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CANADIAN APPLlED MATHEMATICS QUARTERLY Volume 2, Number 4, Fall 1994 REACTION-DIFFUSION EQUATIONS WITH INFINITE DELAY SHIGUI RUAN AND JIANI-IONG WU ABSTRACT. We have developed several results on the ex- istence and asymptotic behavior of nlild solutions to rextion- diffusion systems that have infinite delirys in the nonlinear reaction terms. We find that the semiflow generated by a co- operative and irreducible reaction-tliffilsio system with infi- nite delay is not compact but set-condcrrsing, ancl not strongly order-preserving but quasi strongly ordc~r-1)rest:rviiig. These set-condenseness and quasi strong ordt!r-prosc:rvi~~g proper- ties allow us to use a ~nodificirt,ion, recent.1~ givol~ by Freed- man, Miller and one of the ilutliors of tllis l,iqier, of the well- known monotone dynan~ical systenl tl~c:ory due to Dancer, Hess, Hirsch, Matano, Smith, Tllicme, 1301iCik and TakaE to obtain some results about convergence and stability of solu- tions. Examples of Loth-Volterra co~npetition-diffusion mod- els with distributed delay are given to illustrate the obtained results. 1. Introduction. A variety of m;~tlic;nat,ical inodcls for biological processes are most appropriatcly framed as partial functional diffcr- ential equations. For example, the reaction-tlilfusioi logistic equation with finite delay Accepted for publication on October 4, 199.1. Research was carried out while tlie first ~rutl~or was a Jui~ior Ibllow at the Fields Institute for Research in Matllernatical Sciences i111d WLS ~~artially supported by the Ministry of College and University of Ontario and the NSERC of Canada. Research of the second autl~or pirrtially sul)portetl Ijy tlie NSERC of Canada. This paper was written while the seco~ld aul.llor wi~s ;rll ii~vitc:cl Visiting Scie~ltist to the Fields Institute for Research in M;itl~c~~r:~tic:iil Scic:ncc:s. TIN: ;rutlror woultl like to thank the Institute for its su~port. , AMS Mathematics Subject lasszJ(:utzo~~s. :iSL)35 351t10 341.;30. Key words and phrases. Partial fuiic:tio~~;il t1iffi:rontiai equ:rtions, reaction- diffusion systems with infinite delily, rnonotonicity, illvarianc:e, comparison, set- condensing maps, strongly order-preserving so~niflows, c:orlvergence, stability.

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Page 1: REACTION-DIFFUSION EQUATIONS WITH INFINITE DELAYruan/MyPapers/RuanWu-CAMQ1994.pdfistence and asymptotic behavior of nlild solutions to rextion- diffusion systems that have infinite

CANADIAN APPLlED MATHEMATICS QUARTERLY Volume 2, Number 4, Fall 1994

REACTION-DIFFUSION EQUATIONS WITH INFINITE DELAY

SHIGUI RUAN AND JIANI-IONG WU

ABSTRACT. We have developed several results on the ex- istence and asymptotic behavior of nlild solutions to rextion- diffusion systems that have infinite delirys in the nonlinear reaction terms. We find that the semiflow generated by a co- operative and irreducible reaction-tliffilsio system with infi- nite delay is not compact but set-condcrrsing, ancl not strongly order-preserving but quasi strongly ordc~r-1)rest:rviiig. These set-condenseness and quasi strong ordt!r-prosc:rvi~~g proper-ties allow us to use a ~nodificirt,ion, recent.1~ givol~ by Freed-man, Miller and one of the ilutliors of tllis l,iqier, of the well- known monotone dynan~ical systenl tl~c:ory due to Dancer, Hess, Hirsch, Matano, Smith, Tllicme, 1301iCik and TakaE to obtain some results about convergence and stability of solu- tions. Examples of Loth-Volterra co~npetition-diffusion mod- els with distributed delay are given to illustrate the obtained results.

1. Introduction. A variety of m;~tlic;nat,ical inodcls for biological processes are most appropriatcly framed as partial functional diffcr- ential equations. For example, the reaction-tlilfusioi logistic equation with finite delay

Accepted for publication on October 4, 199.1. Research was carried out while tlie first ~rut l~or was a Jui~ior Ibllow at the Fields

Institute for Research in Matllernatical Sciences i111d WLS ~~art ia l lysupported by the Ministry of College and University of Ontario and the NSERC of Canada.

