Weak Equivalence Transformations for a Class of Models in Biomathematics

Page 1

Research ArticleWeak Equivalence Transformations fora Class of Models in Biomathematics

Igor Leite Freire1 and Mariano Torrisi2

1 Centro de Matematica Computacao e Cognicao Universidade Federal do ABC (UFABC) Rua Santa Adelia 166Bairro Bangu 09210-170 Santo Andre SP Brazil

2 Dipartimento di Matematica e Informatica Universita Degli Studi di Catania Viale Andrea Doria 6 95125 Catania Italy

Correspondence should be addressed to Igor Leite Freire igorfreireufabcedubr

Received 29 November 2013 Accepted 11 January 2014 Published 16 March 2014

Academic Editor Maria Gandarias

Copyright copy 2014 I L Freire and M Torrisi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A class of reaction-diffusion systems unifying several Aedes aegypti population dynamics models is considered Equivalencetransformations are found Extensions of the principal Lie algebra are derived for some particular cases

1 Introduction

In this paper we focus our attention on the following class ofnonlinear advection-reaction-diffusion systems

119906119905= (119891 (119906) 119906

119909)119909+ 119892 (119906 V 119906

119909) 119892

119906119909

= 0

V119905= ℎ (119906 V)

(1)

These systems can describe the evolution of the densities 119906and V of two interacting populations where the balance equa-tion for 119906 takes into account not only the reaction-diffusioneffects but also some advection effects while the balanceequation for density V takes into account only the so-calledreaction terms The advection effects are due to the presencein the function119892 of the gradient 119906

119909and appear when the indi-

viduals of population 119906 feel external stimuli as for instancewind effects or water currents Class (1) can be considered ageneralization of the equations with the typical properties ofthe already known Aedes aegyptimathematical models [1ndash3]We recall them shortly in the following

The Aedes aegypti mosquitos are the main vector ofdengue a viral disease that causes the so-called dengue hem-orrhagic fever characterized by coagulation problems oftenleading the infected individual to death Since the subtropicalzone climate and environmental conditions are favorable tothe development of Aedes aegypti dengue is a serious public

health problem in many countries around the world How-ever due to the global warming the interest in such consid-ered mosquitoes is not restricted to those places affected bythe disease but it is also of interest for those countries whoseweather in the next decades can become similar to the cur-rent environmental found in the subtropical zone Thereforethe interest inmodeling such a vector is not only a theoreticaldeal but also a way for finding methods and alternatives toovercome and control the problems arising from the dispersaldynamics of the mosquitos and consequently the propaga-tion of the disease

The following system

119906119905= (119906119901119906119909)119909minus 2]119906119902119906

119909+

120574

119896

V (1 minus 119906) minus 1198981119906

V119905= 119896 (1 minus V) 119906 minus (119898

2+ 120574) V

(2)

where119901 119902 isin R belongs to class (1) It was introduced in [2] asa generalization of a model studied in [1]

We recall that in (2) as well as in [1] 119906 and V are respec-tively nondimensional densities of winged and aquatic pop-ulations of mosquitoes and 119896 120574 119898

1 and 119898

2are nondimen-

sional in general positive parameters ] isin R Specifically 119896is the ratio between two constants 119896

1and 119896

2 which are

respectively the carrying capacity related to the amount offindable nutrients and the carrying capacity effect dependenton the occupation of the available breeder and 120574 denotes the

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 546083 9 pageshttpdxdoiorg1011552014546083

Page 2

2 Abstract and Applied Analysis

specific rate of maturation of the aquatic form into wingedfemale mosquitoes while 119898

1and 119898

2are respectively the

mortality of winged and themortality of aquatic populationsFinally ] denotes a constant of velocity for flux due to windcurrents that in general generate an advection motion oflarge masses of the winged population and consequently canfacilitate a quick advance of the infestation For furtherdetails see [4] and references therein

Another system belonging to (1) is

119906119905= (119906119901119906119909)119909minus 2]119906119902119906

119909+

120574

119896

V + (

120574

119896

minus 1198981) 119906

V119905= 119896119906 + (119896 minus 119898

2minus 120574) V

(3)

which was also introduced in [2] starting from (2) and due tothe weak interaction between the aquatic and winged popu-lations by modifying the source terms

The first equation of (3) gives the time rate of change ofthe mosquitoes density as a sum of the growth terms (120574119896)Vof the per capita death rate ((120574119896) minus 120583

1)119906 and the diffusive-

advective flux due to the movement of mosquito populationThe second equation gives the corresponding time rate of

change of the density of aquatic population as a sum of thegrowth term 119896119906 of aquatic population due to the new eggdepositions of female mosquitoes with per capita death rate(119896 minus 120583

2)V of aquatic population The term minus120574V represents the

loss due to the change of the aquatic into winged form see[4]

The family of system (1) contains arbitrary functions ornumerical parameters which specifies the individual char-acteristics of phenomena belonging to large subclasses Inthis sense the knowledge of equivalence transformations canprovide us with certain relations between the solutions of dif-ferent phenomena of the same class and allows us to getsymmetries in a quite direct way

Following [5] an equivalence transformation is a nonde-generated change of independent and dependent variables 119905119909 119906 and V into 119909 and V

119909 = 119909 (119909 V)

119905 = 119905 (119909 V)

119906 = 119906 (119909 V)

V = V (119909 V)

(4)

which maps a system of class (1) in another one of the sameclass that is in an equation preserving differential structurebut in general with

119891 () = 119891 (119906) 119892 ( V

119909) = 119892 (119906 V 119906

119909)

ℎ ( V) = ℎ (119906 V)

(5)

Of course in the case119891 () = 119891 (119906) 119892 ( V

119909) = 119892 (119906 V 119906

119909)

ℎ ( V) = ℎ (119906 V)

