Diffusive induced global dynamics and bifurcation in a predator-prey system

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1 Li Advances in Difference Equations :33 DOI /s R E S E A R C H Open Access Diffusive induced global dynamics and bifurcation in a predator-prey system Nadia N Li * * Correspondence: lina3718@163.com Department of Mathematics and Statistics Zhoukou Normal University Zhoukou Henan China Abstract In this paper a diffusive Leslie-type predator-prey model is investigated. The existence of a global positive solution persistence stability of the equilibria and Hopf bifurcation are studied respectively. By calculating the normal form on the center manifold the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. Finally our theoretical results are illustrated by a model with homogeneous kernels and one-dimensional spatial domain. Keywords: existence of a global positive solution; global stability; permanence; Lyapunov function; Hopf bifurcation 1 Introduction Bifurcation phenomena are observed and studied in a variety of fields such as engineering 1 ecological chemical 3 biological 46 control 7 8 and neural networks 913. A classical predator-prey model is investigated as follows: x ẋt=x1 x y ax bx1 ẏt=yδ y x. 1.1 The study shows that model 1.1 has complex bifurcation phenomena and dynamic behavior such as the existence of both Hopf bifurcation and Bogdanov-Takens bifurcation. For more detailed results on model 1.1 one can refer to 14. As we all know the spatial heterogeneity of predator and prey distributions is more or less obvious. The predator or prey can flow from higher density regions to lower density ones which is called normal diffusion. And specifically to the best of our awareness few papers which discuss the dynamic behavior of model 1.1 with diffusion have appeared in the literature. This paper attempts to fill this gap in the literature. To achieve this goal we look at the following model: u t = d u 1 u u1 u v x t 0 lπ 0 t >0 au bu1 v t = d v vδ v x t 0 lπ 0 t >0 u u x t0=v x t0=0 u x t lπ=v x t lπ=0 t >0 u0 x=u 0 x 0 v0 x=v 0 x 0 x 0 lπ The Authors 017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made. 1.

2 Li Advances in Difference Equations :33 Page of 5 where u i = u i x t i =1 = denotes the Laplacian operator d x 1 d are diffusion coefficients of prey and predator respectively. is a bounded subset with sufficiently smooth boundary in R n wherer n is an arbitrary positive-integer N-dimensional space. is the Laplacian operator in the region. The no-flux boundary condition means that the statical environment is isolated. In the biological field the study of the stability and bifurcations of the reaction-diffusion models describing a variety of biological phenomena is one of the fundamental problems which can be conducive to understand the dynamic behavior. Recently the stability of the steady state and that of the non-constant steady state solution 170 are investigated. The main contributions of this article are as follows: a the existence of the global positive solution and the persistence of system 1. are discussed in detail; b the Hopf bifurcation analysis of system 1. is provided; c the Lyapunov function is constructed to complete the proof of global stability of the equilibria. The rest of this article is organized as follows. In Section the existence of a global positive solution of 1. as well as the boundedness are considered. In Section 3 thepersistence of the system is investigated. In Section 4the stability of the constant equilibriathe existence of Hopf bifurcations and the stability of a bifurcating periodic solution around the interior constant equilibrium are investigated. In Section 5 the global stability of the constant equilibria is proved by the Lyapunov function method under certain conditions investigated. Finally simulations are presented to verify our theoretical analysis. Existence of a global positive solution of 1. For convenience we introduce some notations. For x y Rwedenotex y : min{x y} x y := max{x y} x : x 0 x :x 0 and extend these notations to real-valued functions. If is a partially ordered vector space we denote its positive cone by := {v : v 0}. Forp 1 p > N/ let X = {L p }. Assume W =u v X anditsnormis defined as W = u p v p.nowmodel1. can be rewritten as an abstract differential equation dz = Az Hz dt z0 = z 0 t 0.1 where z =ux t vx t T z 0 =u 0 v 0 T A = I 0. 0 I with { DA= u v { C } u ν = v } ν =0 and uv u u Hz= au bu1 vδ 1 v u..3

3 Li Advances in Difference Equations :33 Page 3 of 5 Next we prove the existence of the local solution for.1. First we give out the following lemma. Lemma.1 For every z 0 X the Cauchy problem.1 has a unique maximal local solution z 0 T max ; X C 1 0 T max ; X C 1 0 T max ; DA.4 which satisfies the following Duhamel formula for t 0 T max : zt=e ta p z 0 t 0 e tsa p H zs ds.5 where A p is an operator and the closure of A in X. Moreover if T max < then lim sup T max zt =. Proof Since the operator A p is the closure of A in X it generates an analytic condensed strong continuous operator semi-group e ta p.furthermorefort >0wehave e ta p w e t w w X..6 Moreover we observe that H : DA X is locally Lipschitz on a bounded set. Via Theorem 7..1 in 1 we complete his proof. Lemma. For the initial-boundary problem of model 1. the components of its solution uxt vxt are nonnegative. Proof To prove ux t vx t 0 u t = d 1 u u 1 u u v au bu 1 v t = d 1 v cv δ v u.7 with the boundary and initial conditions u ν = v ν =0 u x0=u 0 v x0=v 0. After multiplying.7byu and v respectively and integrating by parts on the domain weobtain 1 d dt u dx d 1 u dx =0 d v dx d v dx =0. 1 dt Hence for t 0 T max u t v t u 0 v 0 =0. Consequently u x t 0 v x t 0. Now we know that u x t v x t is a solution of 1. and according to Lemma.1 we finally get that ux t 0 vx t 0 t 0 T max.nextweprovetheglobalexistence

