Research Article The Dynamics of a Delayed Predator-Prey ...downloads.hindawi.com/journals/ddns/2015/102597.pdf · Research Article The Dynamics of a Delayed Predator-Prey Model with

Research ArticleThe Dynamics of a Delayed Predator-Prey Model withDouble Allee Effect

Boli Xie12 Zhijun Wang1 Yakui Xue2 and Zhenmin Zhang2

1School of Mechatronic Engineering North University of China Taiyuan Shanxi 030051 China2Department of Mathematics North University of China Taiyuan Shanxi 030051 China

Correspondence should be addressed to Boli Xie bolixie163com

Received 30 June 2015 Accepted 27 July 2015

Academic Editor Carlo Bianca

Copyright copy 2015 Boli Xie et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study the dynamics of a delayed predator-prey model with double Allee effect For the temporal model we showed that thereexists a threshold of time delay in predator-prey interactions when time delay is below the threshold value the positive equilibriumElowast is stable However when time delay is above the threshold value the positive equilibrium Elowast is unstable and period solution willemerge For the spatiotemporal model through numerical simulations we show that the model dynamics exhibit rich parameterspace Turing structures The obtained results show that this system has rich dynamics these patterns show that it is useful for adelayed predator-prey model with double Allee effect to reveal the spatial dynamics in the real model

1 Introduction

The Allee effect named after the ecologist Warder ClydeAllee is a phenomenon in biology characterized by a corre-lation between population size or density and the mean indi-vidual fitness of a population or species [1] Allee effect canoccur whenever fitness of an individual in a small or sparsepopulation decreases as the population size or density alsodeclines [2 3] Allee effect contains two main types strongAllee effect and weak Allee effect A population exhibitinga weak Allee effect will possess a reduced per capita growthrate (directly related to individual fitness of the population)at lower population density or size However even at this lowpopulation size or density the population will always exhibita positive per capita growth rate Meanwhile a populationexhibiting a strong Allee effect will have a critical populationsize or density under which the population growth ratebecomes negativeTherefore when the population density orsize hits a number below this threshold the population willbe doomed

There have been a large group of papers on predator-prey systems with Allee effect [4ndash12] The most usual simple

mathematical example of an Allee effect is given by theequation

119889119873

119889119879= 119903119873(1minus 119873

119870) (119873minus119898) (1)

where119873 denotes the population density 119903 is the intrinsic rateof increase 119870 is the carrying capacity and 119898 is threshold ofthe Allee effectThe population has a negative growth rate for119873 lt 119898 and a positive growth rate for 119873 gt 119898 If 119898 gt 0 (1)is a strong Allee effect if 119898 le 0 (1) is a weak Allee effectHowever two mechanisms of Allee effects acting in the samepopulation interact to produce an overall demographic Alleeeffect in a prey-predator interaction model which can alsobe common and can be complex [13] and their combinedinfluence is termed as double

There are also someworks done on predator-prey systemswith double Allee effect [14ndash16] Gonzalez-Olivares et alfound that the Gause-type predator-prey model with Alleeeffect can induce two limit cycles when the Allee effect iseither strong or weak [14] Huincahue-Arcos and Gonzalez-Olivares found that the Rosenzweig-MacArthur predation

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 102597 8 pageshttpdxdoiorg1011552015102597

2 Discrete Dynamics in Nature and Society

model with double Allee effects may be expressed by differentmathematical formalizations with the form used here theexistence of one limit cycle surrounding a positive equilib-rium point is proved [15] Pal and Saha found that the ratiodependent prey-predator system with a double Allee effectexhibits the bistability and there exists separatrix curve(s) inthe phase plane implying that dynamics of the system arevery sensitive to the variation of the initial conditions [16]However these previous works did not take into account theeffect of space

Time delay plays an important role in many biologicaldynamical systems where time delays have been recognizedto contribute critically to the outcome for prey densitiesunder predation being stable or unstable [17] Time delaydue to gestation is included in some predator-prey modelsbecause generally a duration of 120591 time units elapses betweenthe time when an individual prey is killed and the momentwhen a corresponding increase in the predator populationis realized [18] Furthermore time delays can be used tointroduce oscillations [19 20]

In the present study our objective is to investigate apredator-prey model with double Allee effect and time delayMore specifically the primary objective of the present studyis to investigate the spatial patterns

2 Analysis of Temporal Model

In this section we consider a predator-prey model where theprey population growth is affected by double Allee effectswith time delay The following predator-prey model withdouble Allee effect has been proposed and studied [16]

119889119909

119889=

119903119909

119909 + 119899(1minus 119909

119870) (119909 minus119898) minus

119888119909119910

119909 + 120599119910

119889119910

119889=

1198881119909119910

119909 + 120599119910minus119889119910

(2)

where 119909 and 119910 stand for prey and predator density 119903 is theintrinsic rate of increase 119870 is the carrying capacity 119898 isthreshold of the Allee effect 119888 stands for capturing rate of thepredator 120599 stands for half capturing saturation constant 1198881stands for conversion rate of prey into predators biomass and119889 stands for natural death rate of predator

Following [16] through a nondimensional transforma-tion

119906 =1119870119909

V =120599

119870119910

119905 = 119903

(3)

we arrive at the following equations

119889119906

119889119905=119906 (1 minus 119906) (119906 minus 120574)

119906 + 120579minus

120572119906V119906 + V

119889V119889119905

=120573119906V119906 + V

minus 120575V

(4)

where

120579 =119899

119870

120574 =119898

119870

120572 =119888

119903120599

120573 =1198881119903

120575 =119889

119903

(5)

In the section our objective is to investigate the predator-prey model with double Allee effect and time delay Themodel is given by

119889119906

120597119905=119906 (1 minus 119906) (119906 minus 120574)

119906 + 120579minus

120572119906V (119905 minus 120591)

119906 + V (119905 minus 120591)

119889V120597119905

=120573119906 (119905 minus 120591) V119906 (119905 minus 120591) + V

minus 120575V

(6)

where 120591 gt 0 is a constant delay due to gestationWe analyze model (6) under the initial conditions

119906 (0) gt 0

V (0) gt 0(7)

Next we will discuss the dynamics of model (6) Wedetermined that model (6) andmodel (4) have two boundaryequilibria named 1198640 = (1 0) and 1198641 = (120574 0) and a uniquepositive equilibrium named 119864lowast = (119906

lowast

Vlowast) where

119906lowast

=120573 minus 120572120573 + 120575120572 + 120574120573 + radic(120573 minus 120572120573 + 120575120572)

