analyticalsolutionsofamodifiedpredator-prey

Research ArticleAnalytical Solutions of a Modified Predator-PreyModel through a New Ecological Interaction

Noufe H Aljahdaly 1 and Manar A Alqudah2

1Department of Mathematics Faculty of Sciences and Arts-Rabigh Campus King Abdulaziz University Jeddah Saudi Arabia2Mathematical Science Department Princess Nourah Bint Abdulrahman University PO Box 84428 Riyadh 11671Saudi Arabia

Correspondence should be addressed to Noufe H Aljahdaly nhaljahdalykauedusa

Received 22 May 2019 Revised 20 August 2019 Accepted 17 September 2019 Published 16 October 2019

Academic Editor Manuel F G Penedo

Copyright copy 2019 Noufe H Aljahdaly and Manar A Alqudah -is is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

-e predator-prey model is a common tool that researchers develop continuously to predict the dynamics of the animalpopulation within a certain phenomenon Due to the sexual interaction of the predator in the mating period the males andfemales feed together on one ormore preys-is scenario describes the ecological interaction between two predators and one preyIn this study the nonlinear diffusive predator-prey model is presented where this type of interaction is accounted for -einfluence of this interaction on the population of predators and preys is predicted through analytical solutions of the dynamicalsystem -e solutions are obtained by using two reliable and simple methods and are presented in terms of hyperbolic functionsIn addition the biological relevance of the solutions is discussed

1 Introduction

Many phenomena in natural science biology physics orengineering are studied by developing a mathematical modelthat consists of partial differential equations A prey-predatorsystem is a well-known mathematical model for studying thespeciesrsquo population and density [1 2] Many different in-teractions in this model are very significant phenomena innaturersquos population [3 4] Alqudah [5] modified the diffusivepredator-preymodel in which two predators interact with oneor more preys in the mating period as follows

ut minus D1(Δu) a1u minus a2u2

minus a3uv minus a4uv2

vt minus D2(Δv) a3uv minus a5v minus a6v2

+ a4uv2

(1)

-e biological meaning of each term is presented inTable 1 We assume that ai for all i 1 2 6 are positiveconstants and D D1D2 D1 0 D2 ne 0 which means thatthe predator is moving toward the prey

-is model focuses on the term uv2 which refers to apositive interaction with respect to the predator species like

sexual interaction in the mating period to proliferate thepredators -us the males and females are together and feedon the same prey or more but this term is negative withrespect to the prey In [5] the existence and the uniquenessof the solution of model (1) are studied and the scaling isintroduced to get the dimensionless parameters -usmodel (1) can be expressed as

Ut U minus U2

minus UV minus UV2

Vt VXX minus k2k1V minus k1V2

+ k1VU + k1UV2

(2)

where k1 a4a1

radic a2 and k2 a5(a1k1)One way to understand the applications and dynamic

system is finding the analytical or approximate solutions Inthe field of mathematical computation several methods havebeen developed to find the analytical solutions such as thetanh method [6] the Riccati equation expansion method [7]the (GprimeG2)-expansion method [8 9] the algebraic method[10] and so on In this paper the exact and stable solutionsof the system are obtained using two sufficient methods

HindawiComputational and Mathematical Methods in MedicineVolume 2019 Article ID 4849393 7 pageshttpsdoiorg10115520194849393

(i) the (GprimeG)-expansion method [11] and (ii) the general-ized auxiliary equation method [12]

-is paper is organized as follows Section 2 is the de-scription of two employed methods Section 3 presents theanalytical solutions of the considered dynamical systemSection 4 is the discussion of the results including the bi-ological interpretation -e last section is the conclusion ofthe work and the results

2 Description of Algorithms

-is section shows briefly the steps of applying the function-expansion method for finding analytical solutions We chosetwo methods that are simple and reliable using the as-ymptotic software (MATHEMATICA) which are the(GprimeG)-expansion method and the generalized auxiliaryequation method Let us consider the nonlinear evaluationequation of the form

P1 U V Ut Vt Ux Vx Uxx Vxx ( 1113857 0

P2 U V Ut Vt Ux Vx Uxx Vxx ( 1113857 0(3)

where P1 and P2 are polynomials in U V and their de-rivatives and U(x t) and V(x t) are two unknown func-tions -e methods that are considered in this article havethe common following steps

(i) Transforming the function U(x t) and V(x t) tou1(ξ) and v1(ξ) respectively by applying thetransformation ξ x minus ct where c is the wave speed

-us the system of equations (3) is reduced to theordinary differential equation (ODE)

P1 u1 v1 minus cu1ξ minus cv1ξ u1ξ v1ξ u1ξξ v1ξξ 1113874 1113875 0


(4)

(ii) Assuming the function u1(ξ) and v1(ξ) areexpressed by the following polynomials

u1(ξ) 1113944n

i0αi(F(ξ))

i

v1(ξ) 1113944m

i0βi(F(ξ))

i

(5)

where ngt 0 and mgt 0 are the degrees of the poly-nomial of u1 and v1 respectively We obtain thevalue of n and m by the homogeneous balancetheory [13ndash15]

(iii) Substituting the polynomials (5) into equation (4) toobtain polynomials of the function F(ξ)

(iv) Equating the confections of (F(ξ))i i 1 2 n

(or m) to zero to obtain a system in terms of var-iables αi βi and c

(v) Solving the system to find the value of αi βi and c(vi) Substituting the obtained variables αi βi and the

value of the function F(ξ) into equation (5) toconstruct the solutions of U and V