(6)

an equivalence transformation becomes a symmetry

In this paper we look for certain equivalence transforma-tions for the class of systems (1) in order to find symmetriesfor special systems belonging to (1) and to get informationabout constitutive parameters 119891 119892 and ℎ appearing thereMoreoverwewish to stress that as it is known an equivalencetransformation maps solutions of an equation in solutions ofthe transformed equation [6] Then in order to find solutionsfor a certain equation one can look for the equivalence trans-formations that bring the equation in simpler other oneswhose solutions are well studied see for example [7 8] andreferences inside

The plan of the paper is as follows In the next section weprovide some elements about equivalence transformations InSection 3 we apply these concepts in order to obtain a setof weak equivalence generators In Section 4 after havingintroduced a projection theorem we show how to apply it tofind symmetries of (1) In Section 5 after having introduceda special structure of the advection-reaction function 119892 thatgeneralizes that one used in (3) we find extensions withrespect to the principal Lie algebra Conclusions and finalremarks are given in Section 6

2 Elements on Equivalence Transformations

In the past differential equation literature it is possible to findseveral examples of equivalence transformations The directsearch for the most general equivalence transformationsthrough the finite form of the transformation is connectedwith considerable computational difficulties and quite oftenleads to partial solutions of the problem (eg [9 10])

A systematic treatment to look for continuous equiva-lence transformations by using the Lie infinitesimal criterionwas suggested by Ovsiannikov [11]

In general the equivalence transformations for class (1)can be considered as transformations acting on point of thebasic augmented space

119860 equiv 119905 119909 119906 V 119906119905 119906119909 V119905 V119909 119891 119892 ℎ (7)

The previous elements allow us to consider in the fol-lowing the one-parameter equivalence transformations as agroup of transformations acting on the basic augmentedspace 119860 of the type

119909 = 119909 (119909 V 120576)

119905 = 119905 (119909 V 120576)

119906 = 119906 (119909 V 120576)

V = V (119909 V 120576)

119891 = 119891 (119909 V 119909 V V119909119891 119892

ℎ 120576)

119892 = 119892 (119909 V 119909 V V119909119891 119892

ℎ 120576)

ℎ = ℎ (119909 V 119909 V V119909119891 119892

ℎ 120576)

(8)

which is locally a 119862infin-diffeomorphism depending analyti-

cally on the parameter 120576 in a neighborhood of 120576 = 0 andreduces to the identity transformation for 120576 = 0

Page 3

Abstract and Applied Analysis 3

Following [6 11ndash14] (see also eg [5 15ndash17] ) we considerthe infinitesimal generator of the equivalence transforma-tions (8) of the systems (1) that reads as follows

119884 = 1205851120597119909+ 1205852120597119905+ 1205781120597119906+ 1205782120597V + 120583

1120597119891+ 1205832120597119892+ 1205833120597ℎ (9)

where the infinitesimal components 1205851 1205852 1205781 and 120578

2 aresought depending on 119909 119905 119906 and V while the infinitesimalcomponents 120583119894 (119894 = 1 2 3) are sought at least in principledepending on 119909 119905 119906 V 119906

119905 119906119909 119906119909 V119905 V119909 119891 119892 and ℎ In

order to obtain the determining system which allows us to getthe infinitesimal coordinates 120585119894 120578119894 and 120583

119895 (119894 = 1 2 and 119895 =

1 2 3) we apply the Lie-Ovsiannikov infinitesimal criterionby requiring the invariance with respect to suitable prolon-gations 119884(1) and 119884

(2) of (9) of the following equations

119906119905minus (119891119906

119909)119909minus 119892 = 0

V119905minus ℎ = 0

(10)

together with the invariance of the auxiliary conditions [13 1418 19]

119891119905= 119891119909= 119891V = 119891

119906119909

= 119891119906119905

= 119891V119909

= 119891V119905

= 119892119905= 119892119909= 119892119906119905

= 119892V119905

= 119892V119909

= 0

ℎ119905= ℎ119909= ℎ119906119909

= ℎ119906119905

= ℎV119909

= ℎV119905

= 0

(11)

where 119906 and V are (119905 119909) functions while 119891 119892 and ℎ

are considered as functions depending a priori on (119905 119909 119906

V 119906119905 119906119909 V119905 V119909) All of these functions are assumed to be ana-

lytical The constraints given by (11) characterize the func-tional dependence of 119891 119892 and ℎ

In this paper instead in view of further applications andfollowing [20] wemodify the previous classical procedure bylooking for equivalence transformations whose generatorsare got by solving the determining system obtained from thefollowing invariance conditions

119884(2)[119906119905minus (119891 (119906) 119906

119909)119909minus 119892 (119906 V 119906

119909)]

10038161003816100381610038161003816119906119905minus(119891(119906)119906

119909)119909minus119892(119906V119906

119909)=0

V119905minusℎ(119906V)=0

= 0

(12)

119884(1)

[V119905minus ℎ (119906 V)]

10038161003816100381610038161003816119906119905minus(119891(119906)119906

119909)119909minus119892(119906V119906

119909)=0

V119905minusℎ(119906V)=0

= 0 (13)

As the functional dependences of the parameters 119891 119892 and ℎ

are known a priori we do not require the invariance of theauxiliary conditions (11) In this way we work in a basic aug-mented space 119860 equiv 119905 119909 119906 V 119906

119909 119891 119892 ℎTherefore the120583119894 com-

ponents must be sought at least in principle depending on119909 119905 119906 V 119906

119909 119891 119892 and ℎ

The infinitesimal operators obtained by following thisshortening procedure can generate transformations thatmapequations of our class into new equations of the same classwhere the transformed arbitrary functions may have newadditional functional dependencies Such transformationsare called weak equivalence transformations [13 14]

With respect to the application in biomathematical mod-els equivalence and weak equivalence transformations wereapplied not only to study of tumor models [21 22] but also tothe population dynamics in [20 23 24]

