两类带有非局部扩散项的种群模型的行波解

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20183a¬ÆØ©üa‘kšÛÜ*Ñ‘«+.1Å)Šö6¶“r•“4?JÇƉ;’A^êÆïÄ••‡©•§†ÄåXÚüêƉÆÆÆSc•2015c9–2018c6˜lc8 ìÜŒÆ20183a¬ÆØ©üa‘kšÛÜ*Ñ‘«+.1Å)Šö6¶“r•“4?JÇƉ;’A^êÆïÄ••‡©•§†ÄåXÚüêƉÆÆÆSc•2015c9–2018c6"˜lc8 ThesisforMaster'sdegree,ShanxiUniversity,2018TravelingWavesforTwoTypesofPopulationModelswithNonlocalDiusionStudentNameZhimeiWangSupervisorProf.GuirongLiuMajorAppliedMathematicsFieldofResearchDierentialEquationsandDynamicalSystemsDepartmentSchoolofMathematicalSciencesResearchDuration2015.09-2018.06June,2018 8¹¥©Á‡...............................................................................iAbstract...............................................................................i1˜ÙXØ...........................................................................11Ù‘kšÛÜ*Ñ‘Úž¢«+.Åc)..............................32.1Úó...........................................................................32.2Åc)•35..............................................................42.3Åc)ìC5.............................................................112.4Åc)Ø•35...........................................................192.5ž¢Ú=zÇé•1Å„ÝK•.......................................191nÙ‘kšÛÜ*Ñ‘Úž˜¢D/¾.1Å)......................213.1Úó..........................................................................213.21Å)•35.............................................................22o(†Ð"...........................................................................32ë•©z.............................................................................33ïĤJ.............................................................................36—..................................................................................37‡<{¹9éX•ª.................................................................38«ìÖ................................................................................39ÆØ©¦^Ç(²...............................................................40i