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2 Computers ad Mathematics with Applicatios 6 (011) Cotets lists available at SciVerse ScieceDirect Computers ad Mathematics with Applicatios joural homepage: A ote o Hopf bifurcatios i a delayed diffusive Lotka Volterra predator prey system Shasha Che a, Jupig Shi b, Jujie Wei a, a Departmet of Mathematics, Harbi Istitute of Techology, Harbi, Heilogjiag, , PR Chia b Departmet of Mathematics, College of William ad Mary, Williamsburg, VA, , USA a r t i c l e i f o a b s t r a c t Article history: Received 3 September 010 Received i revised form 6 July 011 Accepted 7 July 011 Keywords: Predator prey model Reactio diffusio Lotka Volterra Delay Stability Hopf bifurcatio The diffusive Lotka Volterra predator prey system with two delays is recosidered here. The stability of the coexistece equilibrium ad associated Hopf bifurcatio are ivestigated by aalyzig the characteristic equatios, ad our results complemet earlier oes. We also obtai that i a special case, a Hopf bifurcatio of spatial ihomogeeous periodic solutios occurs i the system. 011 Elsevier Ltd. All rights reserved. 1. Itroductio Partial fuctioal-differetial equatios have bee proposed as mathematical models for biological pheomea by may researchers i recet years. I the last 15 years especially, the stability/istability ad bifurcatio of equilibrium solutios for reactio diffusio equatios/systems with a delay effect have bee cosidered extesively. The theory of partial fuctioaldifferetial equatios ad the related bifurcatio theory have bee developed for aalyzig various mathematical questios that have arise from models of populatio biology, biochemical reactios ad other applicatios [1 4]. For the models with a sigle populatio, So ad Yag [5] ivestigated the global attractivity of the equilibrium for the diffusive Nicholso s blowflies equatio with Dirichlet boudary coditio; So, Wu ad Yag [6] ad Su et al. [7] also studied the Hopf bifurcatio o the diffusive Nicholso s blowflies equatio with Dirichlet boudary coditio; Yi ad Zou [8] ivestigated the global attractivity of the diffusive Nicholso blowflies equatio with Neuma boudary coditio; Buseberg ad Huag [9], ad Su, Wei ad Shi [10] ivestigated the Hopf bifurcatio of a reactio diffusio populatio model with delay ad Dirichlet boudary coditio, which occurs at the spatially ihomogeeous equilibrium; Davidso ad Gourley [11] (ad also [10]) studied the dyamics of a diffusive food-limited populatio model with delay ad Dirichlet boudary coditio. For multiple-populatio models, there are may results o predator prey systems (see e.g. [1 0]). I this work we cosider the followig Lotka Volterra predator prey system: u(t, x) u(t, x) = d 1 + u(t, x)[r t x 1 a 1 u(t, x) a v(t ν, x)], t > 0, x (0, π), v(t, x) v(t, x) = d + v(t, x)[ r t x + a 3 u(t τ, x) a 4 v(t, x)], t > 0, x (0, π), (1.1) u(t, x) v(t, x) = = 0, t 0, x = 0, π, x x This research was supported by the Natioal Natural Sciece Foudatio of Chia (Nos ad ). Correspodig author. address: weijj@hit.edu.c (J. Wei) /$ see frot matter 011 Elsevier Ltd. All rights reserved. doi: /j.camwa

3 S. Che et al. / Computers ad Mathematics with Applicatios 6 (011) u(t, x) ad v(t, x) are iterpreted as the desities of prey ad predator populatios. Moreover, τ, r 1, r, a, a 3 are positive costats ad ν, a 1, a 4 are o-egative costats. More biological explaatio of the above delayed diffusive Lotka Volterra prey predator system ca be foud i [1]. I [1], Faria assumed that a 3 r 1 a 1 r > 0, ad the system (1.1) has a positive equilibrium a r + a 4 r 1 E = (u, v ) =, a 3r 1 a 1 r. a 1 a 4 + a a 3 a 1 a 4 + a a 3 Due to the difficulties i the aalysis of the characteristic equatios, Faria studied the istability of the equilibrium E ad associated Hopf bifurcatios with some assumptios o the coefficiets i reactio terms. I [4], Wu studied the Hopf