3 Calculation of WeakEquivalence Transformations

In order to avoid long formulas and write 119884(1) and 119884(2) in a

compact way we put

119909 = 1199091 119905 = 119909

2

119906 = 1199061 V = 119906

2

119891 = ℎ1 119892 = ℎ

2 ℎ = ℎ

3

(14)

For this reason system (1) is rewritten as

1199061

1199092 minus ℎ2minus ℎ1

1199061(1199061

1199091)

2

minus ℎ11199061

11990911199091 = 0

1199062

1199092 minus ℎ3= 0

(15)

while the equivalence generator assumes the following form

119884 = 120585119894120597119909119894 + 120578120572120597119906120572 + 120583119860 120597ℎ119860 (16)

where 119894 = 1 2 120572 = 1 2 and 119860 = 1 2 3 Here the summationover the repeated indices is presupposed

After putting

(1199111 1199112 1199113 1199114 1199115) = (119909

1 1199092 1199061 1199062 1199061

1)

(]1 ]2 ]3 ]4 ]5) = (1205851 1205852 1205781 1205782 1205771

1)

(17)

119906120572

119894= 119906120572

119909119894 ℎ

119860

119886= ℎ119860

119911119886 (18)

119863119890

119886= 120597119911119886 + ℎ119860

119886120597ℎ119860 (19)

119863119890

119895= 120597119909119895 + 119906120572

119895120597119906120572 + 119906120572

119894119895120597119906120572

119894

+ sdot sdot sdot (20)

the prolongations 119884(1) and 119884(2) assume the following form

119884(1)

= 119884 + 120577120572

119895120597119906120572

119895

+ 120596119860

119886120597ℎ119860

119886

119884(2)

= 119884(1)

+ 1205771

111205971199061

11

= 119884 + 120577120572

119895120597119906120572

119895

+ 120596119860

119886120597ℎ119860

119886

+ 1205771

111205971199061

11

(21)

where

120577120572

119895= 119863119890

119895120578120572minus 119906120572

119896119863119890

119895120585119896

120577120572

119894119895= 119863119890

119895120577120572

119895minus 119906120572

119894119896119863119890

119895120585119896

120596119860

119886= 119863119890

119886120583119860minus ℎ119860

119887119863119890

119886]119887

(22)

The invariant conditions read

119884(2)1198651= 1205771

2minus 1205832minus 21205771

1ℎ1

11990611199061

1minus (1205961

3) (1199061

1)

2

minus (1205831) 1199061

11minus ℎ11205771

11= 0

(23)

119884(1)1198652= 1205772

2minus 1205833= 0 (24)

Page 4

4 Abstract and Applied Analysis

both under the constraints (15) which are after (18)

1199061

2= ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11 (25)

1199062

2= ℎ3 (26)

The coefficients 12057711 12057712 12057722 and 120596

1

3are given respectively

by

1205771

1= 1205781

1+ (1205781

1199061 minus 1205851

1) 1199061

1minus 1205852

11199061

2+ 1205781

11990621199062

1

minus 1205851

1199061(1199061

1)

2

minus 1205851

11990621199061

11199062

1minus 1205852

11990611199061

11199061

2minus 1205852

11990621199061

21199062

1

1205771

2= 1205781

2minus 1205851

21199061

1+ (1205781

1199061 minus 1205852

2) 1199061

2+ 1205781

11990621199062

2

minus 1205851

11990611199061

11199061

2minus 1205851

11990621199061

11199062

2minus 1205852

1199061(1199061

2)

2

minus 1205852

11990621199061

21199062

2

1205772

2= 1205782

2+ 1205782

11990611199061

2+ (1205782

1199062 minus 1205852

2) 1199062

2minus 1205851

21199062

1

minus 1205851

11990611199062

11199061

2minus 1205851

11990621199062

11199062

2minus 1205852

11990611199061

21199062

2minus 1205852

1199062(1199062

2)

2

1205961

3= 1205831

1199061 + ℎ1

11990611205831

ℎ1 + ℎ2

11990611205831

ℎ2 + ℎ3

11990611205831

ℎ3 minus ℎ1

11990611205781

1199061

(27)

Taking into account (24) and (27) we can write

119884(1)1198652= 1205782

2+ 1205782

11990611199061

2+ (1205782

1199062 minus 1205852

2) 1199062

2minus 1205851

21199062

1minus 1205851

11990611199062

11199061

2

minus 1205851

11990621199062

11199062

2minus 1205852

11990611199061

21199062

2minus 1205852

1199062(1199062

2)

2

minus 1205833= 0

(28)

with the constraints (25) and (26)Then substituting (25) and (26) into (28) we get

1205782

2+ 1205782

1199061 (ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11) + (120578

2

1199062 minus 1205852

2) ℎ3

minus 1205851

21199062

1minus 1205851

11990611199062

1(ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11)

minus 1205851

11990621199062

1ℎ3minus 1205852

1199061 (ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11) ℎ3

minus 1205852

1199062(ℎ3)

2

minus 1205833= 0

(29)

From this condition we obtain the following determiningequations

1205782

2+ (1205782

1199062 minus 1205852

2) ℎ3minus 1205852

1199062(ℎ3)

2

minus 1205833= 0

1205851

2= 0 120585

1

1199061 = 0 120585

1

1199062 = 0

1205782

1199061 = 0 120585

2

1199061 = 0

(30)

Then it follows that

1205851= 1205851(1199091) 120585

2= 1205852(1199091 1199092 1199062) 120578

2= 1205782(1199091 1199092 1199062)

1205781= 1205781(1199091 1199092 1199061 1199062) 120583

3= 1205782

2+ (1205782

1199062 minus 1205852

2) ℎ3minus 1205852

1199062(ℎ3)

2

(31)

and consequently 12058331199061

1

= 0

Following the same procedurewe canwrite the invariancecondition (23) as

119884(2)1198651= 1205781

2+ (1205781

1199061 minus 1205852

2) (ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11)