ContentsAbstractinChinese..................................................................iAbstract...............................................................................iChapter1Preface...................................................................1Chapter2Travelingwavefrontforapopulationmodelwithdelayandnonlocaldiusion...............................................................................32.1Introduction...................................................................32.2Existenceoftravelingwavefront...............................................42.3Asymptoticbehavioroftravelingwavefront...................................112.4Nonexistenceoftravelingwavefront..........................................192.5Theeectsofdelayandtransferratesonminimumtravelingwavespeed......20Chapter3Travelingwaveforaninfectiousdiseasemodelwithspatio-temporaldelayandnonlocaldiusion.......................................................213.1Introduction..................................................................213.2Existenceoftravelingwaves..................................................22Thesummaryandoutlook........................................................32References...........................................................................33Researchachievement..............................................................36Acknowledgements..................................................................37Personalproles.....................................................................38Letterofcommitment..............................................................39Authorizationstatement...........................................................40iii ¥©Á‡¥©Á‡©ïÄüa‘kšÛÜ*Ñ‘«+.1Å).1˜Ù,0‡A*Ñ•§ïĵ9®²¤J,•[ãISuÐı9©̇óŠ.1Ù,ïĹkšÛÜ*Ñ‘Úž¢«+.8>>@tu(x;t)=D[Ju(x;t)u(x;t)]+f(u(x;t);u(x;t1))<1u(x;t))+2v(x;t2);>>:@tv(x;t)=1u(x;t3)2v(x;t);RÙ¥Ju(x;t)=J(xy)u(y;t)dy,u(x;t)•?u¹ÄG<•—Ý;v(x;t)•?Ru·ŽG<•—Ý,1,2L«¹ÄGÚ·ŽGƒm=zÇ,f(u;u)•2)¼ê,D>0´*ÑXê,1;2;3•~ê.Ù$^þe)•{(ÜSchauderØÄ:½ny²,ccž,TXÚ•3Åc).ÏLLaplaceC†,(ÜIkeharaÚnXÚÅc)31?ìC5,¿y²00ž,c'u1üN4O;@2f(0;0)=0ž,c†1Ã'.c'u2;3; 