-e main difference between the methods utilized in thispaper is the value of the function F(ξ)

(1) -e (GprimeG)-expansion method where F GprimeGsatisfies

Fprime minus F2

minus λF minus μ (6)

and G satisfies the second order ODE

GPrime + λGprime + μG 0 (7)

-e function F(ξ) is the general solution of theequation (6) as follows

Table 1 Description of variables and parameters of model (1)

Variable Descriptionu u(x t) Density of preyv v(x t) Density of predatorD1(Δu) Diffusion term of prey speciesD2(Δv) Diffusion term of predator species

D1 and D2Diffusion coefficients which are small positive

constantsa1 Growth rate of the prey

a2Decay rate of prey due to competition on the food

supply between the male and the female

a3Decay rate of prey due to the interaction between one

predator and one prey

a4Decay rate of prey due to the interaction between two

predators and one preya5 Mortality rate of predator

a6Decay rate of predator due to competition on thefood supply between the male and the female

2 Computational and Mathematical Methods in Medicine

F(ξ)

λ2 minus 4μ1113969

2

A1 cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2 sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1 sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2 cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2 for λ2 minus 4μgt 0

minus λ2 + 4μ1113969

2

A1 cos (12)

minus λ2 + 4μ1113969

ξ1113874 1113875 minus A2 sin (12)

minus λ2 + 4μ1113969

ξ1113874 11138751113874 1113875

A1 sin (12)

minus λ2 + 4μ1113969

ξ1113874 1113875 + A2 cos (12)

minus λ2 + 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2 for λ2 minus 4μlt 0

A2

A1 + A2ξ( 1113857minusλ2 for λ2 minus 4μ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

(2) -e generalized auxiliary equation method where Fsatisfies the auxiliary equation

Fprime

h0 + h1F + h2F2 + h3F

3 + h4F4

1113969

(9)

-e general solutions of the auxiliary equation (9) haveseveral types of solutions depending on the value of hi

i 1 2 4 -e following are some types of thesolution and the reader is referred to [10 12] to findmore cases

(I) If h0 h1 0 the solution of equation (9) isexpressed as follows

F(ξ)

2h2sec hh2

1113968ξ( 1113857

δ1

1113968minus h4sec h

h2

1113968ξ( 1113857

h2 gt 0 δ1 gt 0 h3 h4

2h2csc hh2

1113968ξ( 1113857

minus δ1

1113968minus h4csc h

h2

1113968ξ( 1113857

h2 gt 0 δ1 lt 0 h3 h4

minush2

h41 plusmn tan

h2

1113969

21113874 1113875ξ1113874 11138751113874 1113875 h2 gt 0 δ1 0

h2sec h2 (12)h2

1113968x( 1113857

2h2h4

1113968tan h (12)

h2

1113968x( 1113857 minus h3

h2 gt 0

h2sec2 (12)minus h2

1113968x( 1113857

2minus h2h4

1113968tan (12)

minus h2

1113968x( 1113857 minus h3

h2 lt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

where δ1 h23 minus 4h2h4

(II) If h0 h1 h3 0

F(ξ)

minus h2h4( 1113857

1113969sec h

h2

1113968x( 1113857 h2 gt 0 h4 lt 0

minus h2h4( 1113857

1113969sec

minus h2

1113968x( 1113857 h2 lt 0 h4 gt 0

1h4

1113968x

h2 0 h4 gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

(III) If h0 h1 h4 0

F(ξ)

minus h2h3( 1113857sec h2

h2

1113968ξ

21113888 1113889 h2 gt 0

minus h2h3( 1113857secminus h2

1113968ξ

21113888 1113889 h2 lt 0

1h3ξ

2 h2 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

Computational and Mathematical Methods in Medicine 3

3 Analytical Solutions of the DiffusivePredator-Prey System

In this section the analytical solutions of model (1) will befound by the methods that are described in the previoussection First introducing the traveling wave solution ξ

X minus cT reduces model (1) as

minus cu1prime u1 minus u21 minus u1v1 minus u1v

21 (13a)

minus cv1prime vPrime1 minus k1k2v1 minus k1v21 + k1u1v1 + k1u1v

21 (13b)

where U(X T) u1(ξ) V(X T) v1(ξ) and c is the wavespeed -en the equations (13a) and (13b) are combined asfollows

minus c u1prime + v1prime( 1113857 vPrime1 minus k1k2v1 minus k1v21 + u1 minus u

21

minus 1 minus k1( 1113857u1v1 minus 1 minus k1( 1113857u1v21

(14)

Following the steps of the algorithm in Section 2 thesolutions of u1 and v1 can be expressed as follows

u1(ξ) α1F(ξ)

v1(ξ) β1F(ξ)(15)

Substituting equation (15) into equation (14) to obtain

c α1 + β1( 1113857Fprime(x) + β1FPrime(x) + α1 minus β1k1k2( 1113857F(x)

+ α1β1 k1 minus 1( 1113857 minus β21k1 minus α211113872 1113873F2(x)

+ α1β21 k1 minus 1( 1113857F

3(x) 0

(16)

31 7e (GprimeG)-Expansion Method In order to utilize the(GprimeG)-expansion method the definition of Fprime equation (6)is applied to equation (16) and then the following algebraicsystem is obtained

α1(1 minus cλ) + β1 minus cλ + λ2 minus k1k2 + 2μ1113872 1113873 0

minus α21 minus α1 c minus β1 k1 minus 1( 1113857( 1113857 minus β1 c minus 3λ + β1k1( 1113857 0

