含两种竞争食饵和两种捕食者的捕食者食饵模型的定性分析

西北师范大学硕士学位论文含两种竞争食饵和两种捕食者的捕食者-食饵模型的定性分析姓名：王永斌申请学位级别：硕士专业：应用数学指导教师：伏升茂2009-06-8:u1t=4(d1u1+a11u21)+u1(1¡u1¡®u2¡1u3¡2u4);x2­;t0;u2t=4(d2u2+a22u22)+u2(1¡¯u1¡u2¡°1u3¡°2u4);x2­;t0;u3t=4(d3u3+a31u1u3+a32u2u3+a33u23)+u3(¡1+n1u1+n°1u2);x2­;t0;u4t=4(d4u4+a41u1u4+a42u2u4+a44u24)+u4(¡1+n2u1+n°2u2);x2­;t0;@´u1=@´u2=@´u3=@´u4=0;x2@­;t0;ui(x;0)=ui0(i=1;2;3;4);x2­(1),­½RN,´­..(1).(1).Gaglirdo-Nirenberg(1).:;;;;;.iiAbstractInthispaper,globalbehaviorofsolutionsforthefollowingpredator-preymodelwithtwopredatorsandtwocompetitivepreysisinvestigated8:u1t=4(d1u1+a11u21)+u1(1¡u1¡®u2¡1u3¡2u4);x2­;t0;u2t=4(d2u2+a22u22)+u2(1¡¯u1¡u2¡°1u3¡°2u4);x2­;t0;u3t=4(d3u3+a31u1u3+a32u2u3+a33u23)+u3(¡1+n1u1+n°1u2);x2­;t0;u4t=4(d4u4+a41u1u4+a42u2u4+a44u24)+u4(¡1+n2u1+n°2u2);x2­;t0;@´u1=@´u2=@´u3=@´u4=0;x2@­;t0;ui(x;0)=ui0(i=1;2;3;4);x2­;(1)where­isaboundedregioninRNwithsmoothboundary@­,´denotestheoutwardunitnormalvectorontheboundary@­.Thepaperisdividedintothreesections.InSection1,thestabilityofnonnegativeequilibriumpointsforthemodel(1)ofODEtypeisdiscussed.InSection2,theuniformboundednessofglobalsolutionsandthestabilityofthenonnegativeequilibriumpointsforthemodeloftheweaklycoupledreaction-di®usiontypeareconsidered.InSection3,usingmethodofenergyestimatesandGagliardo-Nirenbergtypeinequal-ities,theexistenceanduniformboundednessofnonnegativeglobalsolutionsforthemodel(1)ofcross-di®usiontypeareprovedwhenthespacedimensionisone.Keywords:Di®usion;self-di®usion;cross-di®usion;positiveequilibriumpoint;globalsolution;stability.iii:::i.[1],-.,,,[2]-8:u01(t)=u1(t)(1¡u1(t)¡®u2(t)¡1u3(t)¡2u4(t));u02(t)=u2(t)(1¡¯u1(t)¡u2(t)¡°1u3(t)¡°2u4(t));u03(t)=u3(t)(¡1+n1u1(t)+n°1u2(t));u04(t)=u4(t)(¡1+n2u1(t)+n°2u2(t))(0:1),®,¯,n,i,°i(i=1;2),u1;u2,u3,u4,n,i,°i(i=1;2),®,¯.(0.1),[2-4].,[5]..,(0.1)8:u1t¡d14u1=u1(1¡u1¡®u2¡1u3¡2u4);x2­;t0;u2t¡d24u2=u2(1¡¯u1¡u2¡°1u3¡°2u4);x2­;t0;u3t¡d34u3=u3(¡1+n1u1+n°1u2);x2­;t0;u4t¡d44u4=u4(¡1+n2u1+n°2u2);x2­;t0;@´u1=@´u2=@´u3=@´u4=0;x2@­;t0ui(x;0)=ui0(i=1;2;3;4);x2­;(0:2)­½RN,´@­,di(i=11;2;3;4),ui0(i=1;2;3;4)­.,,,[24;25],(0.2)u1t=4(d1u1+®11u21+®12u1u2+®13u1u3+®14u1u4)+[1¡u1¡®u2¡1u3¡2u4]u1;x2­;t0;u2t=4(d2u2+®21u1u2+®22u22+®23u2u3+®24u2u4)+[1¡¯u1¡u2¡°1u3¡°2u4]u2;x2­;t0;u3t=4(d3u3+®31u1u3+®32u2u3+®33u23)+[¡1+n1u1+n°1u2]u3;x2­;t0;u4t=4(d4u4+®41u1u4+®42u2u4+®44u24)+[¡1+n2u1+n°2u2]u4;x2­;t0;@ui@´(x;t)=0(i=1;2;3;4);x2@­;t0;ui(x;0)=ui0(x)¸0(i=1;2;3;4);x2­;(0:3)­RN,´@­,di(i=1;2;3;4),aii(i=1;2;3;4),aij(i6=j;i;j=1;2;3;4).