2+ 4120573120579120572120575

2120573

Vlowast =(120573 minus 120575) 119906

lowast

120575

(8)

We aim to look for the conditions so that (119906lowast Vlowast) is stablefor the temporal model and is unstable for the spatiotemporalmodel We always assume that (119906lowast Vlowast) is linearly stable withrespect to the perturbation of 119906 and V thus the eigenvaluesof the Jacobian

Discrete Dynamics in Nature and Society 3

119869 = (

11988611 11988612

11988621 11988622

) = (

minus3 (119906lowast)2 + 2119906lowast120574 + 2119906lowast minus 120574

119906lowast + 120579minus119906lowast

(1 minus 119906lowast

) (119906lowast

minus 120574)

(119906lowast + 120579)2 minus

120572 (120573 minus 120575)2

1205732 minus120572120575

2

1205732

(120573 minus 120575)2

120573minus(120573 minus 120575) 120575

120573

) (9)

at (119906lowast Vlowast) must have negative real parts which is equivalentto

tr (119869) = 11988611 + 11988622 lt 0

det (119869) = 1198861111988622 minus 1198861211988621 gt 0(10)

Next we consider small spatiotemporal perturbations ℎand 119901 on a homogeneous steady state 119864lowast(119906lowast Vlowast) Let ℎ = 119906 minus

119906lowast and 119901 = V minus Vlowast then we derive that

119889ℎ

120597119905= 11988611ℎ + 11988612119901 (119905 minus 120591)

119889119901

120597119905= 11988621ℎ (119905 minus 120591) + 11988622119901

(11)

Spatiotemporal perturbations ℎ and 119901 are given by

ℎ = ℎlowast

119890120582119905 cos (119896

119909119909)

119901 = 119901lowast

119890120582119905 cos (119896

119909119909)

(12)

By substitution of this form in (11) we get the followingmatrix equation about eigenvalues

(120582 minus 11988611 minus11988612119890

minus120582120591

minus11988621119890minus120582120591

120582 minus 11988622)(

ℎlowast

119901lowast) = (

00) (13)

Linear system (11) is characterized by the equation

1205822minus (11988611 + 11988622) 120582 + 1198861111988622 minus 1198861211988621119890

minus2120582120591= 0 (14)

If 120582 = 119894120596 is a root of (14) then we have

minus1205962+ 1198861111988622 = 1198861211988621cos (2120596120591)

(11988611 + 11988622) 120596 = 1198861211988621sin (2120596120591) (15)

which leads to

1205964+ (119886

211 + 119886

222) 120596

2+ 119886

211119886

222 minus 119886

212119886

221 = 0 (16)

Then (16) has the solution

1205962119888

=minus (119886

211 + 119886

222) + radic(119886211 + 119886222)

2minus 4 (119886211119886222 minus 119886212119886

221)

2

(17)

From (15) we can obtain

120591119888=

12120596119888

arccosminus120596

2119888+ 1198861111988622

1198861211988621 (18)

Now we investigate the sign of (119889Re(120582)119889120591)|120591=120591119888

Let 120582 =

120590 + 119894120596 be a solution of (14) then

Re (120582) = 1205902minus120596

2minus (11988611 + 11988622) 120590 + 1198861111988622

minus 1198861211988621119890minus2120590120591 cos (2120596120591) = 0

Im (120582) = 2120590120596minus (11988611 + 11988622) 120596 + 1198861211988621119890minus2120590120591 sin (2120596120591)

= 0

(19)

By derivation of 120591 in both sides of (19) notice that120582 = 120590+119894120596 =

119894120596 we can get

119860119889120590

119889120591minus119861

119889120596

119889120591+119862 = 0

119860119889120590

119889120591+119861

119889120596

119889120591+119863 = 0

(20)

where

119860 = minus (11988611 + 11988622) + 21205911198861211988621 cos (2120596120591)

119861 = 2120596minus 21205911198861211988621 sin (2120596120591)

119862 = 21205961198861211988621 sin (2120596120591)

119863 = 21205961198861211988621 cos (2120596120591)

(21)

Thus we can get

119889Re (120582)119889120591

10038161003816100381610038161003816100381610038161003816120591=120591119888

=119889120590

119889120591

10038161003816100381610038161003816100381610038161003816120591=120591119888

= minus119860119862 + 119861119863

1198602 + 1198612 (22)

Binding (15) and (16) we obtain

minus (119860119862+119861119863) = 21205962(119886

211 + 119886

222 + 21205962

) gt 0 (23)

which implies that (119889Re(120582)119889120591)|120591=120591119888

gt 0 Furthermore weget the following conclusions If the delay 120591 is satisfied 120591 = 0then system (6) exhibits a Hopf bifurcation critical 120591 = 120591

119888

When 120591 lt 120591119888 the positive equilibrium 119864

lowast is stable but when120591 gt 120591

119888 the positive equilibrium 119864

lowast is unstable and periodsolution will emerge

We take the following values 120572 = 06 120573 = 08 120574 = minus03120575 = 058 and 120579 = 22 Through calculations we obtain thecritical value 120591

119888= 07397 then 119864

lowast

= (036 01365517241)The initial value is (03 01)

We adopt 120591 = 02 lt 120591119888 From Figure 1 we can see that the

positive equilibrium 119864lowast is stable

We adopt 120591 = 081 gt 120591119888 From Figure 2 we can see that

the positive equilibrium 119864lowast is unstable


0 200 400 600 800 1000The time t

Tim

e ser

ies o

f the

indu

ced

prey

and

pred

ator

005

01

015

02

025

03

035

04

u

v

(a)

025 026 027 028 029 03 031 032 033 034u

0095

01

0105

011

0115

012

(b)

Figure 1 Behavior and phase portrait of system (6) Parameter values are used as 120572 = 06 120573 = 08 120574 = minus03 120575 = 058 120579 = 22 and120591 = 02 lt 120591

119888

0 200 400 600 800 1000The time t

Tim

e ser

ies o

f the

indu

ced

prey

and

pred

ator

005

01

015

02

025

03

035

04

u

v

(a)

02 022 024 026 028 03 032 034u

v

0085

009

0095

01

0105

011

0115

012

0125

(b)

Figure 2 Behavior and phase portrait of system (6) Parameter values are used as 120572 = 06 120573 = 08 120574 = minus03 120575 = 058 120579 = 22 and120591 = 081 gt 120591

