t)=1bδ1(t).
Then taking into account the previous results it reads
(72)λ0(t)=1b[ss+r-1α1+c1Γ1e(1-s-r)Γ1t].
Going to put the previous result in the condition (46) and by separating the variable we get
(73)β(t)=-β0,δ1(t)=ss+r-1α1,λ0(t)=1b[ss+r-1α1],
with α1 and β0 arbitrary constants and provided that the diffusion coefficient D is solution of
(74)uDu=2(1+r-1s)D.
From condition (48) we do not get further restrictions on the infinitesimal components of the symmetry generator but only the following constraint on the reaction function F:
(75)buFu+vFv=F.
Taking into account the arbitrariness of α0 in this case we have got a 3-dimensional Lie algebra. The additional generator is
(76)X3=x∂x+ss+r-1u∂u+1bss+r-1v∂v.
In particular, from (75), setting F=ku+(k-m2-γ)v, we obtain b=1, while from (75) we obtain the power function D=D0u2((r+s-1)/s).Remark 3.
By puttingr=1, the corresponding system of class (1) admits still a 3-dimensional symmetry Lie algebra generated by translations in time and in space and by X3. These results are in agreement with the ones obtained in [2].By considerings+r-1=0 we do not get extension of L𝒫 for D and F arbitrary. We get, instead, the following extension:
(77)X3=st∂t+x∂x+Γ1stu∂u+Γ1stbv∂v,
provided that D=D0=const. and F=(Γ1/b)v.(
2
) Consider λ(t,v)=δ1(t)=0. In this case the only symmetries admitted are the translations in space and time and the form of D and F is arbitrary.


## 5.2.s=2

We analyze this case by beginning with the discussion of (56) from which two cases arise.(
1
)  D(u)Is Arbitrary.  It follows, of course, that δ=0 and β′=2α1 so
(78)β=2α1t+β0
with β0 arbitrary constant.Moreover (46) and (48) become
(79)-2α1G+α1uxGux-λ(v)Gv=0,(80)(λv-2α1)F-λ(v)Fv=0.
But taking into account that
(81)G=ρurux2+Γ1ua+Γ2vb
(79) becomes
(82)-2α1(ρurux2+Γ1ua+Γ2vb)+2α1ρurux2-λ(v)bΓ2vb-1=0
and gives us the following conditions:
(83)2α1Γ1=0,2α1v+λ(v)b=0
from where, taking into account the work hypotheses at the beginning of this section, we get α1=λ=0 with F arbitrary function of u and v.In this case the only admitted symmetries are translations in time and space.(
2
)
Consider
D
u
/
D
=
(
2
α
1
-
β
′
)
/
δ
(
t
,
u
)
. Then by requiring
(84)∂∂t(2α1-β′δ(t,u))=0,
we get
(85)-δ(t,u)β′′-(2α1-β′)δt=0
from where we derive(a)δ=δ(t,u) arbitrary function and 2α1-β′=0 and then
(86)β=2α1t+β0
and Du=0, so we omit this case,(b)(87)δ(t,u)=(2α1-β′)A(u).
In this case
(88)DuD=1A(u),
which implies
(89)D(u)=D0e∫‍du/A(u)≡D0ea(u),
where
(90)a′(u)=1A(u).
Equation (47) after (54), (58), and (81) becomes
(91)-β′′A(u)+((2α1-β′)A′(u)-βt)(ρurux2+Γ1ua+Γ2vb)-(δuuux2)D0e∫‍du/A(u)+-((2α1-β′)A′(u)-α1)ux2ρurux-(2α1-β′)A(u)(rρur-1ux2+aΓ1ua-1)-λ(t,v)bΓ2vb-1=0.
From terms in v we get the following condition:
(92)((2α1-β′)A′(u)-β′)v-λ(t,v)b=0,
which gives us A′′(u)=0 and then A=A1u+A0. Therefore, from (87)
(93)δ(t,u)=(2α1-β′)(A1u+A0),
which implies that
(94)δuu=0.From (92) arise two cases.(i) Considerλ(t,v)=0 and (2α1-β′)A1-β′=0. In this case after having derived from (91) the following additional conditions
(95)A0=0,β′′A1=0,2A1(2α1-β′)(2-r)-2α1=0,A1(2α1-β′)aΓ1=0,
it is a simple matter to ascertain that there does not exist extension of L𝒫.(ii) Considerλ(t,v)=λ0(t)v with
(96)λ0(t)=1b((2α1-β′)A1-β′).Then (91) assumes the following form:
(97)-β′′(A1u+A0)+((2α1-β′)A1-βt)(ρurux2+Γ1ua)+-((2α1-β′)A1-α1)ux2ρurux-(2α1-β′)(A1u+A0)(rρur-1ux2+aΓ1ua-1)=0.Forr≠1, from (97) we obtain the following:
(98)A0=0,β=β1t+β0,(2α1-β1)(1-A1(1+r))=0,Γ1[(2α1-β1)A1(1-a)-β1]=0.From the previous conditions we consider the following subclasses(A) Consider2α1-β1=0 and (2α1-β1)A1(1-a)-β1=0. As a consequence we get
(99)β1=α1=δ=λ=0,
and then there is no extension of L𝒫.(B) For1-A1(1+r)=0 and (2α1-β1)A1(1-a)-β1=0 we get, for a≠1,
(100)A1=11+r,α=β12(2-a-r)x+α0,β=β1t+β0,δ=β11-au,λ=β11ba1-av
and then in this subcase we got an extension by one of L𝒫 given by
(101)X3=t∂t+2-a-r2x∂x+u1-a∂tu+av(1-a)b∂v
provided that D(v) and F(u,v) are solutions of the following equations:
(102)u1-aDu=(1-a-r)D,(a-b(1-a))F-buFu-avFv=0.
For a=1, instead, we conclude that
(103)β1=0,δ=2α11+r,λ=1b2α11+rv.
Therefore, the extension is given by
(104)X3=x∂x+21+ru∂u+1b21+rv
provided that D(u) and F(u,v) are solutions of the following differential equations:
(105)u1+rDu=D,F-buFu-vFv=0.


## 6. Conclusions

In this paper we have considered a class of reaction-diffusion systems with an additional advection term. The studied class includes, as particular cases, all partial differential equation models concerned with theAedes 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. In agreement with some modifications of the Lie-Ovsiannikov algorithm, introduced in [20], we obtained a group of weak equivalence transformations. We have applied these transformations in order to obtain symmetry generators by using a projection theorem. In particular after having obtained the principal Lie algebra, we investigated a specific, but quite general, form for the advection-reaction term G and we derived some extensions of the principal Lie algebra. The specializations of the results for the systems studied in [2] are in agreement with those ones obtained there.



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*Source: 546083-2014-03-16.xml*