+ 1205781

1199062ℎ3minus 1205852

1199062 (ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11) ℎ3

minus 1205832minus 2 120578

1

1+ (1205781

1199061 minus 1205851

1) 1199061

1

minus 1205852

1(ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11) + 1205781

11990621199062

1+

minus1205852

1199062 (ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11) 1199062

1 ℎ1

11990611199061

1

minus (1205831

1199061 + ℎ1

11990611205831

ℎ1 + ℎ2

11990611205831

ℎ2 + ℎ3

11990611205831

ℎ3 minus ℎ1

11990611205781

1199061)

times (1199061

1)

2

minus (1205831) 1199061

11

minus ℎ1(119863119890

11205781

1+ 119863119890

1(1205781

1199061 minus 1205851

1) 1199061

1+ (1205781

1199061 minus 1205851

1) 1199061

11

+ (119863119890

11205781

1199062) 1199062

1+ 1205781

11990621199062

11)

minus ℎ1(minus119863119890

1(1205852

1) (ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11)

minus 1205852

11199061

21minus 119863119890

11205852

1199062 (ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11) 1199062

1

minus 1205852

11990621199061

211199062

1minus 1205852

1199062 (ℎ2+ ℎ1

1199061(1199061

1)

2

+ ℎ11199061

11) 1199062

11

minus1199061

111205851

1) = 0

(32)

Collecting the terms with (ℎ1

1199061)2 and with ℎ

1

1199061 and equat-

ing their respective coefficients to zero it is obtained that

1205852

1= 0 120585

2

1199062 = 0 120578

1

1= 0

1205831

ℎ1 = 2120585

1

1minus 1205852

2 1205781

1199062 = 0 120578

1

1= 0

(33)

Thus from the coefficients of ℎ21199061 and ℎ

3

1199061

1205831

ℎ2 = 1205831

ℎ3 = 0 (34)

Considering the coefficients of 119906111we get

1205831= (21205851

1minus 1205852

2) ℎ1= 0 997904rArr 120583

1

1199061 = 0 (35)

The remaining terms of (32) give the following form for theinfinitesimal component of ℎ2

1205832= 1205781

2+ (1205781

1199061 minus 1205852

2) ℎ2+ (1205851

11minus 1205781

119906111990611199061

1) ℎ1 (36)

Therefore once having taken into account all restrictionsobtained we are finally able to write the following infinites-imal components for the weak equivalence generators

1205851= 120572 (119909

1) 120585

2= 120573 (119909

2)

1205781= 120575 (119909

2 1199061) 120578

2= 120582 (119909

1 1199092 1199062)

1205831= (2120572

1015840minus 1205731015840) ℎ1 1205833= (1205821199062 minus 1205731015840) ℎ3+ 1205821199092

(37)

Page 5

Abstract and Applied Analysis 5

while from (36) we get

1205832= 1205751199092 + (120575

1199061 minus 1205731199092) ℎ2+ (12057211990911199091 minus 120575119906111990611199061

1199091) ℎ1 (38)

where 120572(1199091) 120573(1199092) 120575(1199092 1199061) and 120582(1199091 1199092 1199062) are arbitrary

real functions of their arguments Then going back to theoriginal variables the most general operator of these contin-uous weak equivalence transformations reads

119884 = 120572 (119909) 120597119909+ 120573 (119905) 120597

119905+ 120575 (119905 119906) 120597

119906

+ 120582 (119909 119905 V) 120597V + (21205721015840minus 1205731015840) 119891120597119891

+ (120575119905+ (120575119906minus 120573119905) 119892 + (120572

10158401015840minus 120575119906119906119906119909) 119891) 120597

119892

+ ((120582V minus 1205731015840) ℎ + 120582

119905) 120597ℎ

(39)

4 Symmetries for the System (1)In the next sections in order to carry out symmetries for thesystem (1) we do not use the classical Lie approach Insteadof the mentioned method we apply the projection theoremintroduced in [25] and eventually reconsidered in [13 14 1819] In agreement with these references we can affirm thefollowing

Theorem 1 Let

119884 = 120572 (119909) 120597119909+ 120573 (119905) 120597

119905+ 120575 (119905 119906) 120597

119906

+ 120582 (119909 119905 V) 120597V + (21205721015840minus 1205731015840) 119891120597119891

+ (120575119905+ (120575119906minus 120573119905) 119892 + (120572

10158401015840minus 120575119906119906119906119909) 119891) 120597

119892

+ ((120582V minus 1205731015840) ℎ + 120582

119905) 120597ℎ

(40)

be an infinitesimal equivalence generator for the system (1)then the operator

X = 120572 (119909) 120597119909+ 120573 (119905) 120597

119905+ 120575 (119905 119906) 120597

119906+ 120582 (119909 119905 V) 120597V (41)

which corresponds to the projection of119884 on the space (119909 119905 119906 V)is an infinitesimal symmetry generator of the system (1) if andonly if the constitutive equations specifying the forms of 119891 ℎand 119892 are invariant with respect to 119884

For the system under consideration in general the con-stitutive equations whose invariance must be requested are

119891 = 119863 (119906)

119892 = 119866 (119906 V 119906119909)

ℎ = 119865 (119906 V)

(42)

The request of invariance

119884 (119891 minus 119863 (119906))1003816100381610038161003816(42)

= 0 119884(1)

(119892 minus 119866 (119906 V 119906119909))

10038161003816100381610038161003816(42)

= 0

119884 (ℎ minus 119865 (119906 V))|(42)