119888

3 Analysis of Spatiotemporal Model

In the section our objective is to consider the spatiotemporalsystem with double Allee effect and time delay

120597119906

120597119905=119906 (1 minus 119906) (119906 minus 120574)

119906 + 120579minus

120572119906V (119905 minus 120591)

119906 + V (119905 minus 120591)+ 1198891nabla

2119906

120597V120597119905

=120573119906 (119905 minus 120591) V119906 (119905 minus 120591) + V

minus 120575V+1198892nabla2V

(24)

Similar to the analysis of (6) we consider small spatiotem-poral perturbations ℎ = 119906 minus 119906

lowast and 119901 = V minus Vlowast on ahomogeneous steady state 119864lowast(119906lowast Vlowast) The linearized systemtakes the form

120597ℎ

120597119905= 11988611ℎ + 11988612119901 (119905 minus 120591) + 1198891nabla

2ℎ

120597119901

120597119905= 11988621ℎ (119905 minus 120591) + 11988622119901+1198892nabla

2119901

(25)


010

005

0

minus005

minus010

Re(120582)

05 1 15

k(a)

(b)

(c)

Figure 3 An illustration of model (24) We set the parameter values as (a) 120572 = 06 120573 = 08 120574 = minus03 120575 = 058 120579 = 22 1198891 = 008 1198892 = 1and 120591 = 02 (b) 120572 = 06 120573 = 08 120574 = minus03 120575 = 058 120579 = 22 1198891 = 008 1198892 = 1 and 120591 = 002 (c) 120572 = 06 120573 = 15 120574 = minus03 120575 = 11 120579 = 221198891 = 005 1198892 = 1 and 120591 = 002

01385

0138

01375

0137

01365

0136

01355

0135

(a)

0058

0056

0054

0052

005

0048

0046

0044

(b)

012

011

01

009

008

007

006

005

004

003

002

(c)

011

01

009

008

007

006

005

004

003

(d)

Figure 4 Snapshots of the time evolution of the prey at different instants with 120572 = 06 120573 = 08 120574 = minus03 120575 = 058 120579 = 22 1198891 = 008 1198892 = 1and 120591 = 02 which are in the Turing space (a) 0 iterations (b) 5000 iterations (c) 10000 iterations (d) 100000 iterations


01385

0138

01375

0137

01365

0136

01355

0135

(a)

0104

0102

01

0098

0096

0094

0092

009

0088

0086

0084

(b)

011

01

009

008

007

006

005

(c)

011

01

009

008

007

006

005

004

(d)


By substitution of form (12) in (25) we get the followingmatrix equation about eigenvalues

(120582 minus 11988611 + 1198891119896

2minus11988612119890minus120582120591

minus11988621119890minus120582120591

120582 minus 11988622 + 11988921198962)(

ℎlowast

119901lowast) = (

00) (26)

The characteristic equation for the linear system (25) isgiven by

1205822minus [(11988611 minus1198891119896

2) + (11988622 minus1198892119896

2)] 120582

+ (11988611 minus11988911198962) (11988622 minus1198892119896

2) minus 1198861211988621119890

minus2120582120591= 0

(27)

Spatial patterns form if (27) has root 120582 = 119894120596 whichare called delay-driven spatial patterns Moreover the criticalvalue of the delay 120591 is called the Turing bifurcation If 119894120596 is aroot of (27) then we have

minus1205962+ (11988611 minus1198891119896

2) (11988622 minus1198892119896

2) = 1198861211988621cos (2120596120591)

[(11988611 minus11988911198962) + (11988622 minus1198892119896

2)] 120596 = 1198861211988621sin (2120596120591)

(28)

which leads to

1205964+1205721198961205962+120573119896= 0 (29)

where

120572119896= (11988611 minus1198891119896

2)2+ (11988622 minus1198892119896

2)2

120573119896= (11988611 minus1198891119896

2)2(11988622 minus1198892119896

2)2minus 119886

212119886

221

(30)


1205962sc =

minus120572119896+ radic1205722119896minus 4120573119896

2

(31)


120591sc =1

2120596scarccos

minus1205962sc + (11988611 minus 1198891119896

2) (11988622 minus 1198892119896

2)

1198861211988621 (32)


01525

0152

01515

01505

015

01495

0149

0151

(a)

0109

01089

01089

01089

01089

01089

01088

01088

(b)

01094

01092

0109

01088

01086

01084

01082

(c)

011

01

009

008

007

006

005

004

(d)


Using mathematical calculations a Turing bifurcation isproduced when the following conditions are met

Im (120582) = 0

Re (120582) = 0

at 119896 = 119896119879

= 0(33)

By setting 120573119896min

= 119889111988921198964min minus (119886111198892 + 119886221198891)119896

2min + 1198861111988622 minus

1198861211988621 = 0 we can obtain the critical value of the Turingbifurcation parameter 120579

119879 which is equal to

120579119879=141205732minus 21205721205732

+ 2120573120575120572 minus 21205741205732+ 120572

21205732minus 21205722

120573120575 minus 21205721205732120574 + 120575

21205722+ 2120575120572120574120573 + 120574

21205732

120572120573 (120573 minus 120575) (34)

To see well the effect of cross diffusion and time delaywe plot the dispersion relation keeping the parameter valuesfixed in Figure 3 It can be seen from Figure 3 that Turingmodes Re(120582) gt 0 can be available

4 Pattern Structures

In the following we will perform a series of numericalsimulations on the two-dimensional model (24) using zeroboundary conditions and119872times119873 discrete lattice


For model (24) space and time were approximated usingthe finite difference method and Eulerrsquos method taking thetime step as Δ119905 = 001 the space step as Δℎ = 1 and 119872 =

119873 = 200 The results indicated that Δℎ and Δ119905 are reducedand do not lead to considerable changes in the results

In Figure 4 we set 120572 = 06 120573 = 08 120574 = minus03 120575 = 058120579 = 22 1198891 = 008 1198892 = 1 and 120591 = 02 In this case theinfected populations exhibit stationary labyrinthine patterns

In Figure 5 we set 120572 = 06 120573 = 08 120574 = minus03 120575 = 058120579 = 22 1198891 = 008 1198892 = 1 and 120591 = 002We can see that shortstripe-like pattern and spotted patterns emerge coexist