1üN4O.c'u2üN4~.1nÙ,ïĹkž˜¢ÚšÛÜ*Ñ‘D/¾.8>@tu(x;t)=D[Ju(x;t)u(x;t)]+f(u(x;t);u(x;t1))<1u(x;t))+2v(x;t2);>>:@tv(x;t)=1u(x;t3)2v(x;t);RJu(x;t)=J(xy)u(y;t)dy,u(x;t)andv(x;t)representactivityphaseandstationaryRphaseofpopulationdensity,1>0and2>0areconversionratesbetweenthetwophases,f(u;u)isreproductivefunction,D>0isdiusioncoecient,1;2;3aredelays.Byusingxedpointtheoremandupper-lowersolutionmethod,weprovethatthepopulationsystemhastravelingwavefrontwhencc.ThroughLaplacetransformandIkeharalemma,weprovetheasymptoticbehavioroftravelingwavefrontin1.Itisprovedthatwhen00,cisanmonotoneincreasingfunctionon1;when@2f(0;0)=0,cisindependentof1.cisamonotoneincreasingfunctionon2;3,and1.cisanmonotonedecreasingfunctionon2.Chapter3studyiestheinfectiousdiseasemodelwithspatio-temporaldelayandnonlocaldiusion80•1Å„Ý.Zeldovich;FrankKamenetskii[6]š‚5‡A*ÑXÚ1Å)•35,•½5.Aronson[7],ÚWeinberger[8]ïÄV•œ/,ùpf(u)=u(1u)(u)(2(0;1)).'uFisherïÄ,ØØf(u)´ü•„´V•,ÑUÑe(Ø:ccž,‡A*ÑXÚ•31Å);00;0w(x;t)1:Ù¥>0•~ê,ef÷v˜½^‡,K:0ccž,XÚ•3•˜üN1Å),…(1)=0,(+1)=1.Š•šÛÜž¢;.,©[10]•ÄZ@u(x;t)+1=d
u(x;t)+fu(x;t);h(xy)u(y;t)dy:@t13·^‡e,Œ±†©[9]aq(Ø.5¿Laplace½Â*ÑL«‡N3˜mþÛÜŠ^,‡N£Ä´‘Å,Ï•·‚ïá.I‡•Äy¢µÚ)ÔÆ¿Â,¤±Lee[17]JÑ^È©½1 üa‘kšÛÜ*Ñ‘«+.1Å)šÛÜ*Ñ,©[13,18]ïÄZu(x;t)t=J(xy)u(t;y)dyu(t;x)+f(u(t;x)):RPanXÚ•31Å).šÛÜ*Ñ„Œ±^5£ãD/¾Ú«+*Ñ,Œë•©z[1921].镘ãžm±5,y²1Å)•35˜†´‡A*Ñ•§ïÄ•‡‘K,y²‡A*Ñ•§1Å)•35kõ«•{,~X:ØÄ:•{[22;23],ÝnØ•{[24],ƒ²¡©Û{Ú•ŒŠn[9],ÛÉÄnØ,üNS“{.ÄuþãïÄyG,©òïÄüa‘kšÛÜ*Ñ«+.1Å).2 1Ù‘kšÛÜ*Ñ‘Úž¢«+.Åc)x2.1ÚóCAc5,¹k·Žã‡A*ÑXÚ®²Ñy,•²;.´HadelerÚLewis[25]ïá8<@tu(x;t)=Duxx(x;t)+f(u(x;t))1u(x;t)+2v(x;t);(2.1.1):@tv(x;t)=1u(x;t)2v(x;t):u(x;t);v(x;t)©O•<•?u¹ÄGÚ·ŽGž—Ý,1Ú2L«¹ÄGÚ·ŽGƒm=zÇ,f(u;u)•2)¼ê,D>0.du‡NÈzÏI‡˜ãžm,Ïd,•Ä‘kž¢‡A*ÑXÚ´k¿Â.Hadeler[26]Úe.8<@tu(x;t)=Duxx(x;t)+f(u(x;t);u(x;t))1u(x;t)+2v(x;t);(2.1.2):@tv(x;t)=1u(x;t)2v(x;t);Ù¥f(u(x;t);u(x;t))•2)¼ê,>0.XZhou[22]ïÄšÛÜ*ÑXÚ8>>@tu(x;t)=D[Ju(x;t)u(x;t)]+f(u(x;t);u(x;t))<1u(x;t)+2v(x;t);(2.1.3)>>:@tv(x;t)=1u(x;t)2v(x;t);RÙ¥Ju(x;t)=RJ(xy)u(y;t)dy,ZhouÏLØÄ:½nÚþe)•{:ccž,TXÚ•3Åc);0>@tu(x;t)=D[Ju(x;t)u(x;t)]+f(u(x;t);u(x;t1))<1u(x;t))+2v(x;t2);(2.1.4)>>:@tv(x;t)=1u(x;t3)2v(x;t);RÙ¥Ju(x;t)=RJ(xy)u(y;t)dy,1Ú2L«üGƒm=zÇ,f(u;u)•2)¼ê,D>0,1;2;3•~ê.‰Ñe^‡RR(H1)J(x)0;J(x)=J(x);J(x)dx=1;…é?¿>0,J(x)exdx<+1:RR(H2)f(0;0)=f(K;K)=0;f2C2([0;K]2;R),é?¿u2(0;K),f(u;u)>0,…é?¿(u;v)2[0;K]2,@2f(u;v)0,Ù¥K•~ê.3 üa‘kšÛÜ*Ñ‘«+.1Å)(H3)é?¿(u;v)2[0;K]2,@1f(0;0)u+@2f(0;0)vf(u;v),@1f(K;K)+@2f(K;K)<0.1!,ÏLSchauderØÄ:½nÚþe)•{y²,ccž,XÚ•3Åc).1n!,ÏLLaplaceC†,(ÜIkeharaÚnXÚÅc)31?ìC5.1o!y²00•1Å„Ý.U(
),V(