= 0

(43)

brings us to the following equations

1205831minus 1205781119863119906= 0

1205832minus 1205771

1119866119906119909

minus 1205781119866119906minus 1205782119866V = 0

1205833minus 1205781119865119906minus 1205782119865V = 0

(44)

under the restrictions (42)Then substituting

1205771

1= (1205781

1199061 minus 1205851

1) 1199061

1= (120575119906minus 1205721015840) 119906119909

(45)

and taking into account the constraints we can write (44) as

(21205721015840minus 1205731015840)119863 (119906) minus 120575 (119905 119906)119863

119906= 0 (46)

120575119905+ (120575119906minus 120573119905) 119866 minus (120575

1199061199061199062

119909minus 120572119909119909119906119909)119863 minus (120575

119906minus 1205721015840) 119906119909119866119906119909

minus 120575119866119906minus 120582 (119905 119909 V) 119866V = 0

(47)

((120582V minus 1205731015840) 119865 + 120582

119905) minus 120575119865

119906minus 120582 (119905 119909 V) 119865V = 0 (48)

We recall here that the principal Lie algebra 119871P [5 12] is theLie algebra of the principal Lie group that is the group of theall Lie point symmetries

119883 = 120585 (119909 119905 119906 V)120597

120597119909

+ 120591 (119909 119905 119906 V)120597

120597119905

+ 1205781(119909 119905 119906 V)

120597

120597119906

+ 1205782(119909 119905 119906 V)

120597

120597V

(49)

that leave the system (1) invariant for any form of the func-tions 119863(119906) 119866(119906 V 119906

119909) and 119865(119906 V) In other words we can

remark that the principal Lie algebra is the subalgebra of theequivalence algebra such that any operator 119884 of this subalge-bra leaves the equations 119891 = 119863(119906) 119892 = 119866(119906 V 119906

119909) and ℎ =

119865(119906 V) invariant for any form of the functions 119863(119906)119866(119906 V 119906

119909) and 119865(119906 V) Then we can say [5] the following

Corollary 2 An equivalence operator for the system (1) belongsto the principal Lie algebra 119871P if and only if 120578119894 = 0 120583119895 = 0119894 = 1 2 and 119895 = 1 2 3

Taking Corollary 2 into account from the previous equa-tions (46)-(48) it is a simple matter to ascertain that the 119871P

[5 12] is spanned by the following translation generators

1198830= 120597119905 119883

1= 120597119909 (50)

5 Some Extensions of 119871P

In order to show some extensions of the principal algebrawhich could be of interest in biomathematics we assume thatthe advection-reaction function is of the form

119866 = 120588119906119903119906119904

119909+ Γ1119906119886+ Γ2V119887 (51)

where the parameters 120588 Γ1 Γ2 119903 119904 119886 and 119887 are constitutive

parameters of the considered phenomena

Page 6

6 Abstract and Applied Analysis

This form of 119866 is a generalization of

119866 = minus2]119906119902119906119909+

120574

119896

V + (

120574

119896

minus 1198981) 119906 (52)

appearing in (3) where

120588 = minus2] Γ1= (

120574

119896

minus 1198981) Γ

2=

120574

119896

(53)

Consequently in (51) we must consider Γ2gt 0 and as limit

cases Γ1=0 and 119886 = 0 Moreover in this section we assume

that the value 119904 = 0will not be considered because in this casethe advective effects disappearWe also assume that 119887 = 0Thislast restriction implies that the balance equation of the density119906 depends on the density V Finally for the sake of simplicitywe omit the limit case Γ

1= 0 and assume that the diffusion is

only nonlinear that is119863119906

= 0In the following we continue the discussion of invariance

conditions written in the previous sectionFrom (46) by deriving with respect to 119909 we get

12057210158401015840= 0 (54)

Then

120572 = 1205721119909 + 1205720

(55)

with 1205721and 120572

0arbitrary constants so (46) becomes

(21205721minus 1205731015840)119863 (119906) minus 120575 (119905 119906)119863

119906= 0 (56)

while after having taken (51) into account (47) reads

120575119905+ (120575119906minus 120573119905) (120588119906119903119906119904

119909+ Γ1119906119886+ Γ2V119887)

minus 1198631205751199061199061199062

119909minus (120575119906minus 1205721) 120588119904119906119903119906119904

119909+

minus 120575 (120588119903119906119903minus1

119906119904

119909+ Γ1119886119906119886minus1

) minus 120582 (119905 119909 V) Γ2119887V119887minus1 = 0

(57)

From (57) we get immediately

120582 = 120582 (119905 V) (58)

In the following we analyze separately the case 119904 = 2 andthe case 119904 = 2

51 119904 = 2 From 1199062

119909coefficient we get 120575

119906119906= 0 that is

120575 = 1205751(119905) 119906 + 120575

0(119905) (59)

Therefore from the remaining terms we have

120588119906119903minus1

119906119904

119909[119906 ((1 minus 119904 minus 119903) 120575

1+ 1199041205721minus 1205731015840) minus 119903120575

0]

+ 1205751015840

0+ 1205751015840

1119906 + Γ1((1 minus 119886) 120575

1minus 1205731015840) +

minus 119886Γ11205750119906119886minus1

+ Γ2(1205751minus 1205731015840) V119887 minus Γ

2120582119887V119887minus1 = 0

(60)

As we assumed 119904 = 0 from the coefficient of 119906119904119909in (60) we

conclude that 1205750(119905) = 0 and

1205731015840= (1 minus 119904 minus 119903) 120575

1+ 1199041205721 (61)

Then still from (60) we have the following constraints toconsider

1205751015840

1119906 + Γ1119906119886((1 minus 119886) 120575

1minus 1205731015840) = 0 (62)

(1205751minus 1205731015840) V119887 minus 120582119887V119887minus1 (63)

From (62) two cases are obtained

(i) Case 119886 = 1 Then from (62) we conclude that 1205751= const

and it follows that

1205731015840= (1 minus 119886) 120575

1 (64)