In Figure 6 we set 120572 = 06 120573 = 15 120574 = minus03 120575 = 11120579 = 22 1198891 = 005 1198892 = 1 and 120591 = 002 As time passesregular spotted patterns appear in space and the dynamics ofthe system do not undergo any further changes

5 Discussions

In this paper the dynamics of a delayed predator-prey modelwith double Allee effect were considered First we discussthe temporal model (6) we showed that there exists a Hopfbifurcation threshold 120591

119888of time delay when 120591 lt 120591

119888 the

positive equilibrium 119864lowast of system (6) is stable However

when 120591 gt 120591119888 the positive equilibrium 119864

lowast of system (6)is unstable and period solution will emerge Second wediscuss the spatiotemporal model (24) the spatial patternsvia numerical simulations are illustrated which show thatthe model dynamics exhibit rich parameter space Turingstructures

Althoughmore work is needed in principle it seems thatdelay and diffusion are able to generate many different kindsof spatiotemporal patterns For such reasons we can predictthat delay and diffusion can be considered as an importantmechanism for the appearance of complex spatiotemporaldynamics in other models such as predator-prey model andmutualistic model

Conflict of Interests

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

Acknowledgments

This work is supported by the National Sciences Foundationof China (10471040) and the National Sciences Foundation ofShanxi Province (2009011005-1)

References

[1] F Courchamp J Berec and J GascoigneAllee Effects in Ecologyand Conservation Oxford University Press Oxford UK 2008

[2] F Courchamp T Clutton-Brock and B Grenfell ldquoInversedensity dependence and the Allee effectrdquo Trends in Ecology ampEvolution vol 14 no 10 pp 405ndash410 1999

[3] P A Stephens and W J Sutherland ldquoConsequences of theAllee effect for behaviour ecology and conservationrdquo Trends inEcology and Evolution vol 14 no 10 pp 401ndash405 1999

[4] S V Petrovskii A Y Morozov and E Venturino ldquoAllee effectmakes possible patchy invasion in a predator-prey systemrdquoEcology Letters vol 5 no 3 pp 345ndash352 2002

[5] A Morozov S Petrovskii and B-L Li ldquoBifurcations and chaosin a predator-prey system with the Allee effectrdquo Proceedings ofthe Royal Society B Biological Sciences vol 271 no 1546 pp1407ndash1414 2004

[6] A Morozov S Petrovskii and B-L Li ldquoSpatiotemporal com-plexity of patchy invasion in a predator-prey system with theAllee effectrdquo Journal of Theoretical Biology vol 238 no 1 pp18ndash35 2006

[7] D Hadjiavgousti and S Ichtiaroglou ldquoAllee effect in a prey-predator systemrdquo Chaos Solitons amp Fractals vol 36 no 2 pp334ndash342 2008

[8] CCelik andODuman ldquoAllee effect in a discrete-timepredator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[9] A Verdy ldquoModulation of predator-prey interactions by theAllee effectrdquo Ecological Modelling vol 221 no 8 pp 1098ndash11072010

[10] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[11] L Cai G Chen and D Xiao ldquoMultiparametric bifurcations ofan epidemiological model with strong Allee effectrdquo Journal ofMathematical Biology vol 67 no 2 pp 185ndash215 2013

[12] G-Q Sun L Li Z Jin Z-K Zhang and T Zhou ldquoPatterndynamics in a spatial predator-prey system with Allee effectrdquoAbstract and Applied Analysis vol 2013 Article ID 921879 12pages 2013

[13] L Berec E Angulo and F Courchamp ldquoMultiple Allee effectsand population managementrdquo Trends in Ecology amp Evolutionvol 22 no 4 pp 185ndash191 2007

[14] E Gonzalez-Olivares B Gonzalez-Yaez J Mena Lorca ARojas-Palma and J D Flores ldquoConsequences of double Alleeeffect on the number of limit cycles in a predator-prey modelrdquoComputers amp Mathematics with Applications vol 62 no 9 pp3449ndash3463 2011

[15] J Huincahue-Arcos and E Gonzalez-Olivares ldquoThe Rosenz-weig-MacArthur predation model with double Allee effects onpreyrdquo in Proceedings of the International Conference on AppliedMathematics and Computational Methods in Engineering pp206ndash211 2013

[16] P J Pal and T Saha ldquoQualitative analysis of a predator-preysystem with double Allee effect in preyrdquo Chaos Solitons ampFractals vol 73 pp 36ndash63 2015

[17] A F Nindjin M A Aziz-Alaoui and M Cadivel ldquoAnalysis of apredator-prey model with modified Leslie-Gower and Holling-type II schemes with time delayrdquoNonlinear Analysis RealWorldApplications vol 7 no 5 pp 1104ndash1118 2006

[18] R Xu and L S Chen ldquoPersistence and stability for a two-species ratio-dependent predator-prey system with time delayin a two-patch environmentrdquo Computers amp Mathematics withApplications vol 40 no 4-5 pp 577ndash588 2000

[19] C Bianca and LGuerrini ldquoOn theDalgaard-Strulikmodel withlogistic population growth rate and delayed-carrying capacityrdquoActa Applicandae Mathematicae vol 128 pp 39ndash48 2013

[20] C Bianca andLGuerrini ldquoExistence of limit cycles in the Solowmodel with delayed-logistic population growthrdquo The ScientificWorld Journal vol 2014 Article ID 207806 8 pages 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


model with double Allee effects may be expressed by differentmathematical formalizations with the form used here theexistence of one limit cycle surrounding a positive equilib-rium point is proved [15] Pal and Saha found that the ratiodependent prey-predator system with a double Allee effectexhibits the bistability and there exists separatrix curve(s) inthe phase plane implying that dynamics of the system arevery sensitive to the variation of the initial conditions [16]However these previous works did not take into account theeffect of space

Time delay plays an important role in many biologicaldynamical systems where time delays have been recognizedto contribute critically to the outcome for prey densitiesunder predation being stable or unstable [17] Time delaydue to gestation is included in some predator-prey modelsbecause generally a duration of 120591 time units elapses betweenthe time when an individual prey is killed and the momentwhen a corresponding increase in the predator populationis realized [18] Furthermore time delays can be used tointroduce oscillations [19 20]

In the present study our objective is to investigate apredator-prey model with double Allee effect and time delayMore specifically the primary objective of the present studyis to investigate the spatial patterns