4 Li Advances in Difference Equations :33 Page 4 of 5 of the positive solution for 1.. Via Lemmas.1 and. we only need to show that the solution of model 1. is uniformly bounded i.e. dissipation. For this we first introduce the following lemma. Lemma.3 Suppose that zx t satisfies the following equation: z = d z z1 z x t >0 t z =0 x t >0 n zx0=z 0 x 0 0 x. Then lim t zx t=1for any x. Theorem.4 The solution ux t vx t of 1. satisfies the following inequalities: lim sup t max ux t 1 x lim sup max vx t δ x. t Proof u t d 1 u u1 u x t >0 u =0 x t >0 n ux0=u 0 x x. By the comparison principle and Lemma.3 we obtain easily the first inequality. Therefore there exist T 1 >0andsufficientlysmall0<ɛ 1suchthatux t 1ɛ for any x and t > T 1. For the second equation we have the following inequality: v t d v vδ v x t >0 1ɛ v =0 x t >0 n vx0=v 0 x x. By the comparison principle for the above 0 < ɛ 1 there exists T > T 1 suchthatfor any t > T vx t δ1 ɛ ɛ. For the arbitrariness of ɛ >0we obtain lim sup max vx t δ t x x. As a result we complete this proof. By Lemmas.1. and Theorem.4wecangetthe following theorem. Theorem.5 System 1. has a unique nonnegative and bounded solution Zx t = ux t vx t such that zx t C 0 X C 1 0 ; X C x DA.

5 Li Advances in Difference Equations :33 Page 5 of 5 3 Permanence In this section we show that any nonnegative solution ux t vx t of 1.liesinacertain bounded region as t for all x. We make the following hypothesis: H 0 δ b <1. Theorem 3.1 If H 0 holds the solution uv of 1. satisfies the following inequalities: lim inf t min x ux t 1 δ b δ lim inf min δ1 vx t t x b. Proof From the first equation of 1.andTheorem3.1wehave u t d 1 u u 1u δ1ɛ By the comparison principle and Lemma.3weobtaineasily lim inf t min x ux t 1 δ b. b ɛ. From the second equation of 1. we have v t d v v δ v 1 δ. b This implies lim inf t a δ min δ1 vx t x. From Theorems.4 and 3.1 we can easily obtain the following theorem. Theorem 3. If H 0 holds model 1. is permanent. 4 The stability of equilibria and the existence of Hopf bifurcations 4.1 The existence of equilibrium of model 1. The system has always constant equilibria E By the direct calculation there exists a unique constant positive equilibrium denoted by E u 0 v 0 if and only if the following cubic equation δ au 3 b a u 1bu 1=0 4.1 holds in the interval 0 1. Note that the third-order algebraic equation 4.1 can have one two or three positive roots in the interval 0 1 which can be evaluated by using the root formula of the third-order algebraic equation. Correspondingly system 1. can have one two or three positive equilibria. Regarding the number of positive equilibria in the interval 0 1 of gives the result in detail as follows. Lemma Let = δ b a 3ab 1and =4 3 7a 9a1 b δ b a δ b a3.

6 Li Advances in Difference Equations :33 Page 6 of 5 H1 If >0 then model 1. has a unique positive equilibrium E which is an elementary and anti-saddle equilibrium. H If =0and >0 then the model has two different positive equilibria: an elementary anti-saddle equilibrium E and a degenerate equilibrium E 1. H3 If =0and =0 the model has a unique positive equilibrium E 3 1b 3δ 1b which is a degenerate equilibrium. H4 If <0 δ = a b 3a1 b and a < b <1 then model 1. has three different positive equilibria E1 E and E 3 which are all elementary equilibria and E3 is a saddle. Remark We further consider the following condition H1. We suppose H1 is satisfied throughout the rest of this paper. 4. The existence of Hopf bifurcation In this section we assume that H1 holds. We first assume that E u 0 v 0 isanypointof system 1.. Now we define the real-valued Sobolev space X := { u v H 0 lπ :ux v x x=0lπ =0 } and the complexification of X X C := X ix = {x 1 ix : x 1 x X}. The linearized system of 1.atanypointEu 0 v 0 hastheform L = A B δ δ 4. with the domain D Ls = X C where A =1u 0 u 0v 0 bu 0 au 0 bu 0 1 B = u 0 au 0 bu 0 1. Obviously B >0. FromWu3 we obtain that the characteristic equation of 4. is λy d y L e λ y =0 y domd y It is well known that the eigenvalue problem ϕ = ϕ x 0 lπ:ϕ 0 = ϕ lπ=0 has eigenvalues n = n l n =01...withcorrespondingeigenfunctions ϕx n = cos nπ l n =01... Substituting y = y 1n y k=0 n cos nπ l