β1 α1β1 k1 minus 1( 1113857 + 2( 1113857 0

minus μ α1c + β1(c minus λ)( 1113857 0

(17)

Hence two cases of the value of μ λ c and α1 are realized

(i) Case I

c β41 k1 minus 1( 1113857

2k1 + 2β21 k1 minus 1( 1113857

2+ 4

2β1 k1 minus 1( 1113857 β21 k1 minus 1( 1113857 minus 21113872 1113873

μ β21 k1 minus 1( 1113857k1k2 + 2

2β21 k1 minus 1( 1113857

λ β41 k1 minus 1( 1113857

2k1 + 2β21 k1 minus 1( 1113857

2+ 4

2β31(k1 minus 1)2

α1 minus2

β1 k1 minus 1( 1113857

(18)

(ii) Case II

μ 0

c 1

β1 minus 2β1 k1 minus 1( 1113857( 1113857minus14β21k1 minus

4β21 k1 minus 1( 1113857

2 +3k1

β21 k1 minus 1( 11138573 minus

3β21 k1 minus 1( 1113857

3⎛⎝

minus34β1

β41 k1 minus 1( 11138572k1 + 2β21 k1 minus 1( 1113857

2+ 41113872 1113873

2minus 8β41 k1 minus 1( 1113857

3 β21 k1 minus 1( 1113857k1k2 + 21113872 1113873

β61 k1 minus 1( 11138574

11139741113972

+3k1

2 k1 minus 1( 1113857+

32 minus 2k1

minus 2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

λ 18

2β1k1 +4k1

β1 k1 minus 1( 1113857+

8k1


4β1 k1 minus 1( 1113857

minus8

β31 k1 minus 1( 11138573

⎛⎝

minus 2


2+ 41113872 1113873

2minus 8β41 k1 minus 1( 1113857

3 β21 k1 minus 1( 1113857k1k2 + 21113872 1113873

β61 k1 minus 1( 11138574

11139741113972

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

α1 minus2

β1 k1 minus 1( 1113857

(19)


-e solutions within both cases exist if k1 ne 1 As we seein Figure 1 λ2 minus 4μgt 0 for k1 ne 1 -erefore the solution ofthe system is in terms of the hyperbolic function and pa-rameters k1 β1 and k2 Assuming 0lt k1 lt 1 yields

a4a1

radic lt a2which is not a reasonable case in actual situations-erefore

the solution is considered only for k1 gt 1 (a4a1

radic gt a2) -efollowing is the solution by the (GprimeG)-expansion methodwhich is a kink soliton solution for U and V as we see inFigure 2

U(ξ) minus2

β1 k1 minus 1( 1113857

λ2 minus 4μ1113969

2

A1cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2

V(ξ) β1

λ2 minus 4μ1113969

2

A1cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2

(20)

32 7e Generalized Auxiliary Equation Method -is sub-section presents the solution of the considered system byapplying the auxiliary equation method -us we use thedefinition of Fprime (9) into equation (16) which yields to thefollowing algebraic systems

α1 + β1 0

α1 + β1 h2 minus k1k2( 1113857 0

minus α21 +12β1 3h3 minus 2β1k1( 1113857 + α1β1 k1 minus 1( 1113857 0

β1 2h4 + α1β1 k1 minus 1( 1113857( 1113857 0

β1h1

2 0

(21)

-e solutions of the algebraic system aforementioned giveh1 0 h2 1 + k1k2 h3 (4k1β13) h4 (12)(k1 minus 1)

β21 and h0 k1 β1 and k2 are arbitrary constants Hence theappropriate solution of the auxiliary equation is in terms ofthe hyperbolic function Assuming h0 0 yields the solutionof the problem in the following expression

U(x t) minusβ1 k1k2 + 1( 1113857sec h2 (12)

k1k2 + 1

1113968(x minus ct)( 1113857

2

radic

β21 k1 minus 1( 1113857 k1k2 + 1( 1113857

1113969

tan h (12)k1k2 + 1

1113968(x minus ct)( 1113857 minus 4β1k13( 1113857

V(x t) minus U(x t)

(22)

-e computed solution by the generalized auxiliaryequation method exists if k1 ne 1 and is a soliton solution interms of k1 k2 and β1 (see Figure 3)

4 Discussion

41 Biological Implication -e solution is obtained by the(GprimeG)-expansion method for λgt 2 μradic and with the con-dition a4 gt (a2

2a1) -us this solution is obtained when thedecay rate of the prey population is greater due to the in-teraction between two predators and one prey (a4) than thedecay rate of the prey population due to the competition onthe food supply over the growth rate of the prey (a2

2a1) -esolution in Figure 2 shows the predator population increasesbecause of plentiful prey and sexual interaction while the

prey population decline because of its high consumptionduring the mating period of the predator in a close envi-ronmental area Ultimately the predator population willbecome dominant in the area -is situation is expectedwhen the predators overgraze in themating period where theprey is plentiful

Figure 3 presents the solution by applying the gener-alized auxiliary equation method for k1 ne 1 -is solutiondepicts that in certain environmental area the prey pop-ulation u will grow due to the absence of the predator v Inthe mating period the predators graze in where the preydensity (u) is large and as a result u will decay However theobtained solutions by both considered methods remainconstant away from places of grazing predator during thepredator-mating period and over time