2 Analysis of Temporal Model

In this section we consider a predator-prey model where theprey population growth is affected by double Allee effectswith time delay The following predator-prey model withdouble Allee effect has been proposed and studied [16]

119889119909

119889=

119903119909

119909 + 119899(1minus 119909

119870) (119909 minus119898) minus

119888119909119910

119909 + 120599119910

119889119910

119889=

1198881119909119910

119909 + 120599119910minus119889119910

(2)

where 119909 and 119910 stand for prey and predator density 119903 is theintrinsic rate of increase 119870 is the carrying capacity 119898 isthreshold of the Allee effect 119888 stands for capturing rate of thepredator 120599 stands for half capturing saturation constant 1198881stands for conversion rate of prey into predators biomass and119889 stands for natural death rate of predator

Following [16] through a nondimensional transforma-tion

119906 =1119870119909

V =120599

119870119910

119905 = 119903

(3)

we arrive at the following equations

119889119906

119889119905=119906 (1 minus 119906) (119906 minus 120574)

119906 + 120579minus

120572119906V119906 + V

119889V119889119905

=120573119906V119906 + V

minus 120575V

(4)

where

120579 =119899

119870

120574 =119898

119870

120572 =119888

119903120599

120573 =1198881119903

120575 =119889

119903

(5)

In the section our objective is to investigate the predator-prey model with double Allee effect and time delay Themodel is given by

119889119906

120597119905=119906 (1 minus 119906) (119906 minus 120574)

119906 + 120579minus

120572119906V (119905 minus 120591)

119906 + V (119905 minus 120591)

119889V120597119905

=120573119906 (119905 minus 120591) V119906 (119905 minus 120591) + V

minus 120575V

(6)

where 120591 gt 0 is a constant delay due to gestationWe analyze model (6) under the initial conditions

119906 (0) gt 0

V (0) gt 0(7)

Next we will discuss the dynamics of model (6) Wedetermined that model (6) andmodel (4) have two boundaryequilibria named 1198640 = (1 0) and 1198641 = (120574 0) and a uniquepositive equilibrium named 119864lowast = (119906

lowast

Vlowast) where

119906lowast

=120573 minus 120572120573 + 120575120572 + 120574120573 + radic(120573 minus 120572120573 + 120575120572)

2+ 4120573120579120572120575

2120573

Vlowast =(120573 minus 120575) 119906

lowast

120575

(8)

We aim to look for the conditions so that (119906lowast Vlowast) is stablefor the temporal model and is unstable for the spatiotemporalmodel We always assume that (119906lowast Vlowast) is linearly stable withrespect to the perturbation of 119906 and V thus the eigenvaluesof the Jacobian


119869 = (

11988611 11988612

11988621 11988622

) = (




) (119906lowast

minus 120574)

(119906lowast + 120579)2 minus

120572 (120573 minus 120575)2

1205732 minus120572120575

2

1205732

(120573 minus 120575)2

120573minus(120573 minus 120575) 120575

120573

) (9)


tr (119869) = 11988611 + 11988622 lt 0

det (119869) = 1198861111988622 minus 1198861211988621 gt 0(10)



119889ℎ

120597119905= 11988611ℎ + 11988612119901 (119905 minus 120591)

119889119901

120597119905= 11988621ℎ (119905 minus 120591) + 11988622119901

(11)


ℎ = ℎlowast

119890120582119905 cos (119896

119909119909)

119901 = 119901lowast

119890120582119905 cos (119896

119909119909)

(12)


(120582 minus 11988611 minus11988612119890

minus120582120591

minus11988621119890minus120582120591

120582 minus 11988622)(

ℎlowast

119901lowast) = (

00) (13)


1205822minus (11988611 + 11988622) 120582 + 1198861111988622 minus 1198861211988621119890

minus2120582120591= 0 (14)


minus1205962+ 1198861111988622 = 1198861211988621cos (2120596120591)

(11988611 + 11988622) 120596 = 1198861211988621sin (2120596120591) (15)

which leads to

1205964+ (119886

211 + 119886

222) 120596

2+ 119886

211119886

222 minus 119886

212119886

221 = 0 (16)


1205962119888

=minus (119886

211 + 119886

222) + radic(119886211 + 119886222)

2minus 4 (119886211119886222 minus 119886212119886

221)

2

(17)


120591119888=

12120596119888

arccosminus120596

2119888+ 1198861111988622

1198861211988621 (18)


Let 120582 =


Re (120582) = 1205902minus120596

2minus (11988611 + 11988622) 120590 + 1198861111988622

minus 1198861211988621119890minus2120590120591 cos (2120596120591) = 0


= 0

(19)


119894120596 we can get

119860119889120590

119889120591minus119861

119889120596

119889120591+119862 = 0

119860119889120590

119889120591+119861

119889120596

119889120591+119863 = 0

(20)

where

119860 = minus (11988611 + 11988622) + 21205911198861211988621 cos (2120596120591)

119861 = 2120596minus 21205911198861211988621 sin (2120596120591)

119862 = 21205961198861211988621 sin (2120596120591)

119863 = 21205961198861211988621 cos (2120596120591)

(21)

Thus we can get

119889Re (120582)119889120591

10038161003816100381610038161003816100381610038161003816120591=120591119888

=119889120590

119889120591

10038161003816100381610038161003816100381610038161003816120591=120591119888

= minus119860119862 + 119861119863

1198602 + 1198612 (22)


minus (119860119862+119861119863) = 21205962(119886

211 + 119886

222 + 21205962

) gt 0 (23)



119888






119888= 07397 then 119864

lowast







0 200 400 600 800 1000The time t

Tim

e ser

ies o

f the

indu

ced

prey

and

pred

ator

005

01

015

02

025

03

035

04

u

v

(a)

025 026 027 028 029 03 031 032 033 034u

0095

01

0105

011

0115

012

(b)


119888

0 200 400 600 800 1000The time t

Tim

e ser

ies o

f the

indu

ced

prey

and

pred

ator

005

01

015

02

025

03

035

04

u

v

(a)

02 022 024 026 028 03 032 034u

v

0085

009

0095

01

0105

011

0115

012

0125

(b)


119888



120597119906

120597119905=119906 (1 minus 119906) (119906 minus 120574)

119906 + 120579minus

120572119906V (119905 minus 120591)

119906 + V (119905 minus 120591)+ 1198891nabla

2119906

120597V120597119905

=120573119906 (119905 minus 120591) V119906 (119905 minus 120591) + V


(24)