7 Li Advances in Difference Equations :33 Page 7 of 5 into the characteristic Eq. 4.3 we have A d 1 n B l y 1n δ δ d 1n = λ y l n y 1n y n n =01... Therefore the characteristic Eq. 4.3 is equivalent to detλi M n L=0 4.4 where I is the identity matrix and M n = n l diag{d 1 d } n =01...Itfollowsfrom 4.4 that the characteristic equations for the equilibrium u 0 v 0 are the following sequence of quadratic transcendental equations: n λ τ=λ T n λ D n =0 4.5 where T n =d 1 d n δ A l D n = d 1d n 4 d l 4 1 δ d A n l The eigenvalues are given by δ Bδ A. 4.6 λ 1 = T n ± Tn 4D n n = We make the following hypothesis with fixing δ: H 5 A < ρδ and H 6 Bδ < A < d 1 d δ where ρ = min{1 d 1 d B }. Lemma 4. i Suppose H 1 and H 5 are satisfied. Then all the roots of Eq.4.5 have a negative real part. ii Suppose H 1 and H 6 hold then Eq.4.5 has at least one positive real part. Proof i Obviously by H 5 T n <0for n =01...weknowalltherootsofEq.4.5have negative real parts. ii If H 6 holds we know that D n is monotone increasing with regard to nand lim n D n = and D 0 <0. There exists an integer n 0 1 such that D n <0when n =01...n 0.HencethereareD n <0 Tn 4D n > Tn λ 1 >0and λ <0for n =01...n 0 and D n >0 Tn 4D n < Tn λ 1 >0and λ >0for n n 0.These complete the proof of i and ii. From the discussion we have the following theorem. Theorem 4.3 i Suppose H 1 and H 5 hold the equilibrium E is stable.

8 Li Advances in Difference Equations :33 Page 8 of 5 ii Suppose H 1 and H 6 hold the equilibrium E is unstable. Next we consider the occurrence of Hopf bifurcation of E by regarding A as a bifurcation parameter with fixing δ. In the light of 4.7 it is known that 4.5 has purely imaginary roots if and only if A = A n =d 1 d n δ n = l and D n >0holds. From 4.8 we know that A n is monotone increasing with regard to n and lim A n = and A 0 = δ >0. n Hence there is A n >0for n =01...Bringing A n into D n we get D n = d n 4 d n B δ l 4 l 1 δ. 4.9 If D 0 = δ B 1>0holds then there exists an integer n 1 1 such that D n >0when n = 01...n 1 and D n 0 when n n 1. Let n =n 1 and = {A n 1 > A n > A n 3 > > A 0 }. Lemma 4.4 If B 1>0is satisfied then Eq.4.5 has purely imaginary roots if and only if A. Proof Let λ n A=α n Ai n A n =01...n be the roots of Eq. 4.8satisfyingα n A= 0 n A= D n A. If B 1>0holdsweknowthatD n is monotone decreasing with regard to n andlim n D n = and D 0 > 0. There exists an integer n 1 1suchthatD n >0 when n =01...n. These complete the proof of Lemma 4.4. From expression 4.7we have the following conclusion. Lemma 4.5 Suppose B 1>0is satisfied then all the roots of Eq. 4.5 with A = A n n have at least one root with a positive real part except the imaginary roots ±i D n A n n 01...n. Proof If B 1>0holdsweknowthatD n is monotone decreasing with regard to n and lim n D n =.HencethereareD n <0Tn 4D n > Tn λ 1 >0andλ <0forn > n.in combination with Lemma 4.4 we completetheproof oflemma 4.5. Based on the above analysis we have the following theorem. Theorem 4.6 Suppose B 1>0issatisfied. Then model 1. undergoes a Hopf bifurcation at the origin when A = A n for 0 n n. Moreover: i The bifurcation periodic solution from A = A 0 is spatially homogeneous which coincides with the periodic solution of the corresponding ODE system. ii The bifurcation periodic solution from A = A n is spatially non-homogeneous for 0<n n.

9 Li Advances in Difference Equations :33 Page 9 of Direction and stability of Hopf bifurcation From the previous section we know that model 1.undergoesaHopfbifurcationatthe origin when A = A n n =01...n wheren and A n are defined as in 4.8. In this section we study the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by applying the center manifold theorem and the normal form theorem of partial differential equations. We first transform E of 1. to the origin via the translation: û = u u 0 ˆv v 0 and drop the hats for simplicity of notation then 1.is transformed into u t = d 1 u f δ u v v t = d 4.10 v gδ u v where f A u v=u u 0 u u 0 u u 0 v v 0 au u 0 bu u 0 1 ga u v=v v 0 δ v v 0. u u 0 So we have q := a 0 b 0 = 1 δ q := δi The straightforward computation yields f uuu = v 01abu 0 b u 0 8au 0 bb f vv =0 g uu = δ u 0 g uv = δ u 0 g uuv =4 δ u 0 g uvv = u 0 g vv = Moreover a 0 b 0 = f uu = v 0bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu 01 au 0 bu 01 3 f uv = bu 0 u 0 au 0 bu 01 bu f uuv = 0 u 0bu 0 4au 0 b a u 0 bu 0 1 au 0 bu 01 3 u 0. 1 lπ lπiδδ. 4v 04abu 3 au 0 bu 01 0 b u 0 4au 0 bu 0 au 0 bu v 0abu 0 b u 0 au 0 b au 0 bu 01 4 Q qq = c n d n cos nx = f uu a n f uva n b n f vv b n l g uu a n g uva n b n g vv b n Q q q = e n f n cos nx = f uu a n f uv a n b n a n b n f vv b n l g uu a n g uv a n b n a n b n g vv b n Q qq q = g n h n cos nx = f uuu a n b n f uuv a n b n a n b nf uvv b n a n b n a nf vvv b n b n l. g uuu a n b n g uuv a n b n a n b ng uvv b n a n b n a ng vvv b n b n