From (64) and (61) we obtain

1205721=

119904 + 119903 minus 119886

119904

1205751

120573 = (1 minus 119886) 1205751119905 + 1205730

(65)

with 1205730and 1205751arbitrary constants

The analysis of (63) leads to the following two subcases(1)Consider 120582(V) = 120582

0V with

1205820=

119886

119887

1205751 (66)

Taking into account the previous results and going back to(56) and (48)we get that the system (3) with119866 of the form (51)admits the 3-dimensional Lie algebra spanned by the trans-lations in space and time and by the following additionalgenerator

1198833= (1 minus 119886) 120597

119905+

1

119904

(119904 + 119903 minus 119886) 119909120597119909+ 119906120597119906+

119886

119887

V120597V (67)

provided that119863 and 119865 are solutions of the following differen-tial equations

119906119863119906= ((1 + 119886) + 2

119903 minus 119886

119904

)119863

119887119906119865119906+ 119886V119865V = (119886 minus 119887 (1 minus 119886)) 119865

(68)

(2)Consider 120582(V) = 1205751= 0 In this case the only symme-

tries admitted are translations of the independent variablesand the form of 119863 and 119865 is arbitrary so there is not anextension of the principal Lie algebra

(ii) Case 119886 = 1 In this case from (62) it follows that

1205751015840

1minus Γ11205731015840= 0 (69)

After having substituted (69) into (61) we obtain

120573 (119905) = minus1205730+ 1198881119890(1minus119904minus119903)Γ

1119905

1205751(119905) =

119904

119904 + 119903 minus 1

1205721+ 1198881Γ1119890(1minus119904minus119903)Γ

1119905

(70)

Page 7

Abstract and Applied Analysis 7

with12057211205730 and 119888

1arbitrary constants and once assumed 119904+119903minus

1 = 0 Finally we analyze the contribution of (63) from wherethe following two subcases arise

(1) Consider 120582(119905 V) = 1205820(119905)V with

1205820(119905) =

1

119887

1205751(119905) (71)

Then taking into account the previous results it reads

1205820(119905) =

1

119887

[

119904

119904 + 119903 minus 1

1205721+ 1198881Γ1119890(1minus119904minus119903)Γ

1119905] (72)

Going to put the previous result in the condition (46) and byseparating the variable we get

120573 (119905) = minus1205730

1205751(119905) =

119904

119904 + 119903 minus 1

1205721

1205820(119905) =

1

119887

[

119904

119904 + 119903 minus 1

1205721]

(73)

with 1205721and 120573

0arbitrary constants and provided that the dif-

fusion coefficient119863 is solution of

119906119863119906= 2 (1 +

119903 minus 1

119904

)119863 (74)

From condition (48) we do not get further restrictions on theinfinitesimal components of the symmetry generator but onlythe following constraint on the reaction function 119865

119887119906119865119906+ V119865V = 119865 (75)

Taking into account the arbitrariness of1205720in this casewe have

got a 3-dimensional Lie algebra The additional generator is

1198833= 119909120597119909+

119904

119904 + 119903 minus 1

119906120597119906+

1

119887

119904

119904 + 119903 minus 1

V120597V (76)

In particular from (75) setting 119865 = 119896119906 + (119896 minus 1198982minus 120574)V we

obtain 119887 = 1 while from (75) we obtain the power function119863 = 119863

01199062((119903+119904minus1)119904)

Remark 3 By putting 119903 = 1 the corresponding system ofclass (1) admits still a 3-dimensional symmetry Lie algebragenerated by translations in time and in space and by 119883

3

These results are in agreement with the ones obtained in [2]

By considering 119904 + 119903 minus 1 = 0 we do not get extension of119871P for119863 and119865 arbitraryWe get instead the following exten-sion

X3= 119904119905120597119905+ 119909120597119909+ Γ1119904119905119906120597119906+

Γ1119904119905

119887

V120597V (77)

provided that119863 = 1198630= const and 119865 = (Γ

1119887)V

(2)Consider 120582(119905 V) = 1205751(119905) = 0 In this case the only sym-

metries admitted are the translations in space and time andthe form of119863 and 119865 is arbitrary

52 119904=2 We analyze this case by beginning with the discus-sion of (56) from which two cases arise

(1)119863(119906) Is Arbitrary It follows of course that 120575 = 0 and 1205731015840 =21205721so

120573 = 21205721119905 + 1205730

(78)

with 1205730arbitrary constant

Moreover (46) and (48) become

minus21205721119866 + 120572

1119906119909119866119906119909

minus 120582 (V) 119866V = 0 (79)

(120582V minus 21205721) 119865 minus 120582 (V) 119865V = 0 (80)

But taking into account that

119866 = 1205881199061199031199062

119909+ Γ1119906119886+ Γ2V119887 (81)

(79) becomes

minus21205721(1205881199061199031199062

119909+ Γ1119906119886+ Γ2V119887) + 2120572

11205881199061199031199062

119909minus 120582 (V) 119887Γ

2V119887minus1 = 0

(82)

and gives us the following conditions

21205721Γ1= 0 2120572

1V + 120582 (V) 119887 = 0 (83)

from where taking into account the work hypotheses at thebeginning of this section we get 120572

1= 120582 = 0 with 119865 arbitrary

function of 119906 and VIn this case the only admitted symmetries are translations

in time and space

(2) Consider 119863119906119863 = (2120572

1minus 1205731015840)120575(119905 119906)Then by requiring

120597

120597119905

(

21205721minus 1205731015840

120575 (119905 119906)

) = 0 (84)

we get

minus120575 (119905 119906) 12057310158401015840minus (2120572

1minus 1205731015840) 120575119905= 0 (85)

from where we derive(a) 120575 = 120575(119905 119906) arbitrary function and 2120572

1minus 1205731015840= 0 and

then

120573 = 21205721119905 + 1205730

(86)

and119863119906= 0 so we omit this case

(b)

120575 (119905 119906) = (21205721minus 1205731015840)119860 (119906) (87)