120597ℎ

120597119905= 11988611ℎ + 11988612119901 (119905 minus 120591) + 1198891nabla

2ℎ

120597119901

120597119905= 11988621ℎ (119905 minus 120591) + 11988622119901+1198892nabla

2119901

(25)


010

005

0

minus005

minus010

Re(120582)

05 1 15

k(a)

(b)

(c)


01385

0138

01375

0137

01365

0136

01355

0135

(a)

0058

0056

0054

0052

005

0048

0046

0044

(b)

012

011

01

009

008

007

006

005

004

003

002

(c)

011

01

009

008

007

006

005

004

003

(d)



01385

0138

01375

0137

01365

0136

01355

0135

(a)

0104

0102

01

0098

0096

0094

0092

009

0088

0086

0084

(b)

011

01

009

008

007

006

005

(c)

011

01

009

008

007

006

005

004

(d)



(120582 minus 11988611 + 1198891119896

2minus11988612119890minus120582120591

minus11988621119890minus120582120591

120582 minus 11988622 + 11988921198962)(

ℎlowast

119901lowast) = (

00) (26)


1205822minus [(11988611 minus1198891119896

2) + (11988622 minus1198892119896

2)] 120582

+ (11988611 minus11988911198962) (11988622 minus1198892119896

2) minus 1198861211988621119890

minus2120582120591= 0

(27)


minus1205962+ (11988611 minus1198891119896

2) (11988622 minus1198892119896

2) = 1198861211988621cos (2120596120591)

[(11988611 minus11988911198962) + (11988622 minus1198892119896

2)] 120596 = 1198861211988621sin (2120596120591)

(28)

which leads to

1205964+1205721198961205962+120573119896= 0 (29)

where

120572119896= (11988611 minus1198891119896

2)2+ (11988622 minus1198892119896

2)2

120573119896= (11988611 minus1198891119896

2)2(11988622 minus1198892119896

2)2minus 119886

212119886

221

(30)


1205962sc =


2

(31)


120591sc =1

2120596scarccos

minus1205962sc + (11988611 minus 1198891119896

2) (11988622 minus 1198892119896

2)

1198861211988621 (32)


01525

0152

01515

01505

015

01495

0149

0151

(a)

0109

01089

01089

01089

01089

01089

01088

01088

(b)

01094

01092

0109

01088

01086

01084

01082

(c)

011

01

009

008

007

006

005

004

(d)



Im (120582) = 0

Re (120582) = 0

at 119896 = 119896119879

= 0(33)


= 119889111988921198964min minus (119886111198892 + 119886221198891)119896

2min + 1198861111988622 minus



120579119879=141205732minus 21205721205732

+ 2120573120575120572 minus 21205741205732+ 120572

21205732minus 21205722

120573120575 minus 21205721205732120574 + 120575

21205722+ 2120575120572120574120573 + 120574

21205732

120572120573 (120573 minus 120575) (34)










5 Discussions



119888 the







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119869 = (

11988611 11988612

11988621 11988622

) = (




) (119906lowast

minus 120574)

(119906lowast + 120579)2 minus

120572 (120573 minus 120575)2

1205732 minus120572120575

2

1205732

(120573 minus 120575)2

120573minus(120573 minus 120575) 120575

120573

) (9)


tr (119869) = 11988611 + 11988622 lt 0

det (119869) = 1198861111988622 minus 1198861211988621 gt 0(10)



119889ℎ

120597119905= 11988611ℎ + 11988612119901 (119905 minus 120591)

119889119901

120597119905= 11988621ℎ (119905 minus 120591) + 11988622119901

(11)


ℎ = ℎlowast

119890120582119905 cos (119896

119909119909)

119901 = 119901lowast

119890120582119905 cos (119896

119909119909)

(12)


(120582 minus 11988611 minus11988612119890

minus120582120591

minus11988621119890minus120582120591

120582 minus 11988622)(

ℎlowast

119901lowast) = (

00) (13)


1205822minus (11988611 + 11988622) 120582 + 1198861111988622 minus 1198861211988621119890

minus2120582120591= 0 (14)


minus1205962+ 1198861111988622 = 1198861211988621cos (2120596120591)

(11988611 + 11988622) 120596 = 1198861211988621sin (2120596120591) (15)

which leads to

1205964+ (119886

211 + 119886

222) 120596

2+ 119886

211119886

222 minus 119886

212119886

221 = 0 (16)


1205962119888

=minus (119886

211 + 119886

222) + radic(119886211 + 119886222)

2minus 4 (119886211119886222 minus 119886212119886

221)

2

(17)


120591119888=

12120596119888

arccosminus120596

2119888+ 1198861111988622

1198861211988621 (18)


Let 120582 =


Re (120582) = 1205902minus120596

2minus (11988611 + 11988622) 120590 + 1198861111988622

minus 1198861211988621119890minus2120590120591 cos (2120596120591) = 0


= 0

(19)


119894120596 we can get

119860119889120590

119889120591minus119861

119889120596

119889120591+119862 = 0

119860119889120590

119889120591+119861

119889120596

119889120591+119863 = 0

(20)

where

119860 = minus (11988611 + 11988622) + 21205911198861211988621 cos (2120596120591)

119861 = 2120596minus 21205911198861211988621 sin (2120596120591)

119862 = 21205961198861211988621 sin (2120596120591)

119863 = 21205961198861211988621 cos (2120596120591)

(21)

Thus we can get

119889Re (120582)119889120591

10038161003816100381610038161003816100381610038161003816120591=120591119888

=119889120590

119889120591

10038161003816100381610038161003816100381610038161003816120591=120591119888

= minus119860119862 + 119861119863

1198602 + 1198612 (22)


minus (119860119862+119861119863) = 21205962(119886

211 + 119886

222 + 21205962

) gt 0 (23)



119888






119888= 07397 then 119864

lowast







0 200 400 600 800 1000The time t

Tim

e ser

ies o

f the

indu

ced

prey

and

pred

ator

005

01

015

02

025

03

035

04

u

v

(a)

025 026 027 028 029 03 031 032 033 034u

0095

01

0105

011

0115

012

(b)


119888

0 200 400 600 800 1000The time t

Tim

e ser

ies o

f the

indu

ced

prey

and

pred

ator

005

01

015

02

025

03

035

04

u

v

(a)

02 022 024 026 028 03 032 034u

v

0085

009

0095

01

0105

011

0115

012

0125

(b)