42 Connection with the Previous Studies -e diffusivepredator-prey model has been solved numerically and an-alytically in some papers -e analytical solutions of themodel when both species having the same diffusivities andthe prey density being of the Allee type are computed by the(GprimeG)-expansion method [16] and the results are eithersingle structure or periodic structure Also it was computedby the improved Riccati equation mapping method [17] and

the solutions are presented in three structures single pe-riodic and kink -e same model was solved analytically byutilizing the exp-expansion method [18] and the obtainedsolutions were kink solutions singular kink solutions darksoliton solutions bright soliton solutions soliton solutionssingular soliton solutions multiple soliton-like solutionsand triangular periodic solutions but all these solutions werenot discussed in terms of the actual situation

-e numerical solution of a system of nonlinear Volterradifferential equations governing on the problem of prey andpredator was solved numerically by the Adomian de-composition method [19] RungendashKuttandashFehlberg methodand Laplace Adomian decomposition method [20] and theresults state that the number of predator increases as thenumber of prey decreases

-e present paper studied the predator-prey model withadding the term uv2 and in absence of diffusion among preysD1 0 We obtained that predatorrsquos population increases asthe preyrsquos population decreases in a certain interval and thesolutions are constant out of this interval Comparing theresults obtained in the present study with different modi-fications of the diffusive predator-prey model it can beconcluded that obtained results are new and different

5 Conclusion

In this paper both the (GprimeG)-expansion method and thegeneralized auxiliary equation method have been used tofind new stable analytical solutions of the modified non-linear diffusive predator-prey model by considering the termuv2 To the best of the researchersrsquo knowledge this modifiedmodel has not been solved analytically or numerically inprevious research -e result showed the effect of the termuv2 in the diffusive predator-prey dynamic system whichleads to a decline in the prey population and an increase inthe predator population in the mating period at predatorrsquosdensity points However the idea of this modified systemcan be used in epidemiological researches to constructseveral models For example if we consider u as the infectedhuman body by any disease in many cases we use two typesof medicine which is considered as v2 -us the interactionterm uv2 refers to combatting the epidemic or the disease

02 04 06 08k1

2

4

6

8λ2

ndash4μ

(a)

k1

λ2ndash

4μ

20 25 30

20

30

40

50

(b)

Figure 1 Plot λ2 minus 4μ2 versus k1 of case (I) where β1 and k2 are fixed (a) 0lt k1 lt (0lt a4a1

radic lt a2) (b) k1 gt 1 (a4a1

radic gt a2)

ndash420ndash2

2

0

ndash2

4

ndash420ndash2

4

x

t

U (x t)V (x t)

Figure 2 Plot of the scaled density of the prey for t x [minus 4 4]μ 0 λ 2(β21 + 2)3β1 α1 minus β1 k1(2β

21) + 1 and

k2 (4β41 + 7β21 + 16)(9β21 + 18) where β1 05

ndash4

ndash4

2

2

0

0

ndash2

ndash2

4

4

x

10

05

00

ndash05

ndash10

t

U (x t)V (x t)

Figure 3 Plot of the scaled density of the prey equations (22) fort x [minus 4 4] k2 05 β1 1 c 1 and (k1 3)


through two types of medicine In addition the prey-predator system has been studied widely in the literatureregarding the time delay [21ndash23] In our future work we willstudy time delay for our new modified system with the effectof fluctuation of the biological parameters or fuzzyparameters

Data Availability

-e data supporting this research are from previously re-ported studies which have been cited

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Noufe Aljahdaly implemented the analytical solutions withsoftware and analyzed the data Manar Alqudah proposedthe mathematical model of this paper All authors wrote thispaper read and approved the final manuscript

Acknowledgments

-is research was funded by the Deanship of ScientificResearch at Princess Nourah Bint Abdulrahman Universitythrough the Fast-Track Research Funding Program

References

[1] A J Lotka ldquoElements of physical biologyrdquo Science Progress inthe Twentieth Century vol 21 no 82 pp 341ndash343 1926

[2] V Volterra ldquoVariations and Fluctuations of the Number ofIndividuals in Animal Species Living Togetherrdquo AnimalEcology pp 409ndash448 1926

[3] O P Misra P Sinha and C S Sikarwar ldquoDynamical study ofa prey-predator system with a commensal species competingwith prey species effect of time lag and alternative foodsourcerdquo Computational and Applied Mathematics vol 34no 1 pp 343ndash361 2015

[4] Q Wang Z Liu X Zhang and R A Cheke ldquoIncorporatingprey refuge into a predator-prey system with imprecise pa-rameter estimatesrdquo Computational and Applied Mathematicsvol 36 no 2 pp 1067ndash1084 2017

[5] M A Al Qudah ldquoExistence uniqueness solution of amodified predator-prey modelrdquo Nonlinear Analysis andDifferential Equations vol 4 pp 669ndash677 2016

[6] S A Elwakil S K M A El-labany R Sabry and R SabryldquoModified extended tanh-function method for solving non-linear partial differential equationsrdquo Physics Letters Avol 299 no 2-3 pp 179ndash188 2002

[7] C Yong L Biao and Z Hong-Qing ldquoSymbolic computationand construction of soliton-like solutions to the (2+1)-di-mensional breaking soliton equationrdquo Communications in7eoretical Physics vol 40 no 2 pp 137ndash142 2003

[8] K Zhouzheng ldquo(GprimeG2)-expansion solutions to MBBM andOBBM equationsrdquo Journal of Partial Differential Equationsvol 28 no 2 pp 158ndash166 2015

