Traveling waves of some Holling–Tanner predator–prey system with nonlocal diffusion

Applied Mathematics and Computation 338 (2018) 12–24

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Traveling waves of some Holling–Tanner predator–prey system with nonlocal diffusion Hongmei Cheng a,∗, Rong Yuan b a b

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, PR China School of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR China

a r t i c l e

i n f o

Keywords: Traveling waves Predator–prey model Nonlocal diffusion Schauder’s fixed point theorem Coexistence state

a b s t r a c t This paper is devoted to establish the existence and non-existence of the traveling waves for the nonlocal Holling–Tanner predator–prey model. By applying the Schauder’s fixed point theorem, we can obtain the existence of the traveling waves. Moreover, in order to prove the limit behavior of the traveling waves at infinity, we construct a sequence that converges to the coexistence state. For the proof of the nonexistence of the traveling waves, we use the property of the two-sided Laplace transform. Finally, we give the effect of the nonlocal diffusion term for the traveling waves. © 2018 Elsevier Inc. All rights reserved.

1. Introduction In recent years, the dynamic relationship between predators and their prey has been one of the dominant themes in ecology due to its universal existence and importance. Researchers have established many models due to the different functional response to predation, see [22–24]. The classical Lotka–Volterra model and its modified models have been studied for the stability of equilibria and the existence of traveling waves, see [6,14,17,20,28–30,33] et al. The properties of the model with the Leslie–Gower functional response and its modified models have been discussed, for instance [1,21,25,36], et al. For the predator–prey model

⎧ du(t ) ⎪ ⎨ dt = u(t )(1 − u(t )) − (u(t ))v(t ),   dv(t ) v(t ) ⎪ ⎩ = rv(t ) 1 − , dt u(t )

(1.1)

the second equation means that the intrinsic population growth rate r affects not only the potential increase of the population but also its decrease. If v is greater than u, the population will decline, and the speed of its decline is directly proportional to the intrinsic growth rate. It seems to be a contradiction, but it is realistic since that species of small body size and early maturity have high intrinsic growth rates and also have low survival rates and short lives. Typical example of the functional response (u) is given by Holling-type functional response in [15]. Hsu and Huang [16] have studied the global stability property with three types of functional responses. Tanner [31] has considered the stability of the system

∗

Corresponding author. E-mail addresses: [email protected] (H. Cheng), [email protected] (R. Yuan).

https://doi.org/10.1016/j.amc.2018.04.049 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

(1.1) with (u ) =

13

mu (m > 0, A > 0). A. Ducrot [11] has studied the predator-prey model with the Laplacian diffusion A+u

⎧∂ ⎪ ⎨ ∂ t u(x, t ) = du(x, t ) + u(x, t )(1 − u(x, t )) − (u(x, t ))v(x, t ),   ∂ v(x, t ) ⎪ ⎩ v(x, t ) = v(x, t ) + rv(x, t ) 1 − , ∂t u(x, t )

(1.2)

where d > 0 describes the diffusivity of prey, r denotes the growth rate of predator and (u ) = uπ (u ), π : [0, ∞) → [0, ∞) is of the class C1 such that π (u) > 0 for all u ∈ (0, 1]. He has proved that the system has a spreading speed property and the solution converges towards a generalized transition wave with some determined global mean speed of propagation for the one-dimensional system. Although the standard Laplacian operator can described the movement of individuals under a Brownian process, the movement of individuals which cannot be limited in a small area is often free and random. So recently, various integral operators have been widely used to model the nonlocal diffusion phenomena. For example, an operator of the form

K[u](x ) =

 R

k(x, y )(u(y ) − u(x ))dy

appears in the theory of phase transition, ecology, genetics and neurology, see [2,18,19,32]. Meanwhile, many researchers give more attention on the study of traveling waves of nonlocal reaction diffusion equations. For instance, many authors have obtained the properties of the solution of the reaction–diffusion systems with nonlocal diffusion term, see [3,4,7–10,27,37,38]. In [26], the authors have obtained the existence of the traveling waves of the model without monotone condition. In [5], we have considered the spreading speed properties for the model (1.2) with the fractional diffusion term α (α ∈ (0, 1)). In the present paper, we consider the Holling–Tanner predator-prey model with the classical Lotka–Volterra functional response, that is the following model

⎧∂ ⎪ ⎨ ∂ t u(x, t ) = d1 (J ∗ u(x, t ) − u(x, t )) + u(x, t )(1 − u(x, t )) − β u(x, t )v(x, t ),   ∂ v(x, t ) ⎪ ⎩ v(x, t ) = d2 (J ∗ v(x, t ) − v(x, t )) + rv(x, t ) 1 − , ∂t u(x, t )