bifurcatio of system (1.1) whe ν = 0, a 4 = 0, ad he obtaied that uder some assumptios o the coefficiets, the the system (1.1) ca give rise to a Hopf bifurcatio of spatially ihomogeeous periodic solutios. I this paper we also assume (1.); hece the system (1.1) possesses a positive equilibrium E. I this ote, we obtai two ew results for (1.1) which complemet the oes i [1]. First we show that a similar istability aalysis of E holds whe the two diffusio coefficiets are close to each other but without additioal coditios o the coefficiets i the reactio terms. Secodly we prove that i a special case of a = b = 0 ad d 1 = d, a Hopf bifurcatio of spatially ihomogeeous periodic solutios occurs ad the bifurcatig periodic solutios are ustable. The rest of the paper is orgaized as follows. I Sectio, we aalyze the stability of the positive equilibrium E through the study of the characteristic equatios. We show some results o the distributio of the roots of the characteristic equatios ad these results are supplemetary to the oes i Faria [1]. We also show that the positive equilibrium E ca be destabilized through a Hopf bifurcatio as τ icreases whe the two diffusio coefficiets are close to each other. I Sectio 3, we cosider a special case whe a = b = 0 ad d 1 = d. (1.). Stability aalysis ad bifurcatio I this sectio, we will carry out the aalysis of stability ad Hopf bifurcatio of the system (1.1) ad give some results supplemetary to those of Faria [1]. I [1], after the time-scalig t t/τ, the chage of variables u a 3 u = u, v a u = v, ad droppig the bars for simplificatio, system (1.1) is trasformed ito u(t, x) u(t, x) = τ d 1 + τ u(t, x)[r t x 1 au(t, x) v(t r, x)], t > 0, x (0, π), v(t, x) v(t, x) = τ d + τv(t, x)[ r t x + u(t 1, x) bv(t, x)], t > 0, x (0, π), u(t, x) v(t, x) = = 0, t 0, x = 0, π, x x r = ν/τ, a = a 1 /a 3, b = a 4 /a. The positive equilibrium E = (u, v ) is ow give by u = r + br 1 ab + 1, v = r 1 ar ab + 1, with the assumptios (which we will always assume i the rest of this paper) r 1 > 0, r > 0, r 0, a 0, b 0, r 1 ar > 0. (.) From [1] (5.6 k ), we kow that the characteristic equatios for the equilibrium E are k (λ, τ) = λ + A k τλ + B k τ + C τ e λ(1+r), k = 0, 1,,..., (.3 k ) A k = d 1 k + d k + au + bv, B k = (d 1 k + au )(d k + bv ), ad C = u v. If iσ k (σ k > 0) is a root of Eq. (.3 k ), the we have σ k B kτ = C τ cos σ k (1 + r), σ k A k = C τ si σ k (1 + r), which leads to (.1) ρ 4 + (A k B k)ρ + B k C = 0, (.4)

4 4 S. Che et al. / Computers ad Mathematics with Applicatios 6 (011) So if A k B k = (d 1 + d )k4 + (d 1 au + d bv )k + (a u + b v ) 0, B k C = (d 1k + au ) (d k + bv ) u v, ad ρ = σ k τ. ab 1, the B k C > 0, ad Eqs. (.3 k), k 0, have o imagiary roots, ad if ab < 1, the there is a iteger K 0 such that Eq. (.4) has a positive real root ρ k = 1 [ ] 1/ B k A + k (B k A k ) 4(B k C ) (.5) (.6) if k K 0 ad has o positive real roots if k > K 0. So Eqs. (.3 k ), k > K 0, have o imagiary roots ad each of Eqs. (.3 k ), k K 0, has oly a couple of imagiary roots ±iσ k at τ k σ k ρ k arccos B k = 1 + r + π, = 0, 1,,..., k = 0, 1,..., K 0, τ k = σ k ρ k, = 0, 1,,..., k = 0, 1,..., K 0, if ab < 1 ad a + b > 0. If a = b = 0, the σ 0 ( + 1)π =, = 0, 1,,..., 1 + r τ 0 = σ 0 ρ 0, = 0, 1,,..., ad σ k, τ k (k 1) are the same as those above. If d 1 = d = d, the ad A k = m k + p, B k = m k + m kp + q, m k = dk, p = au + bv, q = abu v, B k A k = m k m kp (a u + b v ), B k C = (m k + m kp + q) C. Hece we have the followig lemma. Lemma.1. Assume that d 1 = d = d. 1. If ab < 1 ad a + b > 0 are satisfied, the > τ k, 0 k K 0, = 0, 1,..... If a = b = 0 is satisfied, the > τ k, 1 k K 0, = 0, 1,.... Proof. Whe d 1 = d, we have ρ = 1 ] k [(p 4q)(4m k + 4m kp + p ) + 4C mk m kp + q p, ρ k B k = 1 ] [(p 4q)(4m k + 4m kp + p ) + 4C 4mk 4m kp p. If p 4q = 0, it is obvious that > τ k, so we assume that p 4q > 0 sice p 4q 0.

