DYNAMICAL BEHAVIORS OF A DELAYED REACTION-DIFFUSION EQUATION. Zhihao Ge

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1 Mathmatical and Computational Applications, Vol. 5, No. 5, pp , 00. Association for Scintific Rsarch DYNAMICAL BEHAVIORS OF A DELAYED REACTION-DIFFUSION EQUATION Zhihao G Institut of Applid Mathmatics School of Mathmatics and Information Scincs Hnan Univrsity, Kaifng, Hnan, P.R. China zhihaog@hnu.du.cn Abstract- In this papr, w driv a dlayd raction-diffusion quation to dscrib a two-spcis prdator-pry systm with diffusion trms and stag structur. By coupling th uniformly approximat approach with th mthod of uppr and lowr solutions, w prov that th travling wav fronts xist, which connct th zro solution with th positiv stady stat. Finally, w draw a conclusion that th xistnc of travling wav fronts for th dlayd raction-diffusion quation is an intrsting and difficult problm. Kywords- Raction-Diffusion Equations, Asymptotical Stability, Travling Wav fronts. INTRODUCTION Dlay ordinary diffrntial quations (also calld by rtardd functional diffrntial quations) hav bn xtnsivly studid by many authors, such as [], [3], [4], [7], [0] and so on. Rcntly, a two-spcis prdator-pry systm dscribd by a dlayd ordinary diffrntial quation was considrd in [9], whr th dlay mans th stag for th pry population. W rmark that th abov mntiond modls did not considr th ffct of diffusion on th stability of th quilibrium and travling wav fronts. Howvr, th spci's diffusion is a natural tndncy to mov into aras of smallr population dnsity. So w follow th normal tchniqu to handl with th diffusion (s [3], [5], [6] and []) to giv th following dlayd raction-diffusion quations u DΔ u = αu( xt, ) γu( xt, ) α u( xt, τ), t u DΔ u = βu( xt, ) au ( xtvxt, ) (, ) + α u( xt, τ) hu( xt, ), t v D Δ 3 v = v( x, t)( r + au( x, t) bv( x, t)), x Ω, t > 0, t u u v = = = 0, x Ω, t > 0, n n n u( x, t) = ϕ( x, t), u( x, t) = ϕ( x, t), v( x, t) = ϕ3( x, t), x Ω, t [ τ,0], ()

2 Z. G 763 N whr Ω R is opn and boundd with smooth boundary Ω, / n is diffrntiation in th dirction of th outward unit normal, u( x, t), u( x, t) rprsnt th immatur and matur pry population dnsitis at x -spac and t -tim, rspctivly; vxt (, ) rprsnts th dnsity of prdator population at x -spac and t -tim; a is th transformation cofficint of matur prdator population; rprsnts th transformation of immatur to matur; is th birth rat of th immatur pry population; h 0 is th harvsting ffort of th pry spcis; β > 0 rprsnts th dath and ovrcrowding rat of th matur pry population. And th constants r > 0, a > 0, b> 0. And th constant Di ( i =,,3) is positiv, and th initial functions ϕ( x,0), ϕ 3( x,0) ar continuous in Ω and ϕ ( x, t) is continuous in Ω [ τ,0]. In this papr, w aim to study th dynamical bhaviors of th systm (). Not that u ( x, t ) and vxt (, ) of th systm () ar indpndnt of u ( x, t), so w obtain th dynamical bhaviors of th systm () by studying th following systm u DΔ u = βu( xt, ) au ( xtu, ) ( xt, ) + α u( xt, τ) hu( xt, ), t u D3Δ u = u( x, t)( r+ au( x, t) bu( x, t)), x Ω, t > 0, t u u = = 0, x Ω, t > 0, n n u( x, t) = ϕ( x, t), u( x,0) = ϕ( x,0), x Ω, t [ τ,0], whr u( x, t), u( x, t) and ϕ( x, t), ϕ ( x, t) dnot u ( x, t), v( x, t) and ϕ( x, t), ϕ 3( x, t) of th systm (), rspctivly. For this singl spci modl of [0], S.A. Gourly and Y. Kuang pointd out that th xistnc of wav front solutions is an opn qustion. Motivatd by th rsults of [0], w study th xistnc of travling wav fronts of th two-spcis dlayd systm (). Th ky ida is to coupl th uniformly approximatd approach introducd by J. Canosa in [] with th mthod of uppr and lowr solutions. Th mthod to construct th uppr and lowr solutions of th systm () is drivd from th ida of []. Th rmaining parts of th papr ar organizd as follows. In sction, w study th locally asymptotical stability of th constant quilibrium and th xistnc of travling wav fronts of th systm (). Finally, w draw a conclusion.. DYNAMICAL BEHAVIORS OF THE SYTEM () () It is asy to chck that th systm () has only thr nonngativ constant γτ solutions: E(0,0), E(( α h) / β,0) and th positiv quilibrium E3( c, c) as α h β r, > whr b ( α h ) + a r ( α ), h β c = r aa + bβ c = aa + bβ. Using th linarization tchniqu ([8] or []) and omitting th dtaild drivation ( []), w hav