(1.3)

where di > 0 (i = 1, 2 ) are diffusion rates for the prey and predator individuals, respectively, J ∗ u(x, t ) = R J (y )u(x − y, t )dy and J ∗ v(x, t ) = R J (y )v(x − y, t )dy represent the standard convolutions with space invariable x, β u (0 < β < 1) denotes the functional response to predation, and r > 0 denotes the growth rate of predator. Throughout this paper, we need the below assumptions of the diffusion kernel J. Assumption 1.1. (J1 ) J is a smooth function in R and satisfies J ∈ C 1 (R ), J (x ) = J (−x ) ≥ 0, R J (x )dx = 1. (J2 ) There exists λ0 ∈ (0, +∞] such that R J (x )e−λx dx < +∞ for any λ ∈ [0, λ0 ), and R J (x )e−λx dx → +∞ as λ → λ0 − 0. In this work, we mainly study the existence and nonexistence of traveling waves which connect the predator free state (1,0) with the coexistence state ( 1+1β , 1+1β ) for the system (1.3). We will obtain that there exists a critical velocity c∗ > 0

such that for c > c∗ , the system (1.3) admits a traveling wave solution with wave speed c; for 0 < c < c∗ , the system (1.3) has no traveling waves with wave speed c. Further, we can deduce that the existence of the traveling waves is independent of spatial movement patterns of the predator and prey. The spreading speed of the traveling waves, however, depends on the movement of the predator. By our analysis, we can confirm that the diffusion rate d2 and the growth rate r of the predator can increase the spreading speed. The main approach for the proof of the existence is to construct a suitable invariant set and then to use the Schauder’s fixed point theorem, see [12,13,26,27]. To overcome the difficulties of the nonlocal diffusion, we construct an invariant cone in a large bounded domain and then pass to the unbounded domain. Finally, we conclude the nonexistence of the traveling waves with speed c < c∗ by using the two-sided Laplace transform. This paper is organized as follows. In the next section, we will give some preliminaries for the proof of our results. In Section 3, we present the proof of the existence of the traveling waves by Schauder’s fixed point theorem under the assumption of the compact support for the kernel function J. In Section 4, we obtain the nonexistence of traveling waves by two-sided Laplace transform. Finally, we give some discussion about the effect of nonlocal diffusion.

2. Some preliminaries In this section, we will give some useful results for the proof of the existence and nonexistence of the traveling waves for the system (1.3). In the sequel, we always assume that the initial equilibrium is (1,0). The traveling wave solution means

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H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

a solution of the form (u(x + ct ), v(x + ct )). Let ξ = x + ct, then (u(ξ ), v(ξ )) satisfies

⎧ ⎨cu (ξ ) = d1 (J ∗ u(ξ ) − u(ξ )) + u(ξ )(1 − u(ξ )) −  β u ( ξ )v ( ξ ), v ( ξ ) ⎩cv (ξ ) = d2 (J ∗ v(ξ ) − v(ξ )) + rv(ξ ) 1 − u(ξ ) .

(2.1)

Linearizing the second equation of (2.1) at the initial point (1,0), we have

cv (ξ ) = d2 (J ∗ v(ξ ) − v(ξ )) + rv(ξ ). Then we can get a characteristic equation

(λ, c ) := d2



R

J (x )e−λx dx − d2 − cλ + r.

(2.2)

By easy calculations, we can establish

(0, c ) = r > 0,

∂ (0, c ) = −c < 0, ∂λ

∂ (λ, c ) = −λ < 0 for all λ > 0, ∂c  ∂ 2 (λ, c ) = d J (x )x2 e−λx dx > 0. 2 ∂λ2 R

In view of the above properties of the function (λ, c), we can get the following lemma. Lemma 2.1. There exists c∗ > 0 and λ∗ > 0 such that

∂ (λ, c )

∂λ

=0

and

(λ∗ , c∗ ) = 0.

(λ∗ ,c∗ )

Furthermore, (i) if 0 < c < c∗ , (λ, c) > 0 for all λ ∈ (0, λ0 ); (ii) if c > c∗ , then the equation (λ, c ) = 0 has two positive real roots λ1 (c), λ2 (c) with 0 < λ1 (c) < λ∗ < λ2 (c) < λ0 , ( · , c) < 0 in (λ1 (c), λ2 (c)) and ( · , c) > 0 in (0, λ1 (c)) ∪ (λ2 (c), λ0 ). Remark 2.1. From Lemma 2.1, we can see that the critical speed c∗ is related to the diffusion rate d2 and the growth rate r of predator. We can conjecture that the movement and the growth rate of predator can increase the spreading speed of the traveling waves, see Section 5. Next, we give the definition of the supersolution and subsolution. Definition 2.1. A function ψ + : R → [0, a] is called a supersolution to the equation

cU  (ξ ) − d[J ∗ U (ξ ) − U (ξ )] − f (U (ξ )) = 0, where d > 0 and f (U ) ∈ C 2 ([0, a], R ) satisfies f (0 ) = f (a ) = 0, f(U) > 0 for all U ∈ (0, a), if it satisfies

c (ψ + ) (ξ ) − d[J ∗ ψ + (ξ ) − ψ + (ξ )] − f (ψ + (ξ )) ≥ 0,

ξ ∈ R.

