On a predator prey system with cross-diffusion representing the tendency of prey to keep away from its predators

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1 Applied Mthemtics Letters On predtor prey system with cross-diffusion representing the tendency of prey to keep wy from its predtors Wonlyul Ko, Kimun Ryu b, Deprtment of Informtion nd Mthemtics, Kore University, Jochiwon, Chungnm , South Kore b Deprtment of Mthemtics Eduction, Cheongju University, Cheongju, Chungbuk , South Kore Received 9 April 2007; received in revised form 9 November 2007; ccepted December 2007 Abstrct In this work, we study predtor prey system with cross-diffusion, representing the tendency of prey to keep wy from its predtors, under the homogeneous Dirichlet boundry condition. Using fixed point index theory, we provide some sufficient conditions for the existence of positive stedy-stte solutions. Furthermore, we investigte the non-existence of positive solutions. c 2008 Elsevier Ltd. All rights reserved. Keywords: Predtor prey interction; Cross-diffusion; Index theory. Introduction In this work, we study the existence nd non-existence of positive solutions to the following Lotk Volterr predtor prey system with cross-diffusion rtes: u α v = u u c uv β u v = dv + c 2 uv in,. on, where R N is bounded domin with smooth boundry ; nd the given coefficients, c i nd d re positive constnts. In system., u nd v re the densities of the prey nd predtors, respectively; the function u represents the growth rte of the prey in the bsence of its predtors; c u is the functionl response of predtor to the prey, nd describes the chnge in the rte of exploittion of the prey by predtor s result of chnge in the prey density; c 2 c is the conversion rte of predtors to the prey; nd d is the deth rte of the predtor. The positive constnts α nd β re cross-diffusion rtes which express the respective popultion fluxes of the prey nd predtors resulting from the presence of the other species, respectively. Biologiclly, the induced cross-diffusion rte α in. represents the tendency of the prey to keep wy from its predtors; nd β represents the tendency of the predtor to chse its prey. Corresponding uthor. E-mil ddresses: kowl@kore.c.kr W. Ko, ryukm@cju.c.kr K. Ryu /$ - see front mtter c 2008 Elsevier Ltd. All rights reserved. doi:0.06/j.ml

2 78 W. Ko, K. Ryu / Applied Mthemtics Letters Previously, there hve been mny studies on the dynmics of strongly coupled rection diffusion systems with competitive cross-diffusion rtes, which ws proposed first by Shigesd et l. in [7]. For exmple, refer to [5,,2, 4,6] nd the references therein. On the other hnd, little ttention hs been given to studying the predtor prey models hving cross-diffusion rtes [9,4,5]. Furthermore, in these studies, the introduced cross-diffusion rtes represent the tendency of predtors to void group defense by the existence of lrge number of prey species, tht is, the predtors diffuse wy from their victims. However, in predtor prey interctions, it is more nturl nd relistic to consider the cses, s in., where the predtors tend to diffuse in the direction where there is higher concentrtion of the prey species, while the prey tends to diffuse in the direction where there is lower concentrtion of its predtors. For detiled biologicl bckground, one cn refer to [3,4,7,3]. We point out tht the diffusion terms given in. re different from the diffusion rtes of the forms [+αvu] nd [ + βuv], which re introduced in the previous works [9,,2,4,6]. For under the homogeneous Neumnn boundry condition, there re only few results on the stbility of non-negtive constnt solutions nd the ppernce of non-constnt stedy sttes to the predtor prey system with cross-diffusion rtes of the forms u ± αv nd v ± βu. For instnce, see [,3,6,8]. Unfortuntely, the existence of non-constnt positive solutions hs not been studied well. In the uthors view, one of the resons why such predtor prey systems hve not been studied thoroughly is the lck of knowledge of mthemticl methods to pply. This work is minly devoted to finding some sufficient conditions for the existence of positive solutions to. by using fixed point index theory. In ddition, we investigte some conditions which give the non-existence of positive solutions of.. This rticle is orgnized s follows. In Section 2, to pply the fixed point index theory, we introduce nother equivlent predtor prey system to., nd then provide some sufficient conditions for the existence of positive stedy-stte solutions. Finlly, in Section 3, we show the non-existence theorem for positive solutions to.. 2. The existence of positive stedy-stte solutions In this section, by using fixed point index theory, we derive some sufficient conditions for the existence of positive solutions to.. Let E be rel Bnch spce nd W the nturl positive cone of E. For y W, define W y = {x E : y + γ x W for some γ > 0} nd S y = {x W y : x W y }. Let y be fixed point of compct opertor A : W W nd L = A y be the Fréchet derivtive of A t y. We sy tht L hs property α on W y if there exists t 0, nd y W y \ S y such tht y tly S y. For n open subset U W, let index W A, U be the Lery Schuder degree deg W I A, U, 0, where I is the identity mp. The fixed point index of A t y in W is defined by index W A, y := indexa, Uy, W, where Uy is smll open neighborhood of y in W. Then the following theorem cn be obtined from the results of [2,0,8]. Theorem 2.. Assume tht I L is invertible on W y. If L hs property α on W y, then index W A, y = 0. To begin with, consider the following coupled system which is equivlent to.: u = + αβ [u u c v αv d + c 2 u] v = + αβ [v d + c 2u + βu u c v] in, on. Multiplying the first eqution by positive constnt δ which will be determined lter, nd then subtrcting it from the second eqution in 2., we hve u = + αβ [u u c v αv d + c 2 u] v δu = + αβ [ + αδv d + c 2u + β δu u c v] in, on