5 S. Che et al. / Computers ad Mathematics with Applicatios 6 (011) Suppose that x = (p 4q)(4m + k 4m kp + p ) + 4 ; the x (p 4q)p + 4C, ad τ k = arccos x x 4 p 4q + π 1+r x x (p 4q) p + q def = g(x). It is easy to verify that if y > z p 4q ad y, z are i the domai of g, the g(y) > g(z). So we ca obtai that if ab < 1 ad a + b > 0 are satisfied, the > τ k, 0 k K 0, = 0, 1,..., ad if a = b = 0 is satisfied, the > τ k, 1 k K 0, = 0, 1,.... It is obvious that τ k > τ k +1, so we have τ 0 = mi{τ k} 0 0 k K 0,=0,1,... if d 1 = d = d, ab < 1, ad a + b > 0. From Lemma.1 ad the cotiuous depedece of τ k o d 1 ad d we have the followig propositio. Propositio.. Assume that (.) holds, ab < 1, ad a + b > 0; the there exists a ϵ(d, a, b, r i, r) such that for ay d 1, d (d ϵ, d + ϵ), τ 0 = mi{τ k} 0 0 k K 0,=0,1,... Suppose that λ k (τ) = µ k (τ) + iσ k (τ) is the root of Eq. (.3 k ) satisfyig µ k (τ k ) = 0, σ k(τ k ) = ±σ k ; the usig the same method as i [1, Theorem 3.] we have the followig trasversality result. Lemma.3. Assume that (.) holds, ab < 1, ad a + b > 0; the µ k (τ k ) > 0 for 0 k K 0 ad = 0, 1,,.... From Lemma.3, Propositio., [1, Theorem 3.3] ad [3, Theorem 3.3.], we have the followig coclusios o the distributio of the roots of Eqs. (.3 k ), k 0. Lemma.4. Assume that (.) holds. 1. Whe ab 1, the all of the roots of Eqs. (.3 k ) (k 0) have egative real parts for τ [0, ).. Whe ab < 1 ad a + b > 0, the for ay d > 0 there exists a ϵ(d, a, b, r i, r) defied i Propositio. such that for ay d 1, d (d ϵ, d + ϵ): (i) If τ [0, τ 0 0 ), the all of the roots of Eqs. (.3 k) (k 0) have egative real parts. (ii) If τ = τ 0 0, the all of the roots of Eq. (.3 0) except ±iσ 0 0 ad Eq. (.3 k) (k 1) have egative real parts. (iii) If τ (τ 0 0, mi(τ 0 1, τ 1 0 )), Eq. (.3 0) has oly oe pair of roots with positive real parts, ad all of the roots of Eqs. (.3 k ) (k 1) have egative real parts. (iv) If τ > mi(τ 0 1, τ 1 0 ), the Eqs. (.3 k) (k 0) have at least two pair of roots with positive real parts. This lemma gives the spectral properties whe d 1 ad d are close to each other, which is complemetary to Theorem 5.1 of Faria [1]. Remarkably i this theorem τ 0 0 equals τ 0 of [1], ad σ 0 0 equals σ 0 of [1]. Here we do ot assume ab(au + bv ) u v as i [1] but we have the additioal assumptio o the diffusio coefficiets d 1 ad d. Spectral properties i Lemma.4 immediately lead to the followig results o the dyamics of system (1.1) (or equivaletly (.1)). Theorem.5. Cosider system (1.1), ad assume that (.) holds. 1. If ab 1, the E is locally asymptotically stable.. If ab < 1 ad a + b > 0, the for ay d > 0 there exists a ϵ(d, a, b, r i, r) defied i Propositio. such that for ay d 1, d (d ϵ, d + ϵ), E is locally asymptotically stable whe τ [0, τ 0 0 ), ad is ustable whe τ > τ 0 0. Furthermore, the system udergoes a Hopf bifurcatio of spatially homogeeous periodic orbits at E whe τ = τ 0 0. From this theorem we kow that whe the diffusio coefficiets for the prey ad predator are very close to each other i the delayed diffusive Lotka Volterra prey predator system, the diffusio terms have o impact o the local stability of the positive equilibrium E. Here we also do ot assume ab(au + bv ) u v as i [1] but we have a additioal assumptio o the diffusio coefficiets d 1 ad d.