(2.3)

Similarly, we can define a subsolution ψ − (ξ ) by reversing the inequality in (2.3). 3. The existence of traveling waves In this section, we always assume c > c∗ and denote λi (c) by λi for i = 1, 2, respectively. Lemma 3.1. The function v+ (ξ ) = eλ1 ξ satisfies the following equation

cv (ξ ) = d2 (J ∗ v(ξ ) − v(ξ )) + rv(ξ ). Lemma 3.2. Let 0 < α < λ1 be sufficiently small and σ > β be large enough. Then the function u− (ξ ) := max{1 − σ eαξ , 1 − β} satisfies

cu (ξ ) ≤ d1 (J ∗ u(ξ ) − u(ξ )) + u(ξ )(1 − u(ξ )) − β u(ξ )V + (ξ ), where

V + (ξ )

= min{eλ1 ξ , 1}.

Proof. For ξ > 0, u− (ξ ) = 1 − β and V + (ξ ) = 1 which implies (3.1) holds. For α1 ln σβ ≤ ξ ≤ 0, u− (ξ ) = 1 − β and V + (ξ ) = eλ1 ξ . By easy calculation, we can verify (3.1) holds For ξ < α1 ln σβ , u− (ξ ) = 1 − σ eαξ and V + (ξ ) = eλ1 ξ . Then we need to prove

−cσ α + d1 σ



R

J (x )e−α x dx − d1 σ + β e(λ1 −α )ξ − βσ eλ1 ξ ≤ 0.

(3.1)

H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

15

Since ξ < α1 ln σβ , it is just needed to prove



−cσ α − d1 σ 1 −





R

J (x )e−α x dx + β e

λ1 −α β α ln σ

≤ 0.

That is,



−cσ α − d1 σ 1 −

 R

J ( x )e

−α x

  λ1α−α β dx + β ≤ 0. σ

(3.2)

For some σ > β large enough and 0 < α < λ1 small enough satisfies σ α = 1 and let σ → ∞, α → 0, we can confirm (3.2) holds. This completes the proof of the lemma.  Lemma 3.3. Let η ∈ (0, min{λ1 , λ2 − λ1 } ) be sufficiently small and some well-chosen M > 1 satisfies

η, c ). Then the function v− (ξ ) = eλ1 ξ (1 − Meηξ ) satisfies cv (ξ ) ≤ d2 (J ∗ v(ξ ) − v(ξ )) + rv(ξ ) − for ξ < η1 ln

λ1 r ≤ −M η (λ1 + 1−β

r (V + (ξ ))2 1−β

(3.3)

1 M.

Proof. Since M > 1, ξ < η1 ln

1 M

< 0. Then we can get



L := cv− (ξ ) − d2 (J ∗ v− (ξ ) − v− (ξ )) − rv− (ξ ) +

r (V + (ξ ))2 1−β

= c[λ1 eλ1 ξ (1 − Meηξ ) + eλ1 ξ (−Mηeηξ )]  −d2 J (x )eλ1 (ξ −x ) (1 − Meη (ξ −x ) )dx − eλ1 (ξ ) (1 − Meηξ ) R

−reλ1 ξ (1 − Meηξ ) +



= eλ1 (ξ ) cλ1 − d2



r e2λ1 ξ 1−β



J (x )e−λ1 x dx + d2 − r R 

 −Me(η+λ1 )ξ c (η + λ1 ) − d2 J (x )e−(η+λ1 )x dx + d2 − r + R

λ1 r ≤ −M η (λ1 + η, c ), we can establish 1−β

By the definition of (λ, c), Lemma 2.1 and

L = (λ1 + η, c )Meξ (λ1 +η ) +

r e2ξ λ1 1−β

≤ e2ξ λ1 (λ1 + η, c )(Meξ (η−λ1 ) − M

λ1 η

).

Since 0 < η < λ2 − λ1 , (λ1 + η, c ) < 0. Then by ξ < η1 ln

L ≤ e2ξ λ1 (λ1 + η, c )(Me

r e2λ1 ξ . 1−β

η−λ1 1 η ln M

It is easily obtained that the function

λ1 η

−M

v− ( ξ )

1 M

and 0 < η < λ1 , we can get

) = 0.

satisfies the inequality (3.3). This completes the proof.



Now, define a function V − (ξ ) = max{v− (ξ ), 0} and a function set



φ (−A ) = u− (−A ), ϕ (−A ) = V − (−A ),

− − +

A = (φ (· ), ϕ (· )) ∈ C ([−A, A], R ) u (ξ ) ≤ φ (ξ ) ≤ 1, V (ξ ) ≤ ϕ (ξ ) ≤ V (ξ ) ,

for any ξ ∈ [−A, A] 

2

where A > max{ η1 ln M, α1 ln σβ }. For any (φ (· ), ϕ (· )) ∈ C ([−A, A], R2 ), define

 φ ( A ), φˆ (ξ ) = φ (ξ ), u− ( ξ ),

 ξ > A, ϕ ( A ), |ξ | ≤ A, ϕˆ (ξ ) = ϕ (ξ ), ξ < −A, V − ( ξ ),

ξ > A, |ξ | ≤ A, ξ < −A.