3 W. Ko, K. Ryu / Applied Mthemtics Letters Let w := v δu; then the following is system equivlent to 2.2, so 2. is obtined: u = + αβ [u + αdδ + δc + αc 2 u c + αc 2 w + αdw] w = + αβ [w d + αδ + c 2 + αδ β δc u + um + M 2 u] in on, where M := d + αδδ + β δ nd M 2 := c 2 + αδδ + c δβ δ. Observe tht, if 2.3 hs positive solution u, w, then. hs positive solution u, v with v = w + δu, nd thus it suffices to show tht 2.3 hs t lest one positive solution. Theorem 2.2. Any positive solution u, w of 2.3 stisfies ux Q nd wx Q 2, where Q := +αδ +c +αδ δ + c α nd Q 2 := α +c +αδ δ + c α. Proof. Multiplying the first nd second equtions in 2.3 by + αδ nd α, respectively, nd then dding the two equtions obtined, we hve [ + αδu + αw] = u + c δu c w in. Assume tht + αδu + αw ttins its positive mximum t x 0 ; then it esily follows tht [ + αδux 0 + αwx 0 ] = ux 0 + c δux 0 c wx 0 0. This implies + c δux 0 c wx 0 0, nd thus ux 0 +c δ nd wx 0 c. Using these fcts, it is esy to see tht { + αδux + αwx} = + αδux 0 + αwx 0 + αδ + c δ + α, c mx x which yields the desired result. For simplicity, tke δ := + d + + d 2 + 4dαβ, 2αd so tht M = 0. Then, since β δ = d+αδδ, M 2 = c 2 + αδδ + c δβ δ = + αδc dδ c2 d c d 2.3 δ. 2.4 For positive constnt P with P +αβ mx{2 + c + αc 2 Q + c + αc 2 Q 2, d + αδ + β δc Q }, define compct opertor A by where Au, w := + P f u, w + Pu f 2 u, w + Pw f u, w := + αβ [u + αdδ + δc + αc 2 u c + αc 2 w + αdw], [ f 2 u, w := w d + αδ + c 2 + αδ β δc u + M 2 u 2]. + αβ, Then, f u, w + Pu nd f 2 u, w + Pw re monotone incresing with respect to u nd w, respectively, for ll u, w [0, Q ] [0, Q 2 ]. For the ske of convenience, the following nottion is introduced. Nottion 2.3. i λ denotes the principl eigenvlue of under the homogeneous Dirichlet boundry condition. ii E := C D C D, where C D := {φ C : φ = 0 on }.