10 Li Advances in Difference Equations :33 Page 10 of 5 So c 0 = f uu f uv b 0 f vv b 0 = v 0bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu 01 au 0 bu 01 3 bu 0 u 0 δ au 0 bu 01 δi d 0 = g uu g uv b 0 g vv b δ 0 =u 0 δ u 0 δ δi e 0 = f uu f uv b 0 b 0 f vv b 0 bu 0 u 0 δ au 0 bu 01 δi f 0 = g uu g uv b 0 b 0 g vv b 0 = v 0bu 0 1 au 0 bu 01 u 0v 0 bu 0 4au 0 b au 0 bu 01 3 δ δi = δ u 0 δ u 0 δ δi δ δi u 0 δ g 0 = f uuu f uuv b 0 b 0 f uvv b 0 b 0 f vvv b 0 b 0 bu 0 u 0bu 0 4au 0 b a u 0 bu 0 1 δi =4 δ δ u 0 au 0 bu 01 3 h 0 = g uuu g uuv b 0 b 0 g uvv b 0 b 0 g vvv b 0 b 0 =6 δ 4 δ δ u 0 u 0 δ u δi 0 δi δ δi δ δi. δi δ δi 4.11 Then we have q Q qq = 1 q Q qq = 1 lπ c0 lπiδδ d 0 = 1 f uu f uv b 0 f vv b 0 g iδδ uu g uv b 0 g vv b 0 = 1 v 0bu 0 1 u 0v 0 bu 0 au 0 b bu au 0 bu 01 au 0 bu u 0 δ au 0 bu 01 δ iδδ u 0 δ δ u 0 δi δi q Q q q = 1 lπ e0 lπiδδ f 0 = 1 f uu f uv b 0 b 0 f vv b 0 g iδδ uu g uv b 0 b 0 g vv b 0 = 1 v 0bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu 01 au 0 bu 01 3 bu 0 u 0 au 0 bu 01 δ δi iδδ δ u 0 δ lπ c 0 iδδ = 1 f uu f uv b 0 f vv b 0 d 0 u 0 δ δi δ δi δ δi u 0 δ δi g iδδ uu g uv b 0 g vv b 0 = 1 v 0bu 0 1 u 0v 0 bu 0 au 0 b bu au 0 bu 01 au 0 bu u 0 δ au 0 bu 01 δi δ iδδ u 0 δ δ u 0 δi q Q qq q = 1 lπ g0 iδδ h 0 = 1 f uuu f uuv b 0 b 0 f uvv b 0 b 0 f vvv b 0 b 0 g iδδ uuu g uuv b 0 b 0 g uvv b 0 b 0 g vvv b 0 b 0 = 1 4 δ u 0 6 δ 4 δ iδδ u 0 u 0 δ u δi δ 0 bu 0 u 0bu 0 4au 0 b a u 0 bu 0 1 δi au 0 bu 01 3 δ δ δi δ δi δi δ δi

11 Li Advances in Difference Equations :33 Page 11 of 5 and H 0 = H 11 = c 0 d 0 e 0 f 0 q a 0 Q qq q a 0 Q qq =0 b 0 q Q q q a 0 b 0 b 0 q a 0 Q q q b 0 =0 which implies W 0 = W 11 =0thatistosay q QW 11 q = q QW 0 q =0. Therefore c 1 A 0 ={ i q Q qq q Q q q 1 q Q qq q } Rec 1 A 0 = Re{ i q Q qq q Q q q 1 q Q qq q } Imc 1 A 0 = Im{ i q Q qq q Q q q 1 q Q qq q } T =Imc 1 A 0 Rec 1δ 0 α δ 0 δ 0 where A 0 = 1 > 0 then we have the following theorem. Theorem 4.7 Suppose B 1>0is satisfied. The Hopf bifurcation at A = A 0 is backward resp. forward; The bifurcation periodic solution from A = A 0 is asymptotically stable resp. unstable if Rec 1 A 0 < 0 resp.>0; The bifurcation periodic solution from A = A 0 is increasing resp. decreasing if T >0 resp.<0. When 0 < j n fora j we have the following theorem. Theorem 4.8 Suppose B 1>0is satisfied. The Hopf bifurcation at A = A j for 0<j n is backward resp.forward; The bifurcation periodic solution from A = A j is asymptotically stable resp. unstable if Rec 1 A j < 0 resp.>0; The bifurcation periodic solution from A = A j is increasing resp. decreasing if T >0 resp.<0for 0<j n where Rec 1 A j is defined in Appendix. 5 Global stability of equilibria In this section we continue to study the global stability of equilibria. Theorem 5.1 Suppose that the hypothesis of Lemma.3 holds then the positive constant equilibrium E u 0 v 0 is globally asymptotically stable if au 0 v 0 Qp v 0 < 1 5.1a a b 1p 1 1 δ b v 0 1 δ b Qp δ 4 1 δ b u 0v 0 p >0 and 5.1b

12 Li Advances in Difference Equations :33 Page 1 of 5 δ 1 δ b 1 b > 0 5.1c where Q =a1 δ b b1 δ b 1. Proof Let ux t vx t be any solution of model 1.. We introduce the Lyapunov function { Vu v= u u 0 ln u 1 } v v 0 ln vv0 dx. u 0 Now we take the derivative of V with regard to t along the trajectory of model 1.. Then V = V 1 V where V 1 = d 1 1 u 0 u { d1 u 0 u udx d 1 v 0 v d v 0 v u v vdx } dx 0 and uv V = u u 0 1u dx v au 0 bu 1 v 1 δ v dx. u Let V = V 1 V 1 where V 1 = u u 0 u 0 u =u u 0 = = u 0 v au 0 bu 0 1 uv dx au bu 1 u 0 u u 0v 0 au bu 1uvau 0 bu 0 1 pp dx { u u 0 u u 0 u0 v pp 0 au bu 1 uv au 0 bu 0 1 } dx { u u 0 1 au0 v pp 0 u v 0 u u 0 au 0 u buu 0 u u u 0 v v 0 } dx = {u u 0 1 au 0v 0 u v 0 au 0 u buu } 0 u u u pp pp 0 v v 0 and V = 1 v v 0 δ v u = 1 v v 0 dx δ v v 0 v 0 u dx