3 764 Dynamical Bhaviors of a Dlayd Raction-diffusion Equation Thorm. Th quilibrium E (0,0) of th systm () is unstabl; if a α h β r, thn th quilibrium E (( γτ α h ) / β,0) is unstabl; if a α h> β r, th positiv quilibrium E3( c, c) is locally asymptotically stabl. Nxt, w study th xistnc of travling wav solution for th infinit spatial x + ) (,. To sk a pair of travling wav fronts of th systm (), w st u( x, t) = φ( s) and u( x, t) = φ( s), whr s = x+ ct and c is th wav spd. Substituting φ () s and φ () s into th systm (), w hav Dφ" cφ' βφ hφ aφ( s) φ( s) + α φ( s cτ) = 0, D3φ" cφ' rφ+ aφφ bφ= 0, φ( ) = 0, φ( + ) = c, φ( ) = 0, φ( + ) = c. Lt θ = /c, for th larg valus of th wav spd, thn θ is a small paramtr. Dnot η = θs = s/ c, undr th transformation φi( s) = ψi( η) ( i =,), thn th systm (3) bcoms θdψ" ψ' βψ aψψ + α ψ( η τ) hψ = 0, θd3ψ" ψ' rψ + aψψ bψ = 0, ψ( ) = 0, ψ( + ) = c, ψ( ) = 0, ψ( + ) = c. Lt ψ ηθ ψ0 θψ ψ ηθ ψ0 θψ (, ) = + +, (, ) = + +, and substitut into (4) and group th sam powrs of θ, dnot ψ ( ) i0 η by ψ i ( η ) ( i =,), rspctivly, thn w hav (3) (4) ψ' = βψ aψψ + α ψ( η τ) hψ, ψ' = rψ + aψψ bψ, ψ( ) = 0, ψ( + ) = c, ψ( ) = 0, ψ( + ) = c. (5) Thorm. If a α h> β r, thn th systm (5) has at last on non-dcrasing T positiv solution ψ = ( ψ ( η), ψ ( η)) C ( R, R ). Proof. To prov th thorm, w nd to chck that a quasi-monoton condition (s [7] or []) holds and show that thr xists a pair of uppr and lowr solutions ( ψ ( η), ψ ( η )) T and ( ψ ( η), ψ ( η )) T. To do that, w dfin th functional f ( ψ) = ( f ( ψ), f ( ψ)) T c c c by fc ( ψ) = βψ (0) aψ(0) ψ(0) + α ψ( τ) hψ(0), fc( ψ) = rψ(0) + aψ(0) ψ(0) bψ(0). (6)

4 Z. G 765 Ltting δ = ( δ, δ) T, for arbitrary φψ, C([ τ,0]; R ) satisfying 0 ψ ( η) φ( η), w asily obtain f ( φ) f ( ψ) + δ( φ(0) ψ(0)) ( δi Β)( φ(0) ψ(0)) 0, (7) c c whr I is a idntity matrix, δ α + βc and δ r + bc. B diag( c a c hc, r bc ), = β Nxt, w show that thr xists a pair of uppr and lowr solutions. To do that, w introduc th following st Dfin whr ψ () ( η) ψ is picwis continuous and nondcrasing in R Γ= ψ = T ψ () lim 0, lim ( ( η) ψ = ψ = c, c) η η + λη λη c c, η 0,, η 0, ψ ( η) = ψ ( η) = c c c, η > 0, c, η > 0,. (8) λ > α. (9) ψ ( η) And it is asy to chck that ψ = Γ. Nxt, w chck that ψ is a pair of uppr ψ ( η) solutions to (5). To do that, w hav two cass Cas i: η > 0. Thn, w hav ( λ + r) c rc r a b 4 ψ '( η) + ψ ( η) ψ ( η) ψ ( η) + ψ ( η) > 0. (0) From (8) and (9), for th cas η > τ it follows that a ψ '( η) α ψ ( η τ) + βψ ( η) + ψ ( η) ψ ( η) + hψ ( η) c c c acc acc βc 4c c 4 λ α β c ( βc+ ac ) c c c c ( λ+ α ) = [ cλ+ ( ) α ] > 0, () and for th cas 0 < η τ, w gt a ψ '( η) α ψ ( η τ) + βψ ( η) + ψ ( η) ψ ( η) + hψ ( η) c βc acc βc 4 ac c 4 λ + + c ( λ α ) > 0. ()