Consider the following initial value problems

cu (ξ ) = d1



R

J (x )φˆ (ξ − x )dx − d1 u(ξ ) + φ (ξ )(1 − u(ξ )) − β u(ξ )ϕ (ξ ),

(3.4)

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H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

c v ( ξ ) = d2





v (ξ ) J (x )ϕˆ (ξ − x )dx − d2 v(ξ ) + rϕ (ξ ) 1 − φ (ξ ) R

 (3.5)

with

u(−A ) = u− (−A ),

v(−A ) = V − (−A ).

(3.6)

By the ODE theory, the problems (3.4)–(3.6) admit a unique solution (uA (· ), vA (· )) satisfying uA (· ) ∈ C 1 ([−A, A] ) and vA (· ) ∈ C 1 ([−A, A] ). Then, we define an operator F = (F1 , F2 ) : A → C ([−A, A] ) by F1 [φ , ϕ ](ξ ) = uA (ξ ) and F2 [φ , ϕ ](ξ ) = vA (ξ ) for ξ ∈ [−A, A]. Lemma 3.4. The operator F maps A into A . Proof. For ∀ (φ ( · ), ϕ ( · )) ∈ A , we should prove that

F1 [φ , ϕ ](−A ) = u− (−A ), F2 [φ , ϕ ](−A ) = V − (−A ), and

u− ( ξ ) ≤ F1 [φ , ϕ ] ( ξ ) ≤ 1, V − ( ξ ) ≤ F2 [φ , ϕ ] ( ξ ) ≤ V + ( ξ )

for any

ξ ∈ [−A, A].

By the definition of the operator F, it is obvious to see

F1 [φ , ϕ ](−A ) = uA (−A ) = u− (−A ), F2 [φ , ϕ ](−A ) = vA (−A ) = V − (−A ). For ξ ∈ [−A, A], we first consider F1 [φ , ϕ ](ξ ). Using the definition of the operator F, it is sufficient to show u− (ξ ) ≤ uA ( ξ ) ≤ 1. According to the definition of φˆ (ξ ), we can know that



d1

J (x )φˆ (ξ − x )dx − d1 − βϕ (ξ ) ≤ −βϕ (ξ ) ≤ 0,

R

which implies that 1 is a super-solution of (3.4). Thus uA (ξ ) ≤ 1 for ξ ∈ [−A, A]. , A], and u− (ξ ) = 1 − σ eαξ for ξ ∈ [−A, A ) with A  = 1 ln β . By the definition of φˆ (ξ ) Note that u− (ξ ) = 1 − β for ξ ∈ [A α σ and Lemma 3.2, we know that



d1

R

J (x )φˆ (ξ − x )dx − d1 (1 − β ) + φ (ξ )(1 − (1 − β )) − β (1 − β )ϕ (ξ )

≥ β (1 − β ) − β (1 − β )V + (ξ ) ≥ 0 , A], and for ξ ∈ [A 

cu− (ξ ) − d1



J (x )φˆ (ξ − x )dx + d1 u− (ξ ) − φ (ξ )(1 − u− (ξ )) + β u− (ξ )ϕ (ξ )   ≤ cu− (ξ ) − d1 J (x )u− (ξ − x )dx + d1 u− (ξ ) + β u− (ξ )V + (ξ ) − φ (ξ )(1 − u− (ξ )) R

R

≤ u− (ξ )(1 − u− (ξ )) − φ (ξ )(1 − u− (ξ )) ≤0 ). Since u (−A ) = u− (−A ), the comparison principle implies that u− (ξ ) ≤ u (ξ ) for ξ ∈ [−A, A]. So for any ξ ∈ (−A, A A A − u (ξ ) ≤ uA (ξ ) ≤ 1 for all ξ ∈ [−A, A]. Then we consider F2 [φ , ϕ ](ξ ). Again using the definition of the operator F, it is sufficient to show that V − (ξ ) ≤ vA (ξ ) ≤ V + (ξ ). First we know that vA (ξ ) ≥ 0 for ξ ∈ [−A, A] according to the maximum principle. Then since u− (ξ ) ≤ φ (ξ ) ≤ 1, ) with A  = 1 ln 1 , and using Lemma 3.3, we have V − (ξ ) ≤ ϕˆ (ξ ) ≤ V + (ξ ) and V − (ξ ) = eλ1 ξ (1 − Meηξ ) for ξ ∈ [−A, A η M 

cV − (ξ ) − d2



ϕ (ξ )

J (x )ϕˆ (ξ − x )dx + d2V − (ξ ) − rϕ (ξ ) + r V − (ξ ) φ (ξ )  r  ≤ cV − (ξ ) − d2 J (x )V − (ξ − x )dx + d2V − (ξ ) − rV − (ξ ) + (V + (ξ ))2 1−β R ≤0 R

). Since v (−A ) = V − (−A ), the comparison principle implies that V − (ξ ) is sub-solution of (3.5) on [−A, A ). for any ξ ∈ [−A, A A , A], V − (ξ ) = 0. So For ξ ∈ [A

V − ( ξ ) ≤ vA ( ξ )

for

ξ ∈ [−A, A].