13 Li Advances in Difference Equations :33 Page 13 of 5 = 1 v v 0 δ v v 0 u = 1 v v 0 δ v v 0 v 0 u u δ = v v 0 v v 0 v 0 dx u u = v v 0 δ v v 0 u v 0 u = v v 0 v v 0 δu v 0 u u = v v 0 v v 0 δu δu 0 u u = v v 0 u v 0 dx u dx dx dx dx δ u v v 0u u 0 dx where p = au bu1p = au 0 bu 0 1a 11 =1 au 0v 0 uv 0 pp a 1 = a 1 = 1 { au 0 ubu 0uu pp δ u } a = 1 u. Next we calculate V as follows: { } a 11 a 1 V = u u 0 v v 0 u u 0 v v 0 T dx. 5. a 1 a It is obvious that dv < 0 if and only if the matrix in integrand 5. is positive definite dt equivalent to a 11 >0andφu v =a 11 a a 1 a 1 >0whereφu v =a 11 a a 1 a 1 = 1 au 0v 0 uv 0 pp 1 u 1 4 { au 0 ubu 0uu pp au 0 ubu 0uu pp δ u δ u } > 0. By condition 5.1a- 5.1c we know a 11 >0andφu v>0.consequentlye is globally asymptotically stable. Thuswecompletethisproof. Next we give out the conditions of the global asymptotic stability of the boundary equilibrium E Theorem 5. The boundary equilibrium E of 1. is globally asymptotically stable if 1 a 1 <4δ δ 5.3 holds. Proof Define V 0 u v= {u 1ln u v} dx. Taking the derivative of V 0 with regard to t along the trajectory of model 1. we have V 0 = V 1 0 V 0

14 Li Advances in Difference Equations :33 Page 14 of 5 where V 1 0 = {d v d } d 1 u u dx dx 0. u u Furthermore we calculate V0 and reach V 0 = { u 1 1u { = u 1 According to condition 5.3 dv 0 dt globally asymptotically stable. uv v δ v au bu 1 u u au bu 1 δ u } dx u 1v δ u v } dx. <0exceptatE 0. By LaSalle s invariance principle E 0 is 6 Numerical simulations In this section to confirm our analytical results found in the previous section some examples and numerical simulations are presented. We use Matlab 01a to simulate and plot numerical graphs. Example 1 If we choose a =0.5b =1δ =0.5 =0d 1 =1d = 1 the conditions of Theorem 3. are satisfied for system 1.. We have that the axial equilibrium E1 0 is globally stable see Figure 1. Example If we choose a =1b =1 =d 1 =1d =10δ = 1 the conditions of Theorem 3.1 are satisfied for system 1.. The positive equilibrium E is globally asymptotically stable see Figure. Example 3 If we choose a =10b =0.1δ =0. =0.01d 1 =0.1d =10thenweknow that system 1. has a unique positive homogeneous equilibrium E which is asymptotically stable see Figure 3. The system undergoes Hopf bifurcation at the equilibrium E see Figure 4. By the formulas derived in the previous section we get c 1 A e e 00 i with the initial value a b Figure 1 Numerical simulations of system 1.fora =0.5 =1d 1 =1d = 100 b =1δ =1..The positive equilibrium E 1 0 of 1. is globally asymptotically stable.

15 Li Advances in Difference Equations :33 a Page 15 of 5 b Figure Numerical simulations of system 1. for a = 1 = d1 = 1 d = 10 b = 1 δ = 1. The positive equilibrium E of 1. is globally asymptotically stable with the initial value a b Figure 3 Numerical simulations of system 1. for a = 7 = 0.01 d1 = 0.1 d = 10 b = 0.1 δ = 0.. The positive equilibrium E is locally asymptotically stable with the initial value a b Figure 4 Numerical simulations of system 1. for a = 10 = 0.01 d1 = 0.1 d = 10 b = 0.1 δ = 0.. The positive equilibrium E of 1.3 becomes unstable and there exist spatially homogeneous periodic solutions with the initial value Conclusions Based on model. we propose a diffusive prey-predator model with Leslie-type sigmoidal functional response and subject to the Neumann boundary conditions which is in the form of model.. The dynamic behavior of the model is investigated. Some important qualitative properties such as the existence of a global positive solution persistence