119888



120597119906

120597119905=119906 (1 minus 119906) (119906 minus 120574)

119906 + 120579minus

120572119906V (119905 minus 120591)

119906 + V (119905 minus 120591)+ 1198891nabla

2119906

120597V120597119905

=120573119906 (119905 minus 120591) V119906 (119905 minus 120591) + V


(24)



120597ℎ

120597119905= 11988611ℎ + 11988612119901 (119905 minus 120591) + 1198891nabla

2ℎ

120597119901

120597119905= 11988621ℎ (119905 minus 120591) + 11988622119901+1198892nabla

2119901

(25)


010

005

0

minus005

minus010

Re(120582)

05 1 15

k(a)

(b)

(c)


01385

0138

01375

0137

01365

0136

01355

0135

(a)

0058

0056

0054

0052

005

0048

0046

0044

(b)

012

011

01

009

008

007

006

005

004

003

002

(c)

011

01

009

008

007

006

005

004

003

(d)



01385

0138

01375

0137

01365

0136

01355

0135

(a)

0104

0102

01

0098

0096

0094

0092

009

0088

0086

0084

(b)

011

01

009

008

007

006

005

(c)

011

01

009

008

007

006

005

004

(d)



(120582 minus 11988611 + 1198891119896

2minus11988612119890minus120582120591

minus11988621119890minus120582120591

120582 minus 11988622 + 11988921198962)(

ℎlowast

119901lowast) = (

00) (26)


1205822minus [(11988611 minus1198891119896

2) + (11988622 minus1198892119896

2)] 120582

+ (11988611 minus11988911198962) (11988622 minus1198892119896

2) minus 1198861211988621119890

minus2120582120591= 0

(27)


minus1205962+ (11988611 minus1198891119896

2) (11988622 minus1198892119896

2) = 1198861211988621cos (2120596120591)

[(11988611 minus11988911198962) + (11988622 minus1198892119896

2)] 120596 = 1198861211988621sin (2120596120591)

(28)

which leads to

1205964+1205721198961205962+120573119896= 0 (29)

where

120572119896= (11988611 minus1198891119896

2)2+ (11988622 minus1198892119896

2)2

120573119896= (11988611 minus1198891119896

2)2(11988622 minus1198892119896

2)2minus 119886

212119886

221

(30)


1205962sc =


2

(31)


120591sc =1

2120596scarccos

minus1205962sc + (11988611 minus 1198891119896

2) (11988622 minus 1198892119896

2)

1198861211988621 (32)


01525

0152

01515

01505

015

01495

0149

0151

(a)

0109

01089

01089

01089

01089

01089

01088

01088

(b)

01094

01092

0109

01088

01086

01084

01082

(c)

011

01

009

008

007

006

005

004

(d)



Im (120582) = 0

Re (120582) = 0

at 119896 = 119896119879

= 0(33)


= 119889111988921198964min minus (119886111198892 + 119886221198891)119896

2min + 1198861111988622 minus



120579119879=141205732minus 21205721205732

+ 2120573120575120572 minus 21205741205732+ 120572

21205732minus 21205722

120573120575 minus 21205721205732120574 + 120575

21205722+ 2120575120572120574120573 + 120574

21205732

120572120573 (120573 minus 120575) (34)










5 Discussions



119888 the







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 1000The time t

Tim

e ser

ies o

f the

indu

ced

prey

and

pred

ator

005

01

015

02

025

03

035

04

u

v

(a)

025 026 027 028 029 03 031 032 033 034u

0095

01

0105

011

0115

012

(b)


119888

0 200 400 600 800 1000The time t

Tim

e ser

ies o

f the

indu

ced

prey

and

pred

ator

005

01

015

02

025

03

035

04

u

v

(a)

02 022 024 026 028 03 032 034u

v

0085

009

0095

01

0105

011

0115

012

0125

(b)


119888



120597119906

120597119905=119906 (1 minus 119906) (119906 minus 120574)

119906 + 120579minus

120572119906V (119905 minus 120591)

119906 + V (119905 minus 120591)+ 1198891nabla

2119906

120597V120597119905

=120573119906 (119905 minus 120591) V119906 (119905 minus 120591) + V


(24)



120597ℎ

120597119905= 11988611ℎ + 11988612119901 (119905 minus 120591) + 1198891nabla

2ℎ

120597119901

120597119905= 11988621ℎ (119905 minus 120591) + 11988622119901+1198892nabla

2119901

(25)


010

005

0

minus005

minus010

Re(120582)

05 1 15

k(a)

(b)

(c)


01385

0138

01375

0137

01365

0136

01355

0135

(a)

0058

0056

0054

0052

005

0048

0046

0044

(b)

012

011

01

009

008

007

006

005

004

003

002

(c)

011

01

009

008

007

006

005

004

003

(d)



01385

0138

01375

0137

01365

0136

01355

0135

(a)

0104

0102

01

0098

0096

0094

0092

009

0088

0086

0084

(b)

011

01

009

008

007

006

005

(c)

011

01

009

008

007

006

005

004

(d)



(120582 minus 11988611 + 1198891119896

2minus11988612119890minus120582120591

minus11988621119890minus120582120591

120582 minus 11988622 + 11988921198962)(

ℎlowast

119901lowast) = (

00) (26)


1205822minus [(11988611 minus1198891119896

2) + (11988622 minus1198892119896

2)] 120582

+ (11988611 minus11988911198962) (11988622 minus1198892119896

2) minus 1198861211988621119890

minus2120582120591= 0

(27)


minus1205962+ (11988611 minus1198891119896

2) (11988622 minus1198892119896

2) = 1198861211988621cos (2120596120591)

[(11988611 minus11988911198962) + (11988622 minus1198892119896

2)] 120596 = 1198861211988621sin (2120596120591)

(28)

which leads to

1205964+1205721198961205962+120573119896= 0 (29)

where

120572119896= (11988611 minus1198891119896

2)2+ (11988622 minus1198892119896

2)2

120573119896= (11988611 minus1198891119896

2)2(11988622 minus1198892119896

2)2minus 119886

212119886

221

(30)


1205962sc =


2

(31)


120591sc =1

2120596scarccos

minus1205962sc + (11988611 minus 1198891119896

2) (11988622 minus 1198892119896

2)

1198861211988621 (32)


01525

0152

01515

01505

015

01495

0149

0151

(a)

0109

01089

01089

01089

01089

01089

01088

01088

(b)