5 766 Dynamical Bhaviors of a Dlayd Raction-diffusion Equation Cas ii: η < 0. Using (8) and (9), w hav a ψ '( η) α ψ ( η τ) + βψ ( η) + ψ ( η) ψ ( η) + hψ ( η) c λη β c ac λ α λη λη ( + + ) > 0, ψ '( η) + rψ ( η) a ψ ( η) ψ ( η) + bψ ( η) c λη r λη r λη ac λη bc λη λ r = ( ) > 0. From Cas i and Cas ii, w know that ψ is an uppr solution to (5). Dfin λη < ξε, η 0, ψ ( η) = ψ ( η) = 0, ε ξε, η 0, (3) whr λτ β + h α ξ λξ 0< ε <, ξ is small nough, 0 γτ < λ < α λ τ. From (3), w gt λη λη λξε, η < 0, λ ξε, η < 0, ψ '( η) = "( ) ψ η = λξε, η 0, λ ξε, η 0. Using (3) and (4), for η τ w hav + + a + h ψ '( η) α ψ ( η τ) βψ ( η) ψ ( η) ψ ( η) ψ ( η) λ( η τ) = λξε α ( ε ξε ) + β( ε ξε ) + h( ε ξε ) < for 0 < η < τ w obtain γτ 0, + + a + h ψ '( η) α ψ ( η τ) βψ ( η) ψ ( η) ψ ( η) ψ ( η) γτ λ( η τ ) = λξε α ξε + β ( ε ξε ) + h( ε ξε ) < 0 (4) (5) (6) if 0 < ε λτ β + h α ξ λ ξ ; for η < 0 w hav a ψ '( η) α ψ ( η τ ) + βψ ( η) + ψ ( η) ψ ( η) + hψ ( η) = λξε α ξε + β( ξε ) + hξε ( ) λη λ η τ λη λη λη λτ ξε [ λ α + ( β + h) ε ] < 0. ψ ( η) So, ψ = is a pair of lowr solutions. ψ ( η) Thrfor, if a α γτ > β r, from [] w know that thr xists at last on solution in th st Γ. Th proof of th thorm is compltd. (7)

6 Z. G CONCLUSION In our work, w prov th xistnc of travling wav fronts for th two-spcis modl for larg valus of th wav spd. Th systm () is a nw modl and th mthod to prov th xistnc of travling wav fronts is also novl, and it is ffctiv to dal with th cas of larg wav spds, which is dsrvd futur study. Acknowldgmnt: Th prsnt work is supportd by National Natural Scinc Foundation of China undr Grant No REFERENCES. J. Canosa, On a nonlinar diffusion quation dscribing population growth, IBM Journal of Rsarch and Dvlopmnt 7, , H. Frdman, Dtrministic Mathmatical Modls in Population Ecology, Marcl Dkkr, Nw York, W. Aillo and H. Frdman, A tim-dlay modl of singl-spcis growth with stag structur, Mathmatical Bioscincs 0, 39-53, Y. Kuang, Dlay Diffrntial Equations with Applications in Population Dynamics, Acadmic Prss, London, C. V. Pao, Dynamics of nonlinar parabolic systms with tim dlays, Journal of Mathmatical Analysis and Applications 98, , C. V. Pao, Systms of parabolic quations with continuous and discrt dlays, Journal of Mathmatical Analysis and Applications 05, 57-85, T. Faria, Stability and bifurcation for a dlayd prdator-pry modl and th ffct of diffusion, Journal of Mathmatical Analysis and Applications 54, , W. Josph, J. H. Wu and X. F. Zou, A raction-diffusion modl for a singl spcis with ag structur. I. travlling fronts on unboundd domains, Proc. R. Soc. London A 457, -3, X. Y. Song and L. S. Chn, Optimal harvsting and stability for a prdator-pry systm with stag structur, Acta Mathmatica Applicata Sinica (English Sris) 8, , S. A. Gourly, Y. Kuang, A stag structurd prdator-pry modl and its dpndnc on maturation dlay and dath rat, Journal of Mathmatical Biology 49, 88-00, A. Boumnir, N. V. Minh, Prron thorm in th monoton itration mthod for travling wavs in dlayd raction-diffusion quations, Journal of Diffrntial Equations 44, , Z. H. G and Y. N. H, Diffusion ffct and stability analysis of a prdator-pry systm dscribd by a dlayd raction diffusion quations, Journal of Mathmatical Analysis and Applications 339, , 008.

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