)üNž,(U;V)•(2.1.4)Åc).ò(u(x;t);v(x;t))=(U();V())“(2.1.4),Œ8<0cU()=D[JU()U()]+f(U();U(c1))1U()+2V(c2);(2.2.1):cV0()=1U(c3)2V();Ù>.^‡•(U(1);V(1))=(0;0);(U(1);V(1))=(K;K0):(2.2.2)d(2.2.1)1‡•§ÚV(1)=0,ŒZ12(s)V()=U(sc3)ecds;(2.2.3)c1?kZc212(c2s)V(c2)=U(sc3)ecds:(2.2.4)c1U(1)=0;U(1)=Kž,d(2.2.4)Œ•V(1)=0;V(1)=K0.ò(2.2.4)“(2.2.1)1˜‡•§,kcU0()=D[JU()U()]+f(U();U(c))U()11Zc2122(c2s)+U(sc3)ecds;(2.2.5)c1Ù>.^‡•U(1)=0;U(1)=K:(2.2.6)N´eÚn¤á.4 1Ù‘kšÛÜ*Ñ‘Úž¢«+.Åc)Ún2.2.1.eU()2C1(R;[0;K])´(2.2.5)†(2.2.6)üNØ~),K(U(x+ct);V(x+ct))´XÚ(2.1.4)üNØ~Åc),Ù¥V()÷v(2.2.3).Ïd,•y²XÚ(2.1.4)•3Åc),I•Ä(2.2.5)†(2.2.6)üNØ~)•35.e¡|^þe)•{ÚSchauderØÄ:½ny²(2.2.5)†(2.2.6))•35.½ÂŽfF:C(R;[0;K])!C(R;R)Z1(s)(F)()=(H)(s)ecds;(2.2.7)c1ùp>maxfj@1f(u;v)jg+D+1,±9H•(u;v)2[0;K]2(H)()=DJ()+(D1)()+f(();(c1))Zc2122(c2s)+(sc3)ecds:(2.2.8)c1´yF÷v:c(F)0()=(F)()+(H)().Ún2.2.2.(H1)(H3)¤á.(i)e1,22C(R;[0;K]),…1()2();2R,K(H1)()(H2)(),(F1)()(F2)(),2R.(ii)e2C(R;[0;K]),…'uüNØ~,KH,F'u•üNØ~.y².(i)d(H2)Œ•,é?¿2R,Ñkf(1();1(c1))f(2();2(c1))+(D1)(1()2())f(1();2(c1))f(2();2(c1))+(D1)(1()2())(D1+maxfj@1f(u;v)jg)(1()2())(u;v)2[0;K]20†Z1J1()J2()=J(y)(1(y)2(y))ds0:1dŽfH½Â,k(H1)()(H2)();2R.?˜Ú(F1)()(F2)();2R.(ii)é?¿2C(R;[0;K]),Ï•'uüNØ~,?1;22R…1<2,d@2f(u;v)0Œ•,f((1);(1c1))f((2);(2c1))f((1);(2c1))f((2);(2c1))0:5 üa‘kšÛÜ*Ñ‘«+.1Å)Rc22(c2s)-g()=1(sc3)ecds,lZc20222(c2+s)g()=ec(sc3)ecds+(c2c3)c1Zc2222(c2+s)(c2c3)ececds+(c2c3)c1=(c2c3)+(c2c3)=0:Ïd,g()'uüNØ~,=g(1)g(2).u´Œ(H)(1)(H)(2).d0(H)()K.lé?¿1;22R,1<2ž,ZZ121(1s)(2s)(F)(1)(F)(2)=(H)(s)ecds(H)(s)ecdsc11ZZ12K(1s)(2s)(2s)[ecec]ds+ecdsc11=0:Ïd,Ún2.2.2¤á.-~ê2(0;).½ÂcB(R;R)=f2C(R;R):supj()jejj<1g;2R…jj=supfj()jejjg,w,(B(R;R);j
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(c;)2y2c12=DyeJ(y)dy+(c1)@2f(0;0)e@R2c2122c212(2+3)c212(2+3)2c(2+3)+++e(c+2)3(c+2)2c+2>0:½Âc:=supfc>0j
(c;)>0;2Rg,´yc2(0;1).
(c;)=0k•˜),P•.?
(c;)=0,@
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(c;(c))e1(c)+
(c; (c))qe 1(c)110:Ïd,()•(2.2.5)þ).2y()•(2.2.5)e).PL=maxfmaxfj@11f(u;v)j;j@12f(u;v)j;j@22f(u;v)jgg;(u;v)2[0;K]214L=lnq;Q(c; )=max1;:(1)1(c)