16 Li Advances in Difference Equations :33 Page 16 of 5 the local stability and global stability of the equilibria are obtained. Hopf bifurcations are explored by analyzing the characteristic equations. Moreover the formulas determining the direction and stability of the bifurcating periodic solutions are derived by using the center manifold and the normal form theory of partial functional differential equations. Some numerical simulations are carried out to support our theoretical analysis. In the future some challengeable attempts to study the Hopf bifurcation steady state solution and Turing-Hopf bifurcation of fractional diffusive in a predator-prey system with or without time delay will be proceeded. Appendix In this section we follow the bifurcation formula 4 to determine the bifurcation direction of the spatially non-homogeneous periodic solutions found in Theorem 4.8. When A = A H j± j N we calculate ReC 1A j. We set q := cos j l x a j b j q := cos j l x T = cos j l x a j b j T = cos j l x 1 δ δi d j l T iδ lπδ iδiδ d j lπδ 3 From 5 we know q Q qq = q Q q q = q Q qq =0whenj N it follows that we calculate q Q w0 q q Q w11q and q C qq q. It is straightforward to compute that ij I L j A j 1 =α1 iα 1 ij δ 4d j B l δ i j A j 4d 1j l ij I L 0 A j 1 =α3 iα 4 1 ij δ B l l δ l where α 1 = δ 4d j A j 4d 1j 4j Bδ / T. i j A j α = δ 4d j A l j 4d 1j l α 3 = δa j 4j Bδ / α 4 = δ A j u. Then we get w 0 = 1 α 1 iα w 11 = 1 δ 4d j l α 5 ij δ 4d j l B δ c n d n i j A j 4d 1j l 1 ij δ B c j δ α 3 iα 4 i j A j d j B e n δ A j 4d 1j cos j f l n l x 1 α 6 cos j l x δ B δ A j c j d j

17 Li Advances in Difference Equations :33 Page 17 of 5 where α 5 =δ 4d j A l j 4d 1j Bδ / α l 6 = δa j Bδ /. c j = f uu f uv b j f vv b j = v 0bu 0 1 au 0 bu 01 u 0v 0 bu 0 4au 0 b au 0 bu 01 3 bu 0 u 0 au 0 bu 01 δ δi d j l = v 0bu 0 1 au 0 bu 01 u 0v 0 bu 0 4au 0 b au 0 bu 01 3 bu 0 u 0 au 0 bu 01 δ i = v 0bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu 01 au 0 bu 01 3 iδ bu 0 u 0 au 0 bu 01 d j = g uu g uv b j g vv b j = δ u 0 4 δ = δ u 0 4 δ = δ u 0 δ u 0 δi d j l u 0 u 0 δ i δ δi d j l u 0 δ i 4δ 3 δ4 u 0 u 0 e j = f uu f uv b j b j f vv b j = v 0bu 0 1 au 0 bu 01 u 0v 0 bu 0 4au 0 b au 0 bu 01 3 bu 0 u 0 au 0 bu 01 δ δi d j l bu 0 u 0δ au 0 bu 01 4iδ 4 u 0 δ δi d j l = v 0bu 0 1 au 0 bu 01 u 0v 0 bu 0 4au 0 b au 0 bu 01 3 f j = g uu g uv b j b j g vv b j = δ u 0 4 δ = δ u 0 δ u 0 δi d j l 8δ 3 δ 4 u 0 u 0 δ δi d j l u 0 g j = f uuu f uuv b j b j f uvv b j b j f vvv b j b j 4iδ 3 u 0 4bu 0 u 0δ au 0 bu 01 δ δi d j l = v 01abu 0 b u 0 8au 0 bb 4v 04abu 3 au 0 bu 01 0 b u 0 4au 0 bu 0 4v 0abu au 0 bu b u 0 au 0 b au 0 bu 01 4 bu 0 u 0bu 0 4au 0 b a u 0 bu 0 1 au 0 bu 01 3 δi d j l = v 01abu 0 b u 0 8au 0 bb 4v 04abu 3 au 0 bu 01 0 b u 0 4au 0 bu 0 au 0 bu 01 3 bu 0 u 0bu 0 4au 0 bδ 3i a u 0 bu 0 1 au 0 bu 01 3 h j = g uuu g uuv b j b j g uvv b j b j g vvv b j b j =6 δ 4 δ δ δ u 0 u 0 δi d j l δi d j l u 0 =6 δ 4δ3 3i δ4 i u 0 u 0 u 0 δ δ δi d j l 4v 0abu 0 b u 0 au 0 b au 0 bu 01 4 δ δi d j l δ δi d j l and f u u f u v g u u g u v g v v f u uu f u uv g u uu g u uv g u vv are given in 4.6. Then we have f u uξ f uv η f uv b j ξ Q w0 q = cos jx g uu ξ g uv η g uv b j ξ l cos jx l f uu τ f uv χ f uv b j τ cos jx g uu τ g uv χ g uv b j τ l and Q w11 q = f uu ξ f uv η f uv b j ξ cos jx g uu ξ g uv η g uv b j ξ l cos jx l f uu τ f uv χ f uv b j τ cos jx g uu τ g uv χ g uv b j τ l

18 Li Advances in Difference Equations :33 Page 18 of 5 with ξ = α 1 iα i α1 α j δ 4d j l η = α 1 iα δ α1 α c j i j A j 4d 1j l τ = α 3 iα 4 α3 α 4 ij δc j Bd j χ = α 3 iα 4 δ α3 α 4 c j i j A j d j ξ = 1 δ 4d j e α 5 l j Bf j η = 1 δ α 5 e j A j 4d 1j f l j τ = 1 δe j Bf j α 6 χ = 1 δ α 6 e j A j f j. c j Bd j d j Notice that for any j N lπ 0 cos jx = lπ l lπ 1nx 0 cos cos jx = lπ l l 4 lπ 0 cos3 jx = 3π l 8 sowe have lπ q Q w0 q = cos jx q l Q w0 q dx 0 = lπ 4 lπ lπ q Q w11 q = cos jx jx l cos l 0 = lπ 4 a j f uuξ f uv η f uv b j ξ b j g uuξ g uv η g uv b j ξ a j f uuτ f uv χ f uv b j τ b j g uuτ g uv χ g uv b j τ q Q w11 q dx a j f uu ξ f uv η f uv b j ξ b j g uu ξ g uv η g uv b j ξ a j f uu τ f uv χ f uv b j τ b j g uu τ g uv χ g uv b j τ lπ q lπ C qq q = cos 3 jx q l C qq q = 3lπ 8 δ 0 δ 3 a j g j b j h j. Since lπa j = iδ lπ b j = iδiδ d j l it follows that Re q 3 C qq q = 16 f uuu δ 3 f uuv δ δ δ g uuv δ g δ 3 uuu δδ δ g uvv