In this case

119863119906

119863

=

1

119860 (119906)

(88)

which implies

119863 (119906) = 1198630119890int119889119906119860(119906)

equiv 1198630119890119886(119906)

(89)

Page 8

8 Abstract and Applied Analysis

where

1198861015840(119906) =

1

119860 (119906)

(90)

Equation (47) after (54) (58) and (81) becomes

minus 12057310158401015840119860 (119906) + ((2120572

1minus 1205731015840)1198601015840(119906) minus 120573

119905) (1205881199061199031199062

119909+ Γ1119906119886+ Γ2V119887)

minus (1205751199061199061199062

119909)1198630119890int119889119906119860(119906)

+

minus ((21205721minus 1205731015840)1198601015840(119906) minus 120572

1) 1199061199092120588119906119903119906119909

minus (21205721minus 1205731015840)119860 (119906) (119903120588119906

119903minus11199062

119909+ 119886Γ1119906119886minus1

)

minus 120582 (119905 V) 119887Γ2V119887minus1 = 0

(91)

From terms in V we get the following condition

((21205721minus 1205731015840)1198601015840(119906) minus 120573

1015840) V minus 120582 (119905 V) 119887 = 0 (92)

which gives us11986010158401015840(119906) = 0 and then119860 = 1198601119906+1198600 Therefore

from (87)

120575 (119905 119906) = (21205721minus 1205731015840) (1198601119906 + 119860

0) (93)

which implies that

120575119906119906

= 0 (94)

From (92) arise two cases(i) Consider 120582(119905 V) = 0 and (2120572

1minus 1205731015840)1198601minus 1205731015840= 0 In this

case after having derived from (91) the following additionalconditions

1198600= 0 120573

101584010158401198601= 0

21198601(21205721minus 1205731015840) (2 minus 119903) minus 2120572

1= 0 119860

1(21205721minus 1205731015840) 119886Γ1=0

(95)

it is a simple matter to ascertain that there does not existextension of 119871P

(ii) Consider 120582(119905 V) = 1205820(119905)V with

1205820(119905) =

1

119887

((21205721minus 1205731015840)1198601minus 1205731015840) (96)

Then (91) assumes the following form

minus 12057310158401015840(1198601119906 + 119860

0) + ((2120572

1minus 1205731015840)1198601minus 120573119905) (1205881199061199031199062

119909+ Γ1119906119886) +

minus ((21205721minus 1205731015840)1198601minus 1205721) 1199061199092120588119906119903119906119909

minus (21205721minus 1205731015840) (1198601119906 + 119860

0) (119903120588119906

119903minus11199062

119909+ 119886Γ1119906119886minus1

) = 0

(97)

For 119903 = 1 from (97) we obtain the following

1198600= 0 120573 = 120573

1119905 + 1205730

(21205721minus 1205731) (1 minus 119860

1(1 + 119903)) = 0

Γ1[(21205721minus 1205731) 1198601(1 minus 119886) minus 120573

1] = 0

(98)

From the previous conditions we consider the followingsubclasses

(A) Consider 21205721minus1205731= 0 and (2120572

1minus1205731)1198601(1minus119886)minus120573

1= 0

As a consequence we get

1205731= 1205721= 120575 = 120582 = 0 (99)

and then there is no extension of 119871P(B) For 1minus119860

1(1+119903) = 0 and (2120572

1minus1205731)1198601(1minus119886)minus120573

1= 0we

get for 119886 = 1

1198601=

1

1 + 119903

120572 =

1205731

2

(2 minus 119886 minus 119903) 119909 + 1205720 120573 = 120573

1119905 + 1205730

120575 =

1205731

1 minus 119886

119906 120582 = 1205731

1

119887

119886

1 minus 119886

V

(100)

and then in this subcase we got an extension by one of 119871P

given by

1198833= 119905120597119905+

2 minus 119886 minus 119903

2

119909120597119909+

119906

1 minus 119886

120597119905119906 +

119886V(1 minus 119886) 119887

120597V (101)

provided that 119863(V) and 119865(119906 V) are solutions of the followingequations

119906

1 minus 119886

119863119906= (1 minus 119886 minus 119903)119863

(119886 minus 119887 (1 minus 119886)) 119865 minus 119887119906119865119906minus 119886V119865V = 0

(102)

For 119886 = 1 instead we conclude that

1205731= 0 120575 =

21205721

1 + 119903

120582 =

1

119887

21205721

1 + 119903

V (103)

Therefore the extension is given by

1198833= 119909120597119909+

2

1 + 119903

119906120597119906+

1

119887

2

1 + 119903

V (104)

provided that 119863(119906) and 119865(119906 V) are solutions of the followingdifferential equations

119906

1 + 119903

119863119906= 119863

119865 minus 119887119906119865119906minus V119865V = 0

(105)

6 Conclusions

In this paper we have considered a class of reaction-diffusionsystems with an additional advection term The studied classincludes as particular cases all partial differential equationmodels concerned with the Aedes aegyptimosquito that havebeen proposed until nowWe have investigated such a systemfrom the point of view of equivalence transformations in thespirit of the Lie-Ovsiannikov algorithm based on the Lie infi-nitesimal criterion In agreement with some modifications ofthe Lie-Ovsiannikov algorithm introduced in [20] weobtained a group of weak equivalence transformations We

Page 9

Abstract and Applied Analysis 9

have applied these transformations in order to obtain sym-metry generators by using a projection theorem In particularafter having obtained the principal Lie algebra we investi-gated a specific but quite general form for the advection-reaction term 119866 and we derived some extensions of theprincipal Lie algebra The specializations of the results forthe systems studied in [2] are in agreement with those onesobtained there