[9] N H Aljahdaly ldquoSome applications of the modified(GprimeG2)-expansion method in mathematical physicsrdquo Resultsin Physics vol 13 Article ID 102272 2019

[10] E Fan ldquoMultiple travelling wave solutions of nonlinearevolution equations using a unified algebraic methodrdquoJournal of Physics A Mathematical and General vol 35no 32 pp 6853ndash6872 2002

[11] M Wang X Li and J Zhang ldquo-e (GprimeG2)-expansionmethod and travelling wave solutions of nonlinear evolutionequations in mathematical physicsrdquo Physics Letters Avol 372 no 4 pp 417ndash423 2008

[12] E Yomba ldquoA generalized auxiliary equation method and itsapplication to nonlinear Klein-Gordon and generalizednonlinear Camassa-Holm equationsrdquo Physics Letters Avol 372 no 7 pp 1048ndash1060 2007

[13] M Wang ldquoExact solutions for a compound KdV-Burgersequationrdquo Physics Letters A vol 213 no 5-6 pp 279ndash2871996

[14] M Wang ldquoSolitary wave solutions for variant Boussinesqequationsrdquo Physics Letters A vol 199 no 3-4 pp 169ndash1721995

[15] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1-5pp 67ndash75 1996

[16] R A Kraenkel K Manikandan and M Senthilvelan ldquoOncertain new exact solutions of a diffusive predator-prey sys-temrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1269ndash1274 2013

[17] H Kim and J H Choi ldquoExact solutions of a diffusivepredator-prey system by the generalized Riccati equationrdquoBulletin of the Malaysian Mathematical Sciences Societyvol 39 no 3 pp 1125ndash1143 2016

[18] M N Alam and C Tunc ldquoAn analytical method for solvingexact solutions of the nonlinear Bogoyavlenskii equation andthe nonlinear diffusive predator-prey systemrdquo AlexandriaEngineering Journal vol 55 no 2 pp 1855ndash1865 2016

[19] J Biazar and R Montazeri ldquoA computational method forsolution of the prey and predator problemrdquo Applied Math-ematics and Computation vol 163 no 2 pp 841ndash847 2005

[20] S Paul S P Mondal and P Bhattacharya ldquoNumerical so-lution of Lotka Volterra prey predator model by using Runge-Kutta-Fehlberg method and Laplace Adomian decompositionmethodrdquo Alexandria Engineering Journal vol 55 no 1pp 613ndash617 2016

[21] D Pal and G S Mahapatra ldquoEffect of toxic substance ondelayed competitive allelopathic phytoplankton system withvarying parameters through stability and bifurcation analy-sisrdquo Chaos Solitons amp Fractals vol 87 pp 109ndash124 2016

[22] D Pal G S Mahapatra and G P Samanta ldquoNew approachfor stability and bifurcation analysis on predator-prey har-vesting model for interval biological parameters with timedelaysrdquo Computational and Applied Mathematics vol 37no 3 pp 3145ndash3171 2018

[23] D Pal G S Mahapatra and G P Samanta ldquoA study ofbifurcation of prey-predator model with time delay andharvesting using fuzzy parametersrdquo Journal of BiologicalSystems vol 26 no 2 pp 339ndash372 2018


Stem Cells International

Hindawiwwwhindawicom Volume 2018


MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



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Volume 2018

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(i) the (GprimeG)-expansion method [11] and (ii) the general-ized auxiliary equation method [12]

-is paper is organized as follows Section 2 is the de-scription of two employed methods Section 3 presents theanalytical solutions of the considered dynamical systemSection 4 is the discussion of the results including the bi-ological interpretation -e last section is the conclusion ofthe work and the results

2 Description of Algorithms

-is section shows briefly the steps of applying the function-expansion method for finding analytical solutions We chosetwo methods that are simple and reliable using the as-ymptotic software (MATHEMATICA) which are the(GprimeG)-expansion method and the generalized auxiliaryequation method Let us consider the nonlinear evaluationequation of the form

P1 U V Ut Vt Ux Vx Uxx Vxx ( 1113857 0

P2 U V Ut Vt Ux Vx Uxx Vxx ( 1113857 0(3)

where P1 and P2 are polynomials in U V and their de-rivatives and U(x t) and V(x t) are two unknown func-tions -e methods that are considered in this article havethe common following steps

(i) Transforming the function U(x t) and V(x t) tou1(ξ) and v1(ξ) respectively by applying thetransformation ξ x minus ct where c is the wave speed

-us the system of equations (3) is reduced to theordinary differential equation (ODE)



(4)

(ii) Assuming the function u1(ξ) and v1(ξ) areexpressed by the following polynomials

u1(ξ) 1113944n

i0αi(F(ξ))

i

v1(ξ) 1113944m

i0βi(F(ξ))

i

(5)

where ngt 0 and mgt 0 are the degrees of the poly-nomial of u1 and v1 respectively We obtain thevalue of n and m by the homogeneous balancetheory [13ndash15]

(iii) Substituting the polynomials (5) into equation (4) toobtain polynomials of the function F(ξ)

(iv) Equating the confections of (F(ξ))i i 1 2 n

(or m) to zero to obtain a system in terms of var-iables αi βi and c

(v) Solving the system to find the value of αi βi and c(vi) Substituting the obtained variables αi βi and the

value of the function F(ξ) into equation (5) toconstruct the solutions of U and V

-e main difference between the methods utilized in thispaper is the value of the function F(ξ)