6 44 S. Che et al. / Computers ad Mathematics with Applicatios 6 (011) A special case I this sectio we will aalyze system (1.1) (or equivaletly (.1)) i the special case whe a = b = 0 ad d 1 = d = d. It ca be easily verified that arccos 1 d k 4 + π τ k = (1 + r), 1 k K 0, ad = 0, 1,,..., C d k 4 τ 0 = ( + 1)π (1 + r) C, = 0, 1,,.... From [1, Theorem 3.4] ad Lemma.1, we obtai: Theorem 3.1. Assume that (.) holds, a = b = 0, ad d 1 = d = d. 1. Whe d C 0, the all of the roots of Eqs. (.3 k ) (k 1) have egative real parts for τ [0, ).. Whe d C < 0: (i) If arccos τ 1 = 0 1 d (1 + r) π > C d the all of the roots of Eqs. (.3 k ) (k 1) have egative real parts for τ [0, τ 0 0 ]. (1 + r) C = τ 0 0, (3.1) (ii) If arccos 1 d τ 1 = 0 (1 + r) π < C d (1 + r) = τ 0, 0 C (3.) the a Hopf bifurcatio of spatially ihomogeeous periodic solutios occurs at E for system (1.1) ad these spatially ihomogeeous periodic solutios are ustable whe τ is ear τ 1 0. Proof. Whe a = b = 0, d 1 = d = d, the characteristic equatios of system (1.1) are λ + A k τλ + B k τ + C τ e λ(1+r) = 0, (k = 0, 1,,...), A k = d k, B k = d k 4, ad C = u v. The Eq. (.4) becomes ρ 4 + m k ρ + d k 4 C = 0. (3.3) If d C 0, the Eq. (3.3) has o positive roots for k 1, so all of the roots of Eqs. (.3 k ) (k 1) have egative real parts for τ [0, ). If d C < 0, the Eq. (3.3) has a positive root for k = 1. So if arccos 1 d (1 + r) > C d π (1 + r) C, we have τ 0 < τ ad τ 1 < τ k 0 0, 1 k K 0, from Lemma.1. The all of the roots of Eqs. (.3 k ) (k 1) have egative real parts for τ [0, τ 0 1 ]. If arccos 1 d (1 + r) π < C d (1 + r), C we have τ 0 > τ 1 0 0, ad whe τ = τ 1 0, Eqs. (.3 k) (k 0) have o pure imagiary roots except k = 1. From [1, Theorem 3.4], we kow that (.3 k ) has at least two roots with positive real parts whe τ > 0. So whe τ = τ 1 0, Eqs. (.3 k) (k 0) have oly oe pair of simple imagiary roots ad at least two roots with positive real parts. The a Hopf bifurcatio of spatially ihomogeeous periodic solutios occurs at E for system (1.1) whe τ = τ 1 0, ad these spatially ihomogeeous periodic solutios are ustable. Remark 3.. I Theorem 5. of [1], Faria gave a sufficiet coditio for all of the roots of Eqs. (.3 k ) (k 1) to have egative real parts for τ [0, τ 0 0 ] whe a = b = 0. I this theorem we give a sufficiet ad ecessary coditio for all of the roots of Eqs. (.3 k ) (k 1) to have egative real parts for τ [0, τ 0 0 ] with the assumptio d 1 = d = d. We also obtai a sufficiet coditio for the occurrece of Hopf bifurcatio of spatially ihomogeeous periodic solutios.