19 Li Advances in Difference Equations :33 Page 19 of 5 and Re q Q w0 q Re q Q w11 q lπ 1 = 4 lπ 1 4 lπ 1 f uv ξ R 4 g δ 3 uu ξ R g uv η R f uuξ R f uv η R δ δ f δ uuξ I f uv η I g δ 3 uu ξ I g uv η I δ 1 δ δ 1 δ f uv ξ I 1 δ δ lπ δ δ δ δ g uv ξ R 4 δ 3 δ 3 δ δ δ δ g uv ξ I δ 3 δ 3 lπ 1 f uuτ R f uv χ R δ g δ 3 uu τ R g uv χ R lπ 1 δ f δ uuτ I f uv χ I g δ 3 uu τ I g uv χ I lπ 1 δ f uv τ R 1 δ δ 1 δ f uv τ I 1 δ δ lπ δ δ δ δ g uv τ R δ 3 δ 3 δ δ δ δ g uv τ I δ 3 δ 3 f uuξ R f uv η R δ g δ 3 uu ξ R g uv η R lπ 1 4 δ f uu ξ δ I f uv η I g δ 3 uu ξ I g uv η I lπ 1 δ f uv ξ R 4 1 δ δ 1 δ f uv ξ I 1 δ δ lπ δ g uv ξ δ δ δ R 4 δ 3 δ 3 δ g uv ξ δ δ δ I δ 3 δ 3 lπ 1 f uuτ R f uv χ R δ g δ 3 uu τ R g uv χ R lπ 1 δ f δ uu τ I f uv χ I g δ 3 uu τ I g uv χ I lπ 1 = 4

20 Li Advances in Difference Equations :33 Page 0 of 5 lπ 1 δ f uv τ R 1 δ δ 1 δ f uv τ I 1 δ δ lπ δ δ δ δ g uv τ R δ 3 δ 3 g uv τ I δ δ 3 δ δ δ 3 δ where we denote = δ d j l precisely and define Ɣ R := Re Ɣ Ɣ I := Im Ɣ for Ɣ = ξ η τ χ. More α 1 ξ R = { α1 α j δ 4d j l δ bu 0 u 0 au 0 bu 0 1 v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu δ bu 0 u 0δ au 0 bu 0 1 α α 1 α B 4δ 3 u 0 δ4 u 0 u 0 { j v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu bu 0 u 0δ au 0 bu 0 1 δ 4d j δ bu 0 u 0 l au 0 bu 0 1 4iδ 4 } B u 0 α ξ I = { α1 α j δ 4d j η R = l 4iδ 3 u 0 δ bu 0 u 0 au 0 bu 0 1 v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu δ bu 0 u 0δ au 0 bu 0 1 α 1 α 1 α B 4δ 3 u 0 δ4 u 0 u 0 { j v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu bu 0 u 0δ au 0 bu 0 1 δ 4d j δ bu 0 u 0 l au 0 bu 0 1 4iδ 4 } B u 0 { δ α 1 α 1 α 4iδ 3 u 0 v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu } }

21 Li Advances in Difference Equations :33 Page 1 of 5 η I = bu 0 u 0δ au 0 bu 0 1 4δ 4 j u 0 4δ 3 u 0 } 4δ 3 u 0 δ4 u 0 { δ α α 1 α j δ u 0 A 4d 1j l iα α 1 α { δ δ bu 0 u 0 au 0 bu 0 1 4δ 3 u 0 δ4 u 0 A j 4d 1j δ l } u 0 4δ 4 u 0 4δ 3 u 0 v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu bu 0 u 0δ au 0 bu 0 1 4δ 4 j u 0 4δ 3 u 0 A j 4d 1j δ l u 0 { α 1 δ δ bu 0 u 0 α1 α au 0 bu 0 1 j δ 4δ 3 u 0 u 0 δ4 u 0 A j 4d 1j 4δ 4 l u 0 4δ 3 u 0 4δ 3 u 0 δ4 u 0 } { α 3 4δ j bu 0 τ R = u 0 α3 α 4 au 0 bu 0 1 δ v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu bu 0 u 0δ au 0 bu B δ 0 1 u 0 α 4 α 3 α 4 } 4δ 3 u 0 δ4 u 0 { j v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu bu 0 u 0δ au 0 bu 0 1 iδ bu 0 δ u 0 au 0 bu 0 1 α 4 τ I = { α3 α 4 j δ B δ bu 0 u 0 au 0 bu 0 1 4δ 4 u 0 v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu δ 3 } u 0 }