(1) -e (GprimeG)-expansion method where F GprimeGsatisfies

Fprime minus F2

minus λF minus μ (6)

and G satisfies the second order ODE

GPrime + λGprime + μG 0 (7)

-e function F(ξ) is the general solution of theequation (6) as follows

Table 1 Description of variables and parameters of model (1)

Variable Descriptionu u(x t) Density of preyv v(x t) Density of predatorD1(Δu) Diffusion term of prey speciesD2(Δv) Diffusion term of predator species

D1 and D2Diffusion coefficients which are small positive

constantsa1 Growth rate of the prey

a2Decay rate of prey due to competition on the food

supply between the male and the female

a3Decay rate of prey due to the interaction between one

predator and one prey

a4Decay rate of prey due to the interaction between two

predators and one preya5 Mortality rate of predator

a6Decay rate of predator due to competition on thefood supply between the male and the female


F(ξ)

λ2 minus 4μ1113969

2

A1 cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2 sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1 sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2 cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875


minus λ2 + 4μ1113969

2

A1 cos (12)

minus λ2 + 4μ1113969

ξ1113874 1113875 minus A2 sin (12)

minus λ2 + 4μ1113969

ξ1113874 11138751113874 1113875

A1 sin (12)

minus λ2 + 4μ1113969

ξ1113874 1113875 + A2 cos (12)

minus λ2 + 4μ1113969

ξ1113874 11138751113874 1113875


A2


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)


Fprime

h0 + h1F + h2F2 + h3F

3 + h4F4

1113969

(9)




F(ξ)

2h2sec hh2

1113968ξ( 1113857

δ1


h2

1113968ξ( 1113857

h2 gt 0 δ1 gt 0 h3 h4

2h2csc hh2

1113968ξ( 1113857

minus δ1


h2

1113968ξ( 1113857

h2 gt 0 δ1 lt 0 h3 h4

minush2

h41 plusmn tan

h2

1113969

21113874 1113875ξ1113874 11138751113874 1113875 h2 gt 0 δ1 0

h2sec h2 (12)h2

1113968x( 1113857

2h2h4

1113968tan h (12)

h2

1113968x( 1113857 minus h3

h2 gt 0

h2sec2 (12)minus h2

1113968x( 1113857

2minus h2h4

1113968tan (12)

minus h2

1113968x( 1113857 minus h3

h2 lt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)


(II) If h0 h1 h3 0

F(ξ)

minus h2h4( 1113857

1113969sec h

h2

1113968x( 1113857 h2 gt 0 h4 lt 0

minus h2h4( 1113857

1113969sec

minus h2

1113968x( 1113857 h2 lt 0 h4 gt 0

1h4

1113968x

h2 0 h4 gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

(III) If h0 h1 h4 0

F(ξ)


h2

1113968ξ

21113888 1113889 h2 gt 0


1113968ξ

21113888 1113889 h2 lt 0

1h3ξ

2 h2 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)






21 (13a)


21 (13b)



21


(14)


u1(ξ) α1F(ξ)

v1(ξ) β1F(ξ)(15)




+ α1β21 k1 minus 1( 1113857F

3(x) 0

(16)




β1 α1β1 k1 minus 1( 1113857 + 2( 1113857 0


(17)


(i) Case I

c β41 k1 minus 1( 1113857

2k1 + 2β21 k1 minus 1( 1113857

2+ 4


μ β21 k1 minus 1( 1113857k1k2 + 2

2β21 k1 minus 1( 1113857

λ β41 k1 minus 1( 1113857

2k1 + 2β21 k1 minus 1( 1113857

2+ 4

2β31(k1 minus 1)2

α1 minus2

β1 k1 minus 1( 1113857

(18)

(ii) Case II

μ 0

c 1


4β21 k1 minus 1( 1113857

2 +3k1


3β21 k1 minus 1( 1113857

3⎛⎝

minus34β1


2+ 41113872 1113873

2minus 8β41 k1 minus 1( 1113857

3 β21 k1 minus 1( 1113857k1k2 + 21113872 1113873

β61 k1 minus 1( 11138574

11139741113972

+3k1

2 k1 minus 1( 1113857+

32 minus 2k1

minus 2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

λ 18

2β1k1 +4k1

β1 k1 minus 1( 1113857+

8k1


4β1 k1 minus 1( 1113857

minus8

β31 k1 minus 1( 11138573

⎛⎝

minus 2


2+ 41113872 1113873

2minus 8β41 k1 minus 1( 1113857

3 β21 k1 minus 1( 1113857k1k2 + 21113872 1113873

β61 k1 minus 1( 11138574

11139741113972

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

α1 minus2

β1 k1 minus 1( 1113857

(19)



a4a1




U(ξ) minus2

β1 k1 minus 1( 1113857

λ2 minus 4μ1113969

2

A1cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2

V(ξ) β1

λ2 minus 4μ1113969

2

A1cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2

(20)


α1 + β1 0

α1 + β1 h2 minus k1k2( 1113857 0


β1 2h4 + α1β1 k1 minus 1( 1113857( 1113857 0

β1h1

2 0

(21)




k1k2 + 1

1113968(x minus ct)( 1113857

2

radic

β21 k1 minus 1( 1113857 k1k2 + 1( 1113857

1113969

tan h (12)k1k2 + 1


V(x t) minus U(x t)

(22)