(3.7)

Then vA (ξ ) ≤ 1 for ξ ∈ [−A, A] according to the inequality





d2

R

J (x )ϕˆ (ξ − x )dx − d2 + rϕ (ξ ) 1 −

1

φ (ξ )



≤ 0.

H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

17

Further since u− (ξ ) ≤ φ (ξ ) ≤ 1, V − (ξ ) ≤ ϕˆ (ξ ) ≤ V + (ξ ) and V + (ξ ) = eλ1 ξ for ξ ∈ [−A, 0 ), and using Lemma 3.1, we have 

cV + (ξ ) − d2



ϕ (ξ )

J (x )ϕˆ (ξ − x )dx + d2V + (ξ ) − rϕ (ξ ) + r V + (ξ ) φ (ξ )   ≥ cV + (ξ ) − d2 J (x )V + (ξ − x )dx + d2V + (ξ ) − rV + (ξ ) = 0 R

R

for any ξ ∈ [−A, 0 ). Since vA (−A ) = V − (−A ), the comparison principle implies that V + (ξ ) is super-solution of (3.5) on [−A, 0 ). For ξ ∈ [0, A], V + (ξ ) = 1. So vA (ξ ) ≤ V + (ξ ) for ξ ∈ [−A, A]. Combining with (3.7), we can easily show that

V − ( ξ ) ≤ vA ( ξ ) ≤ V + ( ξ )

for

ξ ∈ [−A, A].



This ends the proof.

Lemma 3.5. The operator F : A → A is completely continuous. Proof. ∀ (φ ( · ), ϕ ( · )) ∈ A , F1 [φ , ϕ ](ξ ) = uA (ξ ) and F2 [φ , ϕ ](ξ ) = vA (ξ ). The initial-value problem (3.4)–(3.6) admits a unique solution (uA (· ), vA (· )) ∈ C 1 ([−A, A] )× ∈ C 1 ([−A, A] ). By a direct calculation, we have



 (d1 + φ (s ) + βϕ (s ))ds −A    ξ 1 exp − (d1 + φ (s ) + βϕ (s ))ds (d1 fφ (η ) + φ (η ))dη,

uA (ξ ) = u− (−A ) exp − + and

1 c

 ξ −A

1 c

 ξ

c η

   1 ξ ϕ (s ) vA (ξ ) = V − (−A ) exp − ( d2 + r )ds c −A φ (s )    ξ  ξ 1 1 ϕ (s ) + exp − ( d2 + r )ds (d2 gϕ (η ) + rϕ (η ))dη, c −A c η φ (s )

where

f φ (η ) = and

gϕ ( η ) =



−A

−∞



−A −∞

J (η − x )u− (x )dx +

J (η − x )V − (x )dx +



A

−A



J (η − x )φ (x )dx +

A −A

J (η − x )ϕ (x )dx +



+∞ A

 A

+∞

(3.8)

(3.9)

J (η − x )φ (A )dx,

J (η − x )ϕ (A )dx.

We first show F is continuous. ∀ (φ 1 ( · ), ϕ 1 ( · )), (φ 2 ( · ), ϕ 2 ( · )) ∈ A , we have

| f φ1 ( η ) − f φ2 ( η ) |

 A

 +∞





≤

J (η − x )[φ1 (x ) − φ2 (x )]dx

+

J (η − x )[φ1 (A ) − φ2 (A )]dx

−A A ≤ 2 max

x∈[−A,A]

| φ1 ( x ) − φ2 ( x ) | ,

and

| gϕ1 (η ) − gϕ2 (η ) |≤ 2 max |ϕ1 (x ) − ϕ2 (x )|. x∈[−A,A]

By the continuity of the compound function, we can obtain that F is continuous from Eqs. (3.8) and (3.9). Next we prove that F is compacted, that is, for any bounded subset  ⊂ A , F () is precompact. For all (uA , vA ) ∈ F (), there exists (φ , ϕ ) ∈  such that

F[φ , ϕ ](ξ ) = (uA , vA )(ξ ),

∀ξ ∈ [−A, A].

Since (φ , ϕ ) ∈ , (3.8) and (3.9), we deduce

| uA (ξ ) |≤ M1 and | vA (ξ ) |≤ M1 , ∀ ξ ∈ [−A, A], for some constant M1 > 0. That is to say F () is uniformly bounded. Combining the above inequalities and Eqs. (3.4)–(3.5), we deduce that there exists some constant M2 > 0 such that

| uA (ξ ) |≤ M2 and | vA (ξ ) |≤ M2 , ∀ ξ ∈ [−A, A]. So F () is equicontinuous. By Arzela–Ascoli Theorem, we have that F () is precompact. Then F : A → A is completely continuous with respect to the maximum norm. 