Research of the second autl~or pirrtially sul)portetl Ijy tlie NSERC of Canada. This paper was written while the seco~ld aul.llor wi~s ;rll ii~vitc:cl Visiting Scie~ltist to the Fields Institute for Research in M;itl~c~~r:~tic:iil Scic:ncc:s. TIN: ;rutlror woultl like to thank the Institute for its su~por t .

,

AMS Mathematics Subject lasszJ(:utzo~~s.:iSL)35 351t10 341.;30. Key words and phrases. Partial fuiic:tio~~;ilt1iffi:rontiai equ:rtions, reaction-

diffusion systems with infinite delily, rnonotonicity, illvarianc:e, comparison, set- condensing maps, strongly order-preserving so~niflows, c:orlvergence, stability.

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486 S. RUAN AND J . WU

where d, T , T and K are positive constii~its, has been uscd to model an one-dimensional herbivorous populutio11 and has bccn studied by many authors, for example, Buscnbcrg and Huang [5], Fricseckc [8, 91, Gopalsamy, He and Sun [13], Grcen arid Stccll [14], Huang [28], Lin and Khan [33], Luckhaus [35], Mclnory [44, 451, Morita [47], Yosllida [76] and Yoshida and I<ishimoto [77], to nanlc a fcw.

General partial functional differential cqua!.ions wit11 finitc delay llave been extensively studied. We rcfcr to Fitzgibbon [lo], Rankin [55] and Travis and Webb [68, 691 for dctailctl tliscussions on tlic existcncc and asymptotic behavior of solutions; to 1l;unish and Scllappacllcr [32] for necessary conditions to gcncratc Co-sc.lnigroups; to Hale [17] for the convergence to solutions of an ortlinary ful~cliollal tlifl(:rt:ntial ccluation; to Fitzgibbon and Parrott [ l l ] :\ad 13arrot,t [SO] for the lilicrixcd stability; to He 120, 21) for pcriodic ant1 itliilos! l)crio(lic solutions; to Lin, So and Wu [34] for a centcr manifoltl theory; to Hale and Ladeira [19] for the differentiability witli respcct to delays, i ~ a d to Rcy and Mackey [57, 581 for bifurcations, travclil~g wavcs and multistability.

Recently Martin and Smith, in their tlirce corisecutivc papcrs [36-381, have studied partial functional clilfcrcl~tial ccluations in a Banach space. They developed several fundaillcntiil results on the existcnce and asymptotic behavior of solutioris to abstrir~t semilinear functional differential equations with finitc delay and t.hcn apply the results to reaction-diffusion equatiolis which liave fiilitc tiine delays in the ~lonlin- ear reaction terms. By employing tlic rnoriotolic dyna~nical systeni the- ory due to Hirsch [25-271, Matztno [39-421, Sliiith [GI], and Smith arid Thieme [63, 641, they establisllcd suflicicrit c:oliditions lor a rcaction- diffusion equation witli firiitc delay to gc:ncriltc a (eventually) strongly monotone scmiflow on an appropriiitc. sp;lcc ;u~d collc:lutlcd that al- most all orbits convergc to the scl of cquiIi1)riil. ?'hay also cstal~lishcd the existence of an invariant rcrtanglr ant1 ol)t,ilincd certain conipar- ing systems of ordinary functioiiirl cliff~~rc~llt,inl ~(lui\t,ioli~ rclativc to tlic invariant rcctangle. As an application, tlicy collsidcrcd tllc n-species Lotka-Volterra modcl of compctit,ion wit11 tli(l'11siol1 ant1 firiitc delay and developed sufficient conditions for !,lit g1ol)itl i~~;yllil>tt)ti(: stability of t l ~ e coexistence state.

It is well known that distribu!.ctl dclay slloultl I)c used to dcscribc tlle stochastic elcmcnt in tlie dclayctl rcsl)olisc of a, 11iologic:~l process. Tllc

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