4 Discussion











5 Conclusion


02 04 06 08k1

2

4

6

8λ2

ndash4μ

(a)

k1

λ2ndash

4μ

20 25 30

20

30

40

50

(b)



radic gt a2)

ndash420ndash2

2

0

ndash2

4

ndash420ndash2

4

x

t

U (x t)V (x t)


21) + 1 and

k2 (4β41 + 7β21 + 16)(9β21 + 18) where β1 05

ndash4

ndash4

2

2

0

0

ndash2

ndash2

4

4

x

10

05

00

ndash05

ndash10

t

U (x t)V (x t)




Data Availability






Acknowledgments


References





























of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















F(ξ)

λ2 minus 4μ1113969

2

A1 cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2 sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1 sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2 cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875


minus λ2 + 4μ1113969

2

A1 cos (12)

minus λ2 + 4μ1113969

ξ1113874 1113875 minus A2 sin (12)

minus λ2 + 4μ1113969

ξ1113874 11138751113874 1113875

A1 sin (12)

minus λ2 + 4μ1113969

ξ1113874 1113875 + A2 cos (12)

minus λ2 + 4μ1113969

ξ1113874 11138751113874 1113875


A2


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)


Fprime

h0 + h1F + h2F2 + h3F

3 + h4F4

1113969

(9)




F(ξ)

2h2sec hh2

1113968ξ( 1113857

δ1


h2

1113968ξ( 1113857

h2 gt 0 δ1 gt 0 h3 h4

2h2csc hh2

1113968ξ( 1113857

minus δ1


h2

1113968ξ( 1113857

h2 gt 0 δ1 lt 0 h3 h4

minush2

h41 plusmn tan

h2

1113969

21113874 1113875ξ1113874 11138751113874 1113875 h2 gt 0 δ1 0

h2sec h2 (12)h2

1113968x( 1113857

2h2h4

1113968tan h (12)

h2

1113968x( 1113857 minus h3

h2 gt 0

h2sec2 (12)minus h2

1113968x( 1113857

2minus h2h4

1113968tan (12)

minus h2

1113968x( 1113857 minus h3

h2 lt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)


(II) If h0 h1 h3 0

F(ξ)

minus h2h4( 1113857

1113969sec h

h2

1113968x( 1113857 h2 gt 0 h4 lt 0

minus h2h4( 1113857

1113969sec

minus h2

1113968x( 1113857 h2 lt 0 h4 gt 0

1h4

1113968x

h2 0 h4 gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

(III) If h0 h1 h4 0

F(ξ)


h2

1113968ξ

21113888 1113889 h2 gt 0


1113968ξ

21113888 1113889 h2 lt 0

1h3ξ

2 h2 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)






21 (13a)


21 (13b)



21


(14)


u1(ξ) α1F(ξ)

v1(ξ) β1F(ξ)(15)




+ α1β21 k1 minus 1( 1113857F

3(x) 0

(16)




β1 α1β1 k1 minus 1( 1113857 + 2( 1113857 0


(17)


(i) Case I

c β41 k1 minus 1( 1113857

2k1 + 2β21 k1 minus 1( 1113857

2+ 4


μ β21 k1 minus 1( 1113857k1k2 + 2

2β21 k1 minus 1( 1113857

λ β41 k1 minus 1( 1113857

2k1 + 2β21 k1 minus 1( 1113857

2+ 4

2β31(k1 minus 1)2

α1 minus2

β1 k1 minus 1( 1113857

(18)

(ii) Case II

μ 0

c 1


4β21 k1 minus 1( 1113857

2 +3k1


3β21 k1 minus 1( 1113857

3⎛⎝

minus34β1


2+ 41113872 1113873

2minus 8β41 k1 minus 1( 1113857

3 β21 k1 minus 1( 1113857k1k2 + 21113872 1113873

β61 k1 minus 1( 11138574

11139741113972

+3k1

2 k1 minus 1( 1113857+

32 minus 2k1

minus 2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

λ 18

2β1k1 +4k1

β1 k1 minus 1( 1113857+

8k1


4β1 k1 minus 1( 1113857

minus8

β31 k1 minus 1( 11138573

⎛⎝

minus 2


2+ 41113872 1113873

2minus 8β41 k1 minus 1( 1113857

3 β21 k1 minus 1( 1113857k1k2 + 21113872 1113873

β61 k1 minus 1( 11138574

11139741113972

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

α1 minus2

β1 k1 minus 1( 1113857

(19)



a4a1




U(ξ) minus2

β1 k1 minus 1( 1113857

λ2 minus 4μ1113969

2

A1cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2

V(ξ) β1

λ2 minus 4μ1113969

2

A1cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2

(20)


α1 + β1 0

α1 + β1 h2 minus k1k2( 1113857 0


β1 2h4 + α1β1 k1 minus 1( 1113857( 1113857 0

β1h1

2 0

(21)




k1k2 + 1

1113968(x minus ct)( 1113857

2

radic

β21 k1 minus 1( 1113857 k1k2 + 1( 1113857

1113969

tan h (12)k1k2 + 1


V(x t) minus U(x t)

(22)


4 Discussion











5 Conclusion


02 04 06 08k1

2

4

6

8λ2

ndash4μ

(a)

k1

λ2ndash

4μ

20 25 30

20

30

40

50

(b)



radic gt a2)

ndash420ndash2

2

0

ndash2

4

ndash420ndash2

4

x

t

U (x t)V (x t)