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H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

Theorem 3.1. The operator F has a fixed point in A . Proof. By the definition of A , it is easy to verify that A is closed and convex. Thus, according to Lemma 3.5 and Schauder’s fixed point theorem, there exists (u∗A (· ), v∗A (· )) ∈ A such that

(u∗A (ξ ), v∗A (ξ )) = F[u∗A , v∗A ](ξ ),

∀ξ ∈ [−A, A].



To obtain the existence of the traveling waves, we need another assumption of the kernel function J . (J3 ) The kernel function J is compact support. For the sake of convenience, we use uA ( · ) and vA (· ) instead of u∗A (· ) and v∗A (· ), respectively. Lemma 3.6. Assume (J1 ) − (J3 ) hold, then there exists a positive constant C such that

uA C1,1 ([−B,B]) < C and vA C1,1 ([−B,B]) < C for any B > 0 satisfying B < A. Proof. By Theorem 3.1, (uA (· ), vA (· )) satisfies

cuA (ξ ) = d1 and



R

J (x )uˆ A (ξ − x )dx − d1 uA (ξ ) + uA (ξ )(1 − uA (ξ )) − β uA (ξ )vA (ξ ),



 vA ( ξ ) J (x )vˆ A (ξ − x )dx − d2 vA (ξ ) + rvA (ξ ) 1 − , A ( ξ ) = d2 uA ( ξ ) R 

c v where

 uˆA (ξ ) =

uA ( A ), uA ( ξ ), u− ( ξ ),

(3.10)

 ξ > A, vA ( A ), |ξ | ≤ A, vˆ A (ξ ) = vA (ξ ), ξ < −A, V − ( ξ ),

(3.11)

ξ > A, |ξ | ≤ A, ξ < −A.

Following 1 − β ≤ uA (ξ ) ≤ 1 and 0 ≤ vA (ξ ) ≤ 1 for ξ ∈ [−B, B], we have



d1

d1

J (x )uˆA (ξ − x )dx + |uA (ξ )| c R c 1 β + |uA (ξ )(1 − uA (ξ ))| + |uA (ξ )||vA (ξ )| c c ( 2d1 + 1/4 + β ) ≤ , c

|uA (ξ )| ≤

and

|vA (ξ )| ≤ ≤





d2

r

v (ξ )

d2 J (x )vˆ A (ξ − x )dx + |vA (ξ )| +

vA (ξ )(1 − A )

c R c c uA ( ξ ) 2d2 + r/4 . c

Thus there exists some constant C1 > 0 such that

uA C1 ([−B,B]) < C1 and vA C1 ([−B,B]) < C1 . Obviously,

|uA (ξ ) − uA (η )| < C1 |ξ − η| and |vA (ξ ) − vA (η )| < C1 |ξ − η|

(3.12)

for any ξ , η ∈ [−B, B]. In view of (3.10), we have





c|uA (ξ ) − uA (η )| ≤ d1 J (x )(uˆ A (ξ − x ) − uˆA (η − x ))dx + d1 |uA (ξ ) − uA (η )| R

+ |uA (ξ )(1 − uA (ξ )) − uA (η )(1 − uA (η ))| + β|uA (ξ )vA (ξ ) − uA (η )vA (η )| := d1U1 + d1U2 + U3 + β U4 . By the conditions (J1 ) and (J3 ), we have









J (x )uˆA (ξ − x )dx − J (x )uˆA (η − x )dx

R R

 + R

 +R



=

J (x )uˆ A (ξ − x )dx − J (x )uˆ A (η − x )dx

−R −R

U1 =

(3.13)

H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

19

 ξ +R

 η +R



=

J (ξ − x )uˆ A (x )dx − J (η − x )uˆA (x )dx

ξ −R η−R

 ξ +R  η+R  η−R

 η +R



= ( + + )J (ξ − x )uˆA (x )dx − J (η − x )uˆ A (x )dx

η +R η−R ξ −R η−R

 ξ +R

 η−R





≤

J (ξ − x )uˆ A (x )dx

+

J (ξ − x )uˆ A (x )dx

η +R ξ −R

 η+R



+

(J (ξ − x ) − J (η − x ))uˆA (x )dx

η−R   ≤ 2 J L∞ + RL |ξ − η|, where R is the radius of sptJ and L is the Lipschitz constant of the function J,

U3 = |uA (ξ ) − u2A (ξ ) − uA (η ) + u2A (η )| ≤ |uA (ξ ) − uA (η )| + |u2A (ξ ) − u2A (η )| ≤ 3|uA ( ξ ) − uA ( η )|, and