(c; 1(c))+4LeqQ(c;r),K<0.>ž,k()=0;ž,k()=e1(c)qe 1(c).l,>ž,D[J()()]+f(();(c1))1()Zc2122(c2s)0+(sc3)ecdsc()c1f(0;0)=0:8 1Ù‘kšÛÜ*Ñ‘Úž¢«+.Åc)ž,D[J()()]+f(();(c1))1()Zc2122(c2s)0+(sc3)ecdsc()c1Ze1(c)fc(c)+De1(c)yJ(y)dy1+@f(0;0)111R1(c)c1121(c)c(2+3)+@2f(0;0)e+egc1(c)+2Zqe 1(c)fc (c)+De 1(c)yJ(y)dy1+@f(0;0)111R 1(c)c112 1(c)c(2+3)+@2f(0;0)e+egc 1(c)+2+f(();(c))e1(c)@f(0;0)+@f(0;0)e1(c)c1112+qe 1(c)@f(0;0)+@f(0;0)e 1(c)c112=
(c;(c))e1(c)
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))2C([X;X];R2)S()'()S();()'()I();:X>>1212>>:;82[X;X]½Â:88>>S();<X;>>();<X;<<1()='1();XX;2()='2();XX;(3.2.4)>>>>::'1(X);>X;'2(X);>X:?Øe>Š¯K:8<0R+1cS()=DS1J(y)1(y)dy+(DS+d+(K2)())S():0R+1cI()=DI1J(y)2(y)dy(DI++d+r)I()+"1()(K2)()(3.2.5)S(X)=S(X);I(X)=(X)=maxf0;I(X)g:(3.2.6)d‡©•§)•3•˜5½n[36]Œ±(3.2.5)-(3.2.6)k•˜),P•(S;I),…÷v(S;I)2C2([X;X];R).Ïd,½ÂŽfF=(F1;F2):X!C([X;X];R),F('1;'2)=(S;I);('1;'2)2X:e¡y²ŽfFkØÄ:.Ún3.2.3.e(A1)(A3)¤á,KŽfF:X!X.y².?('1;'2)2X,w,F1('1;'2)(X)=S(X)=S(X);F2('1;'2)(X)=(X).-=1lnS0,2[;X]ž,S()=0;0´(3.2.5)1˜ªe);1k12[X;1)ž,S()=S0ke,Z+1cS0()DJ(y)(y)dy+(D+d+(K)())S()S1S21Z+1cS0()DJ(y)S(y)dy+(D+d+(KI)())S()SS10:25 üa‘kšÛÜ*Ñ‘«+.1Å)2[X;X]ž,kZ+10cS()DSJ(y)1(y)dy+(DS+d+(K2)())S()1Z+10cS()DSJ(y)S(y)dy+(DS+d+(KI)())S()10;¤±2[X;X]ž,S()S()S().-=1ln1,2[X;]ž,()=2M2I();1()='1()S();2()='2()();…Z+1cI0()DJ(y)(y)dy+(D++d+r)I()"()(K)()I2I121Z+1cI0()DJ(y)I(y)dy(D++d+r)I()"()(K)()II1210:2[2;X]ž,()=0.w,I()0:2[X;X]ž,kZ+10cI()DIJ(y)2(y)dy+(DI++d+r)I()"1()(K2)()1Z+10cI()DIJ(y)I(y)dy+(DI++d+r)I()"S()(KI)()10;¤±2[X;X]ž,()I()I().nþ¤ã,Ún3.2.3¤á.Ún3.2.4.ŽfF:X!XëY.y².?('1;'2);('e1;'e2)2X,-F('1;'2)=(S;I);F('e1;'e2)=(S;eIe).d(3.2.5),Z1S()=S(X)exp[DS+d+(K2)(s)]dscXZZ11+exp[DS+d+(K2)(s)]ds(DSJ1()+)d(3.2.7)cXc†1I()=(X)exp(DI++d+r)(+X)cZ11+exp(DI++d+r)()[DIJ2()+"1()K2()]d:cXc(3.2.8)26 1nÙ‘kšÛÜ*Ñ‘Úž˜¢D/¾.1Å)<Xž,1()=e1()=S().>Xž,1()='1(X);e1()='e1(X).é?¿('1;'2);('e1;'e2)2X.l,Z+1J(y)[(y)e(y)]dy(J1)()(Je1)()=111ZXjJ(y)[1(y)e1(y)]dyjXZ+1+J(y)[1(X)e1(X)]dyX2maxj'1(y)'e1(y)j;y2[X;X](J2)()(Je2)()2maxj'2(y)'e2(y)j;y2[X;X]†Z+1Z+1K(y;s)[(ycs)e(ycs)]dyds(K2)()(Ke2)()=2201ZZ+1x1xt=K(x;)[2(t)e2(t)]dtds11ccZZ+1X1xtK(x;)[2(t)e2(t)]dtds1XccZZ+1x1xt+K(x;)[2(X)e2(X)]dtds1Xcc2maxj'2(y)'e2(y)j;y2[X;X]1()K2())e1()Ke2())=1()K2())1()Ke2())+1()Ke2())e1()Ke2()j1()jK2())Ke2())+1()e1()Ke2()2Smaxj'(y)'e(y)j+2e1L(c;)maxj'(y)'e(y)j011111y2[X;X]y2[X;X]†DIJ2()+"1()K2()DIJe2()"e1()Ke2()DIJ2()Je2()+"1()K2()e1()Ke2()2Dmaxj'(y)'e(y)j+2"[S+e1L(c;)]maxj'(y)'e(y)j:I220111y2[X;X]y2[X;X]d(3:2:7);(3:2:8)ŒŽfFëY.