21) + 1 and

k2 (4β41 + 7β21 + 16)(9β21 + 18) where β1 05

ndash4

ndash4

2

2

0

0

ndash2

ndash2

4

4

x

10

05

00

ndash05

ndash10

t

U (x t)V (x t)




Data Availability






Acknowledgments


References





























of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























21 (13a)


21 (13b)



21


(14)


u1(ξ) α1F(ξ)

v1(ξ) β1F(ξ)(15)




+ α1β21 k1 minus 1( 1113857F

3(x) 0

(16)




β1 α1β1 k1 minus 1( 1113857 + 2( 1113857 0


(17)


(i) Case I

c β41 k1 minus 1( 1113857

2k1 + 2β21 k1 minus 1( 1113857

2+ 4


μ β21 k1 minus 1( 1113857k1k2 + 2

2β21 k1 minus 1( 1113857

λ β41 k1 minus 1( 1113857

2k1 + 2β21 k1 minus 1( 1113857

2+ 4

2β31(k1 minus 1)2

α1 minus2

β1 k1 minus 1( 1113857

(18)

(ii) Case II

μ 0

c 1


4β21 k1 minus 1( 1113857

2 +3k1


3β21 k1 minus 1( 1113857

3⎛⎝

minus34β1


2+ 41113872 1113873

2minus 8β41 k1 minus 1( 1113857

3 β21 k1 minus 1( 1113857k1k2 + 21113872 1113873

β61 k1 minus 1( 11138574

11139741113972

+3k1

2 k1 minus 1( 1113857+

32 minus 2k1

minus 2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

λ 18

2β1k1 +4k1

β1 k1 minus 1( 1113857+

8k1


4β1 k1 minus 1( 1113857

minus8

β31 k1 minus 1( 11138573

⎛⎝

minus 2


2+ 41113872 1113873

2minus 8β41 k1 minus 1( 1113857

3 β21 k1 minus 1( 1113857k1k2 + 21113872 1113873

β61 k1 minus 1( 11138574

11139741113972

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

α1 minus2

β1 k1 minus 1( 1113857

(19)



a4a1




U(ξ) minus2

β1 k1 minus 1( 1113857

λ2 minus 4μ1113969

2

A1cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2

V(ξ) β1

λ2 minus 4μ1113969

2

A1cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2

(20)


α1 + β1 0

α1 + β1 h2 minus k1k2( 1113857 0


β1 2h4 + α1β1 k1 minus 1( 1113857( 1113857 0

β1h1

2 0

(21)




k1k2 + 1

1113968(x minus ct)( 1113857

2

radic

β21 k1 minus 1( 1113857 k1k2 + 1( 1113857

1113969

tan h (12)k1k2 + 1


V(x t) minus U(x t)

(22)


4 Discussion











5 Conclusion


02 04 06 08k1

2

4

6

8λ2

ndash4μ

(a)

k1

λ2ndash

4μ

20 25 30

20

30

40

50

(b)



radic gt a2)

ndash420ndash2

2

0

ndash2

4

ndash420ndash2

4

x

t

U (x t)V (x t)


21) + 1 and

k2 (4β41 + 7β21 + 16)(9β21 + 18) where β1 05

ndash4

ndash4

2

2

0

0

ndash2

ndash2

4

4

x

10

05

00

ndash05

ndash10

t

U (x t)V (x t)




Data Availability






Acknowledgments


References





























of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















a4a1




U(ξ) minus2

β1 k1 minus 1( 1113857

λ2 minus 4μ1113969

2

A1cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2

V(ξ) β1

λ2 minus 4μ1113969

2

A1cos h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2sin h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

A1sin h (12)

λ2 minus 4μ1113969

ξ1113874 1113875 + A2cos h (12)

λ2 minus 4μ1113969

ξ1113874 11138751113874 1113875

minusλ2

(20)


α1 + β1 0

α1 + β1 h2 minus k1k2( 1113857 0


β1 2h4 + α1β1 k1 minus 1( 1113857( 1113857 0

β1h1

2 0

(21)




k1k2 + 1

1113968(x minus ct)( 1113857

2

radic

β21 k1 minus 1( 1113857 k1k2 + 1( 1113857

1113969

tan h (12)k1k2 + 1


V(x t) minus U(x t)

(22)


4 Discussion











5 Conclusion


02 04 06 08k1

2

4

6

8λ2

ndash4μ

(a)

k1

λ2ndash

4μ

20 25 30

20

30

40

50

(b)



radic gt a2)

ndash420ndash2

2

0

ndash2

4

ndash420ndash2

4

x

t

U (x t)V (x t)


21) + 1 and

k2 (4β41 + 7β21 + 16)(9β21 + 18) where β1 05

ndash4

ndash4

2

2

0

0

ndash2

ndash2

4

4

x

10

05

00

ndash05

ndash10

t

U (x t)V (x t)




Data Availability






Acknowledgments


References





























of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























5 Conclusion


02 04 06 08k1

2

4

6

8λ2

ndash4μ

(a)

k1

λ2ndash

4μ

20 25 30

20

30

40

50

(b)



radic gt a2)

ndash420ndash2

2

0

ndash2

4

ndash420ndash2

4

x

t

U (x t)V (x t)


21) + 1 and

k2 (4β41 + 7β21 + 16)(9β21 + 18) where β1 05

ndash4

ndash4

2

2

0

0

ndash2

ndash2

4

4

x

10

05

00

ndash05

ndash10

t

U (x t)V (x t)




Data Availability






Acknowledgments


References





























of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















Data Availability






Acknowledgments


References





























of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















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