U4 = |uA (ξ )vA (ξ ) − uA (η )vA (η )| ≤ |uA (ξ ) − uA (η )||vA (ξ )| + |uA (η )||vA (ξ ) − vA (η )| ≤ |uA ( ξ ) − uA ( η )| + |vA ( ξ ) − vA ( η )|. Combining (3.12) and (3.13), we obtain that there exists some constant L2 > 0 such that

|uA (ξ ) − uA (η )| ≤ L2 |ξ − η|. Then applying with (3.11), we establish





c|vA (ξ ) − vA (η )| ≤ d2 J (x )(vˆ A (ξ − x ) − vˆ A (η − x ))dx + d2 |vA (ξ ) − vA (η )|

R



vA ( ξ ) v (η )

+r

vA (ξ )(1 − ) − vA (η )(1 − A )

uA ( ξ ) uA ( η )

:= d2V1 + d2V2 + rV3 . Similar to the argument of U1 , we deduce









J (x )vˆ A (ξ − x )dx − J (x )vˆ A (η − x )dx

R R   ≤ 2 J L∞ + RL |ξ − η|.

V1 =

Then, by (3.12), (3.14) with

 2 

vA (ξ ) v2A (η )



V3 = (vA (ξ ) − vA (η )) − − uA ( ξ ) uA ( η )

2

vA (ξ )uA (η ) − v2A (η )uA (ξ )



≤ |vA ( ξ ) − vA ( η )| +

uA ( ξ )uA ( η ) ≤

|vA ( ξ ) − vA ( η )| +

|v2A (ξ ) − v2A (η )| |v2A (η )||uA (η ) − uA (ξ )| + |uA ( ξ )| |uA ( ξ )uA ( η )|

≤

|vA ( ξ ) − vA ( η )| +

2 1 |u ( η ) − uA ( ξ )| |vA ( ξ ) − vA ( η )| + 1−β ( 1 − β )2 A

≤

3−β 1 |u ( η ) − uA ( ξ )|, |vA ( ξ ) − vA ( η )| + 1−β ( 1 − β )2 A

we establish

|vA (ξ ) − vA (η )| ≤ L2 |ξ − η| for any ξ , η ∈ [−B, B]. So there exists a constant C > 0 such that

uA C1,1 ([−B,B]) < C and vA C1,1 ([−B,B]) < C, for any B < A.



(3.14)

20

H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

Theorem 3.2. Assume (J1 ) and (J3 ) hold. For any c > c∗ , there exists a pair function (u(ξ ), v(ξ )) satisfying (2.1), u(−∞ ) = 1 1, v(−∞ ) = 0, and if lim sup u(ξ ) < 1 and lim inf v(ξ ) > 0, then u(+∞ ) = v(+∞ ) = . 1 + β ξ → + ∞ ξ →+∞ ∞ such that An > max{ η1 ln M, α1 ln σβ } for each n and lim An = +∞. For every Proof. Choosing an increasing sequence {An }+ n=1 n→+∞ ∗ c > c , we have (uAn , vAn ) ∈ An satisfying Lemma 3.6 and Eqs. (3.10) and (3.11) for ξ ∈ [−An , An ]. According to the estimates for the sequence {(uAn , vAn )} in Lemma 3.6, we can extract a subsequence denoted by {(uAn , vAn )}k∈N , tending towards k

(u, v ) ∈ C 1 (R ) in the following topologies uAn → u and k

k

1 vAnk → v in Cloc (R ) as k → +∞.

By the assumption of the kernel function J and applying the dominated convergence theorem, we can establish



lim

k→+∞ R

and

 lim

k→+∞ R

J (x )uˆ An (ξ − x )dx =



k

J (x )vˆ An (ξ − x )dx =

R



k

R

J (x )u(ξ − x )dx = J ∗ u(ξ )

J (x )v(ξ − x )dx = J ∗ v(ξ )

for any ξ ∈ R. Then, it is easy to show that (u, v ) satisfies system (2.1) and

u− ( ξ ) ≤ u ( ξ ) ≤ 1,

V − ( ξ ) ≤ v ( ξ ) ≤ V + ( ξ ).

By the above inequality, we can obtain u(−∞ ) = 1, v(−∞ ) = 0, and

1 − β ≤ lim inf u(ξ ) ≤ lim sup u(ξ ) ≤ 1, 0 ≤ lim inf v(ξ ) ≤ lim sup v(ξ ) ≤ 1. ξ →+∞

ξ →+∞

ξ →+∞

(3.15)

ξ →+∞

Next we deduce the asymptotic behavior of the solution (u(ξ ), v(ξ )) at +∞. Since lim sup u(ξ ) < 1 and lim inf v(ξ ) > 0, ξ →+∞

ξ →+∞

there exist ξ > ξ 0 and δ0 ∈ (0, 1 − β ) such that u(ξ ) ≤ 1 − βδ0 and v(ξ ) ≥ δ0 for any ξ > ξ 0 . Then by (3.15), we have that

δ0 < u(ξ ) ≤ 1 − βδ0 and δ0 ≤ v(ξ ) ≤ 1

(3.16)

for any ξ > ξ 0 . Now, we introduce a sequence {γ n }n ≥ 0 defined by

 γ0 = 1, γ1 = δ0 , 1 − γn+1 = γn , β

n ≥ 1.