22 Li Advances in Difference Equations :33 Page of 5 bu 0 u 0δ au 0 bu 0 1 α 3 α 3 α 4 B δ u 0 4δ 3 u 0 δ4 } u 0 { j v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu bu 0 u 0δ au 0 bu 0 1 iδ bu 0 δ u 0 au 0 bu 0 1 { α 3 δ χ R = α3 α 4 bu 0 u 0δ au 0 bu 0 1 B 4iδ 4 u 0 4iδ 3 } u 0 v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu δ 4 j u 0 4δ 3 u 0 A j δ 4δ 3 u 0 u 0 δ4 } u 0 { α 4 δ δ bu 0 u 0 α3 α 4 au 0 bu 0 1 j δ 4δ 3 u 0 u 0 δ4 u 0 4iδ 4 } A j χ I = u 0 { δ α 4 α 3 α 4 4iδ 3 u 0 v 0bu 0 1 au 0 bu 0 1 u 0v 0 bu 0 4au 0 b au 0 bu bu 0 u 0δ au 0 bu 0 1 4δ 4 j u 0 4δ 3 u 0 A j δ 4δ 3 u 0 u 0 δ4 } u 0 { α 3 δ δ bu 0 u 0 α3 α 4 au 0 bu 0 1 j δ 4δ 3 u 0 u 0 δ4 u 0 4iδ 4 } A j. u 0 4iδ 3 u 0 So far we have Re C 1 A j = Re q 1 Q w11 q = Re q Q w0 q { lπ 1 4 f uuξ R f uv η R δ δ 3 1 Re q C qq q g uu ξ R g uv η R

23 Li Advances in Difference Equations :33 Page 3 of 5 lπ 1 4 δ f uu ξ δ I f uv η I g δ 3 uu ξ I g uv η I lπ 1 δ f uv ξ R 4 1 δ δ 1 δ f uv ξ I 1 δ δ lπ δ g uv ξ δ δ δ R 4 δ 3 δ 3 δ g uv ξ δ δ δ I δ 3 δ 3 lπ 1 f uuτ R f uv χ R δ g δ 3 uu τ R g uv χ R lπ 1 δ f δ uu τ I f uv χ I g δ 3 uu τ I g uv χ I lπ 1 δ f uv τ R 1 δ δ 1 δ f uv τ I 1 δ δ lπ δ δ δ δ g uv τ R δ 3 δ 3 δ δ δ δ } g uv τ I δ 3 δ 3 1 { lπ 1 4 f uuξ R f uv η R δ g δ 3 uu ξ R g uv η R lπ 1 4 δ f δ uuξ I f uv η I g δ 3 uu ξ I g uv η I lπ 1 δ f uv ξ R 4 1 δ δ 1 δ f uv ξ I 1 δ δ lπ δ δ δ δ g uv ξ R 4 δ 3 δ 3 δ δ δ δ g uv ξ I δ 3 δ 3 lπ 1 f uuτ R f uv χ R δ g δ 3 uu τ R g uv χ R lπ 1 δ f δ uuτ I f uv χ I g δ 3 uu τ I g uv χ I lπ 1 δ f uv τ R 1 δ δ 1 δ f uv τ I 1 δ δ

24 Li Advances in Difference Equations :33 Page 4 of 5 lπ g uv τ R δ δ 3 δ δ δ g uv τ I δ 3 1 { 3 16 f uuu 3 δ 3 16 f uuv δ δ 16 δ 3 16 g uuv δ δ 3 δ δ 3 δ δ } δ g δ 3 uuu δδ δ g uvv }. Thus the bifurcating periodic solution is supercritical resp. subcritical if 1 α A H j ReC 1A H j < 0 resp. > 0. Acknowledgements This work was supported by the National Natural Science Foundation of China No the Natural Science and Technology Foundation of Henan Province No.15A the Education Department Foundation of Henan Province No. 15A and 13A and the Research Innovation Project of ZhouKou Normal University no. ZKNUA Declarations The author declares that there is no conflict of interests regarding the publication of this paper. Competing interests The author declares that she has no competing interests. Authors contributions The author contributed equally to the writing of this paper. The author read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 14 March 017 Accepted: 10 August 017 References 1. Kalmár-Nagy T Stépán G Moon FC: Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dyn Rietkerk M Dekker SC de Ruiter PC van de Koppel J: Self-organized patchiness and catastrophic shifts in ecosystems. Science Field RJ Noyes RM: Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys Angeli D Ferrell JE Sontag ED: Detection of multistability bifurcations and hysteresis in a large class of biological positive-feedback systems. Proc. Natl. Acad. Sci. USA Xiao M Cao J: Hopf bifurcation and non-hyperbolic equilibrium in a ratio-dependent predator-prey model with linear harvesting rate. Math. Comput. Model Huang C Cao J Xiao M Alsaedi A Alsaadi FE: Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders. Appl. Math. Comput Xu W Hayat T Cao J Xiao M: Hopf bifurcation control for a fluid flow model of Internet congestion control systems via state feedback. IMA J. Math. Control Inf Xu W Cao J Xiao M: Bifurcation analysis of a class of n 1-dimension Internet congestion control systems. Int. J. Bifurc. Chaos Appl. Sci. Eng Cheng Z Wang Y Cao J: Stability and Hopf bifurcation of a neural network model with distributed delays and strong kernel. Nonlinear Dyn Xu W Cao J Xiao M Ho DWC Wen G: A new framework for analysis on stability and bifurcation in a class of neural networks with discrete and distributed delays. IEEE Trans. Cybern Huang C Meng Y Cao J Alsaedi A Alsaadi FE: New bifurcation results for fractional BAM neural network with leakage delay. Chaos Solitons Fractals Huang C Cao J Xiao M: Hybrid control on bifurcation for a delayed fractional gene regulatory network. Chaos Solitons Fractals Huang C Cao J Xiao M Alsaedi A Hayat T: Bifurcations in a delayed fractional complex-valued neural network. Appl. Comput. Math Huang J Ruan S Song J: Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response. J. Differ. Equ

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