7 S. Che et al. / Computers ad Mathematics with Applicatios 6 (011) Refereces [1] J. Wu, Theory ad Applicatios of Partial Fuctioal-Differetial Equatios, Spriger, New York, [] X. Ya, Stability ad Hopf bifurcatio for a delayed prey predator system with diffusio effects, Appl. Math. Comput. 19 (007) [3] T. Ereux, Applied Delay Differetial Equatios, i: Surveys ad Tutorials i the Applied Mathematical Scieces, vol. 3, Spriger, New York, 009. [4] F.A. Davidso, S.A. Gourley, The effects of temporal delays i a model for a food-limited, diffusig populatio, J. Math. Aal. Appl. 61 (001) [5] T. Faria, Hopf bifurcatio for a delayed predator prey model ad the effect of diffusio, i: Noliear Aalysis ad its Applicatios to Differetial Equatios, i: Progr. Noliear Differetial Equatios Appl., vol. 43, Birkhäuser, Lisbo, 001, pp [6] Z. Ge, Y. He, Diffusio effect ad stability aalysis of a predator prey system described by a delayed reactio diffusio equatios, J. Math. Aal. Appl. 339 (008) [7] X. Ya, C. Zhag, Asymptotic stability of positive equilibrium solutio for a delayed prey predator diffusio system, Applied Mathematical Modellig 34 (010) [8] Z. Ge, Y. He, Travelig wavefrots for a two-species predator prey system with diffusio terms ad stage structure, Applied Mathematical Modellig 33 (009) [9] T. Yi, X. Zou, Global attractivity of the diffusive Nicholso blowflies equatio with Neuma boudary coditio: a o-mootoe case, J. Differetial Equatios 45 (008) [10] G. Hu, W. Li, Hopf bifurcatio aalysis for a delayed predator prey system with diffusio effects, Nol. Aal. Real World Appl. 11 (010) [11] J.K. Hale, L. Verduy, M. Sjoerd, Itroductio to fuctioal-differetial equatios, i: Applied Mathematical Scieces, vol. 99, Spriger-Verlag, New York, [1] S. Buseberg, W. Huag, Stability ad Hopf bifurcatio for a populatio delay model with diffusio effects, J. Differetial Equatios 14 (1996) [13] T. Faria, Stability ad bifurcatio for a delayed predator prey model ad the effect of diffusio, J. Math. Aal. Appl. 54 () (001) [14] Y. Su, J. Wei, J. Shi, Hopf bifurcatios i a reactio diffusio populatio model with delay effect, J. Differetial Equatios 47 (009) [15] H. Thieme, X. Zhao, A o-local delayed ad diffusive predator prey model, Nol. Aal. Real World Appl. (001) [16] Y. Wag, Asymptotic behavior of solutios for a class of predator prey reactio diffusio systems with time delays, J. Math. Aal. Appl. 38 (007) [17] Y. Kuag, Delay Differetial Equatios with Applicatios i Populatio Dyamics, Academic Press, Bosto, [18] J.W.-H. So, Y. Yag, Dirichlet problem for the diffusive Nicholso s blowflies equatio, J. Differetial Equatios 150 (1998) [19] Y. Su, J. Wei, J. Shi, Bifurcatio aalysis i a delayed diffusive Nicholso s blowflies equatio, Noliear Aal. Real World Appl. 11 (010) [0] J.W.-H. So, J. Wu, Y. Yag, Numerical steady state ad Hopf bifurcatio aalysis o the diffusive Nicholso s blowflies equatio, Appl. Math. Comput. 111 (000)