é?¿2[X;X],S();I();S0();I0()k.,F(X);,ŽfF:X!X´ëY.27 üa‘kšÛÜ*Ñ‘«+.1Å)dX½ÂŒ•,X´k.4à8,(ÜÚn3.2.3,Ún3.2.4,ÚSchauderØÄ:½nŒ•,•3(SX;IX)2X,¦(SX();IX())=F(SX;IX)();2[X;X]:-!1Œ±(3.2.1))•35.•y²(3.2.1))•3,kéSX;IX3C1;1([X;X])ŠO.½Â:no1;110C([X;X])=u2C([X;X])u;u´LipschitzëYùp0ju0(x)u0(y)jkukC1;1([X;X])=maxjuj+maxjuj+sup:x2[X;X]x2[X;X]x;y2[X;X];x6=yjxyjÚn3.2.5.e(A4)¤á,Ké?¿½Y>0(Y0,¦kSXkC1;1([Y;Y])max1ln1;1lnS0:Mky².(SX;IX)÷v:Z+1cS0()=DJ(y)Sb(y)dy+(D+d+(KIb)())S();(3.2.9)XSXSXX1Z+1cI0()=DJ(y)Ib(y)dy(D++d+r)I()+"S()(KIb)();(3.2.10)XIXIXXX1ùp88>>SX(X);>X;>>IX(X);>X;<>>>::S();<X;();<X:2[Y;Y]ž,SX()S()=S0;IX()I(Y)=e1Y,u´kZ+1jS0()jDSJ(y)Sb(y)dy++DS+djS()j+(KIbXXXX)()SX()c1ccc+[2DS+d+e1YL(c;1)]S0=c28 1nÙ‘kšÛÜ*Ñ‘Úž˜¢D/¾.1Å)ÚZ+1jI0()jDIJ(y)Ib(y)dy+DI++d+rjI()j+"(KIbXXXX)()SX()c1ccZ+1DI1Y1yDI++d+r1Y"S0L(c;1)1YeJ(y)edy+e+ec1ccR+11yDI1J(y)edy+(DI++d+r)+"S0L(c;1)1Y=e:c•3~êC1(Y)>0,¦kSXkC1([Y;Y])0,¦é?¿;2[Y;Y],ÑkjS0()S0()jC(Y)jj:XX2d(3.2.10)ŒZ+1cjI0()I0()jDJ(y)[Ib(y)Ib(y)]dy+(D++d+r)jI()XXSXXIX1IX()j+"(KIbX)())SX()(KIbX)())SX();(3.2.16)Z+RJS4=J(y)[IbX(y)IbX(y)]dyRJZZ+RJ+RJ=J(x)IbX(x)dxJ(x)IbX(x)dxRJRJZZRJ+RJJ(x)IbX(x)dx+J(x)IbX(x)dxRJ+RJZ+RJ+jJ(x)J(x)jIbX(x)dxRJ1(Y+RJ)1(YRJ)1(Y+RJ)kJkL1e+kJkL1e+2LJRJejj:(3.2.17)30 1nÙ‘kšÛÜ*Ñ‘Úž˜¢D/¾.1Å)Ïd,é?¿;2[Y;Y],ÑkjI0()I0()jC(Y)jj:XX2nþ¤ã,•3C(Y)>0,¦é?¿;2[Y;Y](Ymax1ln1;1lnS0:Mk"½n3.2.1e(A1)(A4);R0=d(+d+r)>1¤á,Ké?¿c>c,XÚ(3.2.1)•31Å)(S(x+ct);I(x+ct)),…lim(S();I())=(S0;0),Ù¥=x+ct;2R.!1noy².én2N,À4OêfXg+1,…÷vX>max1ln1;1lnS0;nn=1nMklimXn=+1:e2[Xn;Xn],K•3(SXn;IXn)2Xn;¦(3:2:9);(3:2:10)¤á,…n!1÷vÚn3:2:5.Àf(SXn;IXn)gff(SXn;IXn)gk2N,…Âñu(S;eIe)2C1(R);kk3C1(R)¥,k!+1ž,SXn!S;IeXn!Ie.Ï•JÚK3R´;|,¤±lockkZ+1Z+1limJ(y)SbXn(y)dy=J(y)Se(y)dy=JSe();k!+11k1Z+1Z+1limJ(y)IbXn(y)dy=J(y)ISf(y)dy=JIe();k!+11k1Z+1Z+1Z+1Z+1limK(y;s)IbXn(ycs)dyds=K(y;s)Ie(ycs)dyds=KIe():k!+101k01?˜ÚŒ,(S;eIe)÷v(3.2.1)Úmaxf0;Skeg=S()Se()S()=S;00maxf0;e1(1Me)g=()Ie()I()=e1:R+1…1I()d<+1.!1ž,Se()=S0;Ie()=0.l,(S;I)´(3.2.1)1Å).½n3.2.1¤á.31 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ÆØ©¦^Ç(²ÆØ©¦^Ç(²<)ìÜŒÆk'3,¦^ÆØ©5½,=:Æk3¿•I[k'Å'½ÅxØ©E<‡Ú>f©,#NØ©Ú/,Œ±æ^K<,<½×£Ãã•,®?ÆØ©.Ó¿ì܌ƌ±^ØÓ•ª3ØÓxNþuL,DÂØ©ܽܩSN.—ÆØ©3)—„ÅdÆ.Šö¶:“¶:20cF40