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors would like to thank FAPESP for financial support(Grants 201120072-0 and 201119089-6) Mariano Torrisiwould also like to thank CMCC-UFABC for its warm hospi-tality and GNFM (Gruppo Nazionale per Fisica-Matematica)for its support He also thanks the support from the Univer-sity of Catania through PRA

References

[1] L T Takahashi N AMaidanaW Castro Ferreira Jr P Pulinoand H M Yang ldquoMathematical models for the Aedes aegyptidispersal dynamics travellingwaves bywing andwindrdquoBulletinof Mathematical Biology vol 67 no 3 pp 509ndash528 2005

[2] I L Freire and M Torrisi ldquoSymmetry methods in mathemat-ical modeling of Aedes aegypti dispersal dynamicsrdquo NonlinearAnalysis Real World Applications vol 14 no 3 pp 1300ndash13072013

[3] I L Freire and M Torrisi ldquoTransformacoes de equivalenciafracas para uma classe de modelos em biomatematicardquo inAnaisdo Congresso deMatematica Aplicada e Computacional (CMAC-CO rsquo13) 2013

[4] I L Freire andM Torrisi ldquoSimilarity solutions for systems aris-ing fromanAedes aegyptimodelrdquoCommunications inNonlinearScience and Numerical Simulation vol 19 no 4 pp 872ndash8792014

[5] N H Ibragimov M Torrisi and A Valenti ldquoPreliminary groupclassification of equations V

119905119905= 119891(119909 V

119909)V119909119909

+ 119892(119909 V119909)rdquo Journal

of Mathematical Physics vol 32 no 11 pp 2988ndash2995 1991[6] I Lisle Equivalence transformations for classes of differential

equations [PhD thesis] University of British Columbia 1992[7] M Senthilvelan M Torrisi and A Valenti ldquoEquivalence trans-

formations and differential invariants of a generalized nonlinearSchrodinger equationrdquo Journal of Physics A vol 39 no 14 pp3703ndash3713 2006

[8] M L Gandarias M Torrisi and R Tracina ldquoOn some differ-ential invariants for a family of diffusion equationsrdquo Journal ofPhysics A vol 40 no 30 pp 8803ndash8813 2007

[9] J-P Gazeau and P Winternitz ldquoSymmetries of variable coef-ficient Korteweg-de Vries equationsrdquo Journal of MathematicalPhysics vol 33 no 12 pp 4087ndash4102 1992

[10] P Winternitz and J-P Gazeau ldquoAllowed transformations andsymmetry classes of variable coefficient Korteweg-de Vriesequationsrdquo Physics Letters A vol 167 no 3 pp 246ndash250 1992

[11] LVOvsiannikovGroupAnalysis ofDifferential Equations Aca-demic Press New York NY USA 1982

[12] I S Akhatov R K Gazizov and N K Ibragimov ldquoNonlocalsymmetries Heuristic approachrdquo Journal of SovietMathematicsvol 55 no 1 pp 1401ndash1450 1991

[13] M Torrisi and R Tracina ldquoEquivalence transformations andsymmetries for a heat conduction modelrdquo International Journalof Non-Linear Mechanics vol 33 no 3 pp 473ndash487 1998

[14] V Romano and M Torrisi ldquoApplication of weak equivalencetransformations to a group analysis of a drift-diffusion modelrdquoJournal of Physics A vol 32 no 45 pp 7953ndash7963 1999

[15] N H Ibragimov CRC Handbook of Lie Group Analysis of Dif-ferential Equations CRC Press Boca Raton Fla USA 1996

[16] M Molati and C M Khalique ldquoLie group classification of ageneralized Lane-Emden type system in two dimensionsrdquo Jour-nal of Applied Mathematics vol 2012 Article ID 405978 10pages 2012

[17] C M Khalique F M Mahomed and B P Ntsime ldquoGroup clas-sification of the generalized Emden-Fowler-type equationrdquoNonlinear Analysis Real World Applications vol 10 no 6 pp3387ndash3395 2009

[18] M Torrisi andR Tracina ldquoEquivalence transformations for sys-tems of first order quasilinear partial differential equationsrdquo inModern Group Analysis VI Developments in Theory Computa-tion andApplication NH Ibragimov and FMMahomed EdsNew Age International 1996

[19] M Torrisi R Tracina and A Valenti ldquoA group analysisapproach for a nonlinear differential system arising in diffusionphenomenardquo Journal of Mathematical Physics vol 37 no 9 pp4758ndash4767 1996

[20] M Torrisi and R Tracina ldquoExact solutions of a reaction-diffusion systems for Proteus mirabilis bacterial coloniesrdquo Non-linear Analysis Real World Applications vol 12 no 3 pp 1865ndash1874 2011

[21] G Gambino A M Greco and M C Lombardo ldquoA groupanalysis via weak equivalence transformations for a model oftumour encapsulationrdquo Journal of Physics A vol 37 no 12 pp3835ndash3846 2004

[22] N H Ibragimov and N Safstrom ldquoThe equivalence group andinvariant solutions of a tumour growth modelrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 9 no1 pp 61ndash68 2004

[23] M Torrisi and R Tracina ldquoSecond-order differential invariantsof a family of diffusion equationsrdquo Journal of Physics A vol 38no 34 pp 7519ndash7526 2005

[24] M Torrisi and R Tracina ldquoOn a class ofd reaction diffusion sys-tems E quivalence transformations and symmetriesrdquo inAsymp-totic Methods in NonlinearWave Phenomena T Ruggeri andMSammartino Eds pp 207ndash216 2007

[25] N H Ibragimov and M Torrisi ldquoA simple method for groupanalysis and its application to a model of detonationrdquo Journal ofMathematical Physics vol 33 no 11 pp 3931ndash3937 1992

Page 10

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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ProbabilityandStatistics

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Advances in

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Complex AnalysisJournal of

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OptimizationJournal of

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Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Advances in

DecisionSciences

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Discrete MathematicsJournal of

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Stochastic AnalysisInternational Journal of