01094

01092

0109

01088

01086

01084

01082

(c)

011

01

009

008

007

006

005

004

(d)



Im (120582) = 0

Re (120582) = 0

at 119896 = 119896119879

= 0(33)


= 119889111988921198964min minus (119886111198892 + 119886221198891)119896

2min + 1198861111988622 minus



120579119879=141205732minus 21205721205732

+ 2120573120575120572 minus 21205741205732+ 120572

21205732minus 21205722

120573120575 minus 21205721205732120574 + 120575

21205722+ 2120575120572120574120573 + 120574

21205732

120572120573 (120573 minus 120575) (34)










5 Discussions



119888 the







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










010

005

0

minus005

minus010

Re(120582)

05 1 15

k(a)

(b)

(c)


01385

0138

01375

0137

01365

0136

01355

0135

(a)

0058

0056

0054

0052

005

0048

0046

0044

(b)

012

011

01

009

008

007

006

005

004

003

002

(c)

011

01

009

008

007

006

005

004

003

(d)



01385

0138

01375

0137

01365

0136

01355

0135

(a)

0104

0102

01

0098

0096

0094

0092

009

0088

0086

0084

(b)

011

01

009

008

007

006

005

(c)

011

01

009

008

007

006

005

004

(d)



(120582 minus 11988611 + 1198891119896

2minus11988612119890minus120582120591

minus11988621119890minus120582120591

120582 minus 11988622 + 11988921198962)(

ℎlowast

119901lowast) = (

00) (26)


1205822minus [(11988611 minus1198891119896

2) + (11988622 minus1198892119896

2)] 120582

+ (11988611 minus11988911198962) (11988622 minus1198892119896

2) minus 1198861211988621119890

minus2120582120591= 0

(27)


minus1205962+ (11988611 minus1198891119896

2) (11988622 minus1198892119896

2) = 1198861211988621cos (2120596120591)

[(11988611 minus11988911198962) + (11988622 minus1198892119896

2)] 120596 = 1198861211988621sin (2120596120591)

(28)

which leads to

1205964+1205721198961205962+120573119896= 0 (29)

where

120572119896= (11988611 minus1198891119896

2)2+ (11988622 minus1198892119896

2)2

120573119896= (11988611 minus1198891119896

2)2(11988622 minus1198892119896

2)2minus 119886

212119886

221

(30)


1205962sc =


2

(31)


120591sc =1

2120596scarccos

minus1205962sc + (11988611 minus 1198891119896

2) (11988622 minus 1198892119896

2)

1198861211988621 (32)


01525

0152

01515

01505

015

01495

0149

0151

(a)

0109

01089

01089

01089

01089

01089

01088

01088

(b)

01094

01092

0109

01088

01086

01084

01082

(c)

011

01

009

008

007

006

005

004

(d)



Im (120582) = 0

Re (120582) = 0

at 119896 = 119896119879

= 0(33)


= 119889111988921198964min minus (119886111198892 + 119886221198891)119896

2min + 1198861111988622 minus



120579119879=141205732minus 21205721205732

+ 2120573120575120572 minus 21205741205732+ 120572

21205732minus 21205722

120573120575 minus 21205721205732120574 + 120575

21205722+ 2120575120572120574120573 + 120574

21205732

120572120573 (120573 minus 120575) (34)










5 Discussions



119888 the







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01385

0138

01375

0137

01365

0136

01355

0135

(a)

0104

0102

01

0098

0096

0094

0092

009

0088

0086

0084

(b)

011

01

009

008

007

006

005

(c)

011

01

009

008

007

006

005

004

(d)



(120582 minus 11988611 + 1198891119896

2minus11988612119890minus120582120591

minus11988621119890minus120582120591

120582 minus 11988622 + 11988921198962)(

ℎlowast

119901lowast) = (

00) (26)


1205822minus [(11988611 minus1198891119896

2) + (11988622 minus1198892119896

2)] 120582

+ (11988611 minus11988911198962) (11988622 minus1198892119896

2) minus 1198861211988621119890

minus2120582120591= 0

(27)


minus1205962+ (11988611 minus1198891119896

2) (11988622 minus1198892119896

2) = 1198861211988621cos (2120596120591)

[(11988611 minus11988911198962) + (11988622 minus1198892119896

2)] 120596 = 1198861211988621sin (2120596120591)

(28)

which leads to

1205964+1205721198961205962+120573119896= 0 (29)

where

120572119896= (11988611 minus1198891119896

2)2+ (11988622 minus1198892119896

2)2

120573119896= (11988611 minus1198891119896

2)2(11988622 minus1198892119896

2)2minus 119886

212119886

221

(30)


1205962sc =


2

(31)


120591sc =1

2120596scarccos

minus1205962sc + (11988611 minus 1198891119896

2) (11988622 minus 1198892119896

2)

1198861211988621 (32)


01525

0152

01515

01505

015

01495

0149

0151

(a)

0109

01089

01089

01089

01089

01089

01088

01088

(b)

01094

01092

0109

01088

01086

01084

01082

(c)

011

01

009

008

007

006

005

004

(d)



Im (120582) = 0

Re (120582) = 0

at 119896 = 119896119879

= 0(33)


= 119889111988921198964min minus (119886111198892 + 119886221198891)119896

2min + 1198861111988622 minus



120579119879=141205732minus 21205721205732

+ 2120573120575120572 minus 21205741205732+ 120572

21205732minus 21205722

120573120575 minus 21205721205732120574 + 120575

21205722+ 2120575120572120574120573 + 120574

21205732

120572120573 (120573 minus 120575) (34)










5 Discussions



119888 the







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01525

0152

01515

01505

015

01495

0149

0151

(a)

0109

01089

01089

01089

01089

01089

01088

01088

(b)

01094

01092

0109

01088

01086

01084

01082

(c)

011

01

009

008

007

006

005

004

(d)



Im (120582) = 0

Re (120582) = 0

at 119896 = 119896119879

= 0(33)


= 119889111988921198964min minus (119886111198892 + 119886221198891)119896

2min + 1198861111988622 minus



120579119879=141205732minus 21205721205732

+ 2120573120575120572 minus 21205741205732+ 120572

21205732minus 21205722

120573120575 minus 21205721205732120574 + 120575

21205722+ 2120575120572120574120573 + 120574

21205732

120572120573 (120573 minus 120575) (34)










5 Discussions



119888 the







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















5 Discussions



119888 the







Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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