(0.3),,(0.3),N=1(0.3).,­=(0;1),u1t=(d1u1+®11u21+®12u1u2+®13u1u3+®14u1u4)xx+[1¡u1¡®u2¡1u3¡2u4]u1;x2(0;1);t0;2u2t=(d2u2+®21u1u2+®22u22+®23u2u3+®24u2u4)xx+[1¡¯u1¡u2¡°1u3¡°2u4]u2;x2(0;1);t0;u3t=(d3u3+®31u1u3+®32u2u3+®33u23)xx+[¡1+n1u1+n°1u2]u3;x2(0;1);t0;u4t=(d4u4+®41u1u4+®42u2u4+®44u24)xx+[¡1+n2u1+n°2u2]u4;x2(0;1);t0;uix=0(i=1;2;3;4);x=0;1;t0;ui(x;0)=ui0(x)¸0(i=1;2;3;4);x2(0;1)(0:4).3.(0.2)-(0.4),.(0.2)(0.3)(),(0.3).(0.4).:QT=­£(0;T);1·p+1.u2Wkp(­)j®j·k®=(®1;®2;¢¢¢;®n);D®u2Lp(­);kukWkp(­)=(R­§j®j·kjD®ujpdx)1p,j¢jk;p=k¢kWkp(0;1);j¢jp=k¢kLp(0;1).u2W2;1p(QT)u;uxi;uxixj(i;j=1;2;¢¢¢;n);ut2Lp(QT),kukW2;1p(QT)=(RQT(jujp+jDujp+jD2ujp+jutjp)dxdt)1p.3x1-8:u01(t)=u1(t)(1¡u1(t)¡®u2(t)¡1u3(t)¡2u4(t))u02(t)=u2(t)(1¡¯u1(t)¡u2(t)¡°1u3(t)¡°2u4(t))u03(t)=u3(t)(¡1+n1u1(t)+n°1u2(t))u04(t)=u4(t)(¡1+n2u1(t)+n°2u2(t))(0:1)®,¯,n,i,°i(i=1;2).f1(u)=1¡u1¡®u2¡1u3¡2u4;f2(u)=1¡¯u1¡u2¡°1u3¡°2u4;f3(u)=¡1+n1u1+n°1u2;f4(u)=¡1+n2u1+n°2u2:,fi(i=1;2;3;4)¹R4+Lipschitz,¹R4+=f(u1;u2;u3;u4)ju1;u2;u3;u4¸0g,(0.1),,(0.1).,(0.1)E0=(0;0;0;0)E1=(1;0;0;0)E2=(0;1;0;0)E3=(0;u2;0;u4)E4=(0;·u2;^u3;0)E5=(u1;0;0;¶u4)E6=(·u1;0;·u3;0)E7=(^u1;¶u2;0;0),u2=1n°2;u4=n°2¡1n°22(n°21);·u2=1n°1;^u3=n°1¡1n°21(n°11);u1=1n2;¶u4=n2¡1n22(n21);·u1=1n1;·u3=n1¡1n21(n11);^u1=1¡®1¡®¯;¶u2=1¡¯1¡®¯(0®;¯1):(H1)0®n°i1;0¯ni1(i=1;2);(H2)(1¡®¯)+n2(®¡1)+n°2(¯¡1)0;(0.1)E8=(~u1;~u2;0;~u4)~u1=2(n°2¡1)+°2®¡n°2)detA2;~u2=°2(n2¡1)+2(¯¡n2)detA2;~u4=(1¡®¯)+n2(®¡1)+n°2(¯¡1)detA2.4x1-detAi=n[¡°2i+(®+¯)i°i¡2i](i=1;2):(H1),(H3)(1¡®¯)+n1(®¡1)+n°1(¯¡1)0;(0.1)E9=(µu1;µu2;µu3;0)µu1=1(n°1¡1)+°1(®¡n°1)detA1;µu2=°1(n1¡1)+1(¯¡n°2)detA1;µu3=(1¡®¯)+n1(®¡1)+n°1(¯¡1)detA1:(H4)°22;1°1;°2°1;12®;(H5)n¸maxf®°2(1¡2)(1°2¡2°1)(°2¡2);1(1¡2)(1°2¡2°1)(1¡°1)g;(0.1)E¤=(u¤1;u¤2;u¤3;u¤4),u¤i=hi=m(i=1;2;3;4);m=n2(1°2¡2°1)2;h1=n(1°2¡2°1)(°2¡°1);h2=n(1°2¡2°1)(1¡2);h3=n(2¡1)(®°2¡2)+n(°2¡°1)(¯2¡°2)+n2(1°2¡2°1)(°2¡2);h4=n(2¡1)(1¡®°1)+n(°2¡°1)(°1¡¯1)+n2(1°2¡2°1)(1¡°1):(H4),(H5)°1=1,°2=2,1=3=2,2=1=2,®=1=5,¯=1=6,n=6=5:(0.1).1.1(u1;u2;u3;u4)(0.1),[0;T),0ui(t)·Mi(i=1;2;3;4),t2[0;T),MiM1=maxf1;u1(0)g;M2=maxf1;u2(0)g;M3;M4(0.1)ui(0)(i=1;2;3;4).,(0.1)[0,T).du1dt·u1(1¡u1),M1=maxf1;u1(0)g,d(u1¡M1)dt+u1(u1¡M1)·0;u1(0)¡M1·0:u1·M1.,u2·M2.z=nu1+nu2+u3+u4,(0.1)dzdt=nu1(1¡u1¡®u2)+nu2(1¡¯u1¡u2)¡(u3+u4)5x1-·n(u1¡u21)+n(u2¡u22)¡(u3+u4)·2n¡z;(1.1)z(t)·maxf2n;nu1(0)+nu2(0)+u3(0)+u4(0)g,t2[0;T).(0.1).,T=+1.1.1(0.1).1.2(1)E0.(2)¯1ni1(i=1;2),E1;¯1ni1(i=1;2),E1.(3)®1n°i1(i=1;2),E2;®1n°i1(i=