4 80 W. Ko, K. Ryu / Applied Mthemtics Letters iii N := N Q N Q, where N Q := {φ C D : φ < mx{q, Q 2 } + in }. iv W := K K, where K := {φ C D : 0 φx, x }. v N := N W. Now, to mke the opertor A positive in N, stisfying the following condition is imposed throughout this section: c 2 d c d β. More precisely, if 2.5 holds, then M 2 0 follows from the fct tht β δ = d+αδδ > 0 nd 2.4, which implies the positivity of the opertor A. In ddition, note tht 2.3 is equivlent to u, w = Au, w. Therefore, it suffices to prove tht A hs positive fixed point in N to show tht 2.3 hs positive solution. We point out tht 2.3 hs no semi-trivil solutions when exctly one of the species is bsent. Thus, we only need to clculte index W A, N nd index W A, 0, 0 to investigte the existence of positive stedy-stte solutions. Lemm 2.4. Assume tht 2.5 holds; then index W A, N =. Proof. Define homotopy A θ : E E by A θ u, w = + P θ f u, w + Pu θ f 2 u, w + Pw for θ [0, ]. Then, every fixed point of A θ is in N but not on N. Applying the homotopy invrince nd normliztion properties of the index, we cn conclude tht index W A, N = index W A 0, N =. In the following lemm, recll tht δ = +d+ +d 2 +4dαβ 2αd. Lemm 2.5. Assume tht 2.5 holds. If λ < +αdδ +αβ, then index WA, 0, 0 = 0. Proof. By simple clcultion, we hve W 0,0 = W, S 0,0 = {0, 0} nd + αdδ + αβ + P αd + αβ L := A 0, 0 = + P 0 Assume tht Lφ, ψ T = φ, ψ T for φ, ψ T W, tht is, φ = + αdδ + αβ φ + αd + αβ ψ d + αδ ψ = ψ in, + αβ φ, ψ = 0, 0 on. d + αδ + αβ + P Then, the strong mximum principle obviously llows us to conclude tht ψ 0 in. Let ϕ > 0 be the principl eigenfunction corresponding to λ. Multiplying ϕ by the first eqution fter substituting ψ = 0 in 2.6, nd then integrting it on, we hve 0 = ϕ φ + + αdδ + αβ φ dx =. φ ϕ + + αdδ + αβ ϕ dx = ϕφ λ + + αdδ dx. + αβ Since λ < +αdδ +αβ, φ K nd ϕ > 0 in, it follows tht φ 0 in. This shows tht I L is invertible on W. Furthermore, it is esy to see tht L hs property α. More precisely, for t := λ + P/ +αdδ +αβ P, it is esy to check tht t 0,, φ, 0 W 0,0 \ S 0,0, nd φ, 0 T tlφ, 0 T S 0,0. Therefore, Theorem 2. leds to the conclusion tht index W A, 0, 0 = 0. By Lemms 2.4 nd 2.5, since index W A, N index W A, 0, 0, we hve the following theorem which provides sufficient conditions for the existence of positive solutions to..

5 W. Ko, K. Ryu / Applied Mthemtics Letters Theorem 2.6. Assume tht c 2 d c d δ = +d+ +d 2 +4dαβ 2αd. β. Then. hs t lest one positive solution provided tht λ < +αdδ +αβ for Corollry 2.7. i If β min{ c 2 d c d, λ αλ }, then. hs t lest one positive solution. ii If λ αλ < β < min{ c 2 d c d, λ +d λ }, then. hs t lest one positive solution. αλ 2 Proof. i Since the condition β λ αλ is equivlent to +αβλ, it is esy to see tht λ < +αdδ +αβ is stisfied, nd thus the desired result follows from Theorem 2.6. ii Since λ αλ < β, we hve 2λ + αβ > 2 > d, so tht 2λ + αβ + d > 0. In ddition, since λ + αβ αdδ = 2 2λ + αβ + d + d 2 + 4dαβ nd [2λ + αβ + d] 2 + d 2 4dαβ = 4αβ + λ 2 αβ λ λ + d, it follows from nother given ssumption β < λ +d λ αλ 2 tht λ < +αdδ +αβ. Therefore, Theorem 2.6 leds to the conclusion tht. hs t lest one positive solution. Remrk 2.8. In view of Corollry 2.7i, we conclude tht there exists positive constnt ˆβ := ˆβ, d, α, c i, λ such tht. hs positive solution provided tht β ˆβ. Biologiclly, this implies tht the prey nd predtor species my coexist when the prey cn survive lone without its predtor i.e., > λ nd its corresponding intrinsic growth rte is greter thn some level i.e., > d c 2, provided tht the cross-diffusion β which is induced on the prey by its predtor is sufficiently smll. 3. The non-existence of positive stedy-stte solutions In this section, some sufficient conditions for the non-existence of positive solutions of. re provided. Theorem 3.. i If λ mx{, c 2 + α c d}, then. hs no positive solution. ii There exists positive constnt β := β, d, α, λ such tht. hs no positive solution provided tht β β. iii There exists positive constnt α := α, d, β, λ such tht. hs no positive solution provided tht α α. Proof. i Let u, v be positive solution of.. Multiplying the first nd second equtions in. by u nd v, respectively, nd then integrting these equtions on, we hve α u vdx = u 2 u c vdx u 2 dx, 3. β u vdx = v 2 d + c 2 udx v 2 dx. Since. is exctly equivlent to 2.3 when δ = 0, it follows from Theorem 2.2 tht the positive solution u, v stisfies u + αc nd v + α c α. Applying the Poincré inequlity to the first eqution in 3., we hve α u vdx u 2 u c vdx λ u 2 dx = λ u 2 dx u 2 u + c vdx < 0. The lst inequlity follows from the given ssumption λ. Therefore, using the Poincré inequlity gin nd nother given ssumption c 2 + c α d λ, the following contrdiction is derived from the second eqution in 3.:

6 82 W. Ko, K. Ryu / Applied Mthemtics Letters < β u vdx = v 2 d + c 2 udx v 2 dx v 2 d + c 2 u λ dx v d 2 + c 2 + αc λ dx 0. ii Suppose, by contrdiction, tht. hs positive solution u, v. Multiplying the first nd second equtions in. by v nd u, respectively, nd then integrting these equtions over, we hve u vdx = uv u c vdx α v 2 dx, u vdx = uv d + c 2 udx + β u 2 dx. The bove two identities yield α v 2 dx + β u 2 dx + d uvdx = uv + c 2 u + c v dx. 3.2 Applying the Poincré inequlity to the left-hnd side of 3.2, we hve α v 2 dx + β u 2 dx + d uvdx αλ v 2 dx + βλ u 2 dx + d uvdx u αλ v 2 dx + βλ u 2 2 dx + d 2ɛ + ɛv2 dx 2 + dɛ = αλ v 2 dx + βλ + d u 2 dx ɛ for n rbitrry positive constnt ɛ. Now, fix positive constnt ɛ 0 with ɛ 0 2αλ +d ; then it follows obviously tht 3.3 is non-negtive for β β := +d 2ɛ 0 λ. This is contrdiction to the fct tht the right-hnd side of 3.2 is negtive. iii This cn be shown like in the proof of ii. Thus, the proof is omitted. Remrk 3.2. In view of Theorem 3., we my conclude tht if the cross-diffusion rte of the prey or its predtor is sufficiently lrge, then the prey nd predtor species cnnot coexist. In other words, the lrge cross-diffusion coefficients α nd β tend to men no positive coexistence. Acknowledgements The uthors thnk the nonymous referee for his/her vluble comments nd suggestions for improving the content of this rticle. The first uthor W. Ko would like to thnk Professor Y. Du for his wrm hospitlity, mthemticl dvice, nd encourgement in Austrli. The first uthor s work ws supported by the Kore Reserch Foundtion Grnt funded by the Koren Government MOEHRD KRF C References [] J. Chttopdhyy, P.K. Tpswi, Effect of cross-diffusion on pttern formtion nonliner nlysis, Act Appl. Mth [2] E.N. Dncer, On the indices of fixed points of mppings in cones nd pplictions, J. Mth. Anl. Appl [3] B. Dubey, B. Ds, J. Hussin, A predtor prey interction model with self nd cross-diffusion, Ecol. Modelling [4] M. Frks, Two wys of modelling cross-diffusion, Nonliner Anl [5] M.E. Gurtin, Some mthemticl models for popultion dynmics tht led to segregtion, Qurt. Appl. Mth /75 9. [6] D. Horstmnn, Remrks on some Lotk Volterr type cross-diffusion models, Nonliner Anl. RWA [7] J. Jorne, Negtive ionic cross diffusion coefficients in electrolytic solutions, J. Theoret. Biol

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