Obviously, the sequences {γ 2n }n ≥ 0 and {γ2n+1 }n≥0 are adjacent. They converge to

γ1 < γ3 < · · · < γ2n+1 < · · · <

1 1+β

and satisfy for each n ≥ 0,

1 < · · · < γ2n < · · · < γ2 < γ0 . 1+β

Next we will verify

γ2n+1 ≤ u(ξ ) ≤ γ2n+2 and γ2n+1 ≤ v(ξ ) ≤ γ2n for all n ≥ 0 and ξ > ξ0 .

(3.17)

We establish (3.17) by induction on n, the case n = 0 being immediate from (3.16). Assume (3.17) is valid for all n ≥ 1 and let us prove (3.17) holds true for n + 1. Firstly, since u(ξ ) ≤ γ2n+2 , then v(ξ ) satisfies



v (ξ ) cv (ξ ) − d2 (J ∗ v(ξ ) − v(ξ )) − r v(ξ ) 1 − γ2n+2 



≤ 0 for

ξ ≥ ξ0 .

That is to say that v(ξ ) is the subsolution of the equation 



cv (ξ ) − d2 (J ∗ v(ξ ) − v(ξ )) − r v(ξ ) 1 −

v (ξ ) γ2n+2



= 0 for

ξ ≥ ξ0 .

(3.18)

Since γ2n+2 is a solution of Eq. (3.18), we can deduce v(ξ ) ≤ γ2n+2 for all ξ ≥ ξ 0 . Then u(ξ ) satisfies 

cu (ξ ) − d1 (J ∗ u(ξ ) − u(ξ )) − u(ξ )(1 − u(ξ )) + βγ2n+2 u(ξ ) ≥ 0, for

ξ ≥ ξ0 .

This yields that u(ξ ) is the supersolution of the equation 

cu (ξ ) − d1 (J ∗ u(ξ ) − u(ξ )) − u(ξ )(1 − u(ξ )) + βγ2n+2 u(ξ ) = 0, for

ξ ≥ ξ0 .

Thus applying 1 − γ2n+3 = βγ2n+2 , we establish u(ξ ) ≥ γ2n+3 for ξ ≥ ξ 0 . Using the same arguments as above and u(ξ ) ≥ γ2n+3 , one can easy to conclude v(ξ ) ≥ γ2n+3 and further show u(ξ ) ≤ γ2n+4 . Thus (3.17) holds true for n + 1.

H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

21

Letting n → +∞ of (3.17), we can obtain that

u (ξ ) ≡

1 , 1+β

v (ξ ) ≡

1 1+β

for ξ ≥ ξ 0 . So u(+∞ ) = v(+∞ ) =

1 . 1+β



4. Nonexistence of traveling waves In this section, we will establish the nonexistence of traveling waves for (2.1) when 0 < c < c∗ . Theorem 4.1. Assume that (J1 ) and (J2 ) hold. For 0 < c < c∗ , there exist no the traveling waves (u(ξ ), v(ξ )) of the system (2.1) satisfying (u(−∞ ), v(−∞ )) = (1, 0 ) and (u(∞ ), v(∞ )) =

1 , 1 1+β 1+β

.

Proof. Suppose there exists a traveling wave solution (u(ξ ), v(ξ )) of system (2.1) satisfying the limit behavior at infinity. Since u(−∞ ) = 1 and v(−∞ ) = 0, there exist ξˆ < 0 and  1 > 0 such that u(ξ ) ≥ 1 − 1 and v(ξ ) ≤ 1 for any ξ ≤ ξˆ. Therefore, we deduce



v (ξ ) u (ξ )

cv (ξ ) = d2 (J ∗ v(ξ ) − v(ξ )) + rv(ξ ) 1 −

r (1 − 21 ) v (ξ ) 1 − 1

≥ d2 (J ∗ v(ξ ) − v(ξ )) + for any ξ ≤ ξˆ. Let V (ξ ) =

 ξ −∞

c v ( ξ ) ≥ d2



(4.1)

v(η )dη for any ξ ∈ R. Integrating two sides of inequality (4.1) from −∞ to ξ with ξ ≤ ξˆ, we can establish





ξ −∞

J ∗ v ( η )d η − V ( ξ )

r (1 − 21 ) V ( ξ ). 1 − 1

+

(4.2)

By Fubini theorem, we have

 ξ −∞

J ∗ v ( η )d η =

 ξ  −∞

 =

R

 =

R

R

J (x ) J (x )

J (x )v(η − x )dxdη  ξ −∞

v ( η − x )d η d x

 ξ −x −∞

v(η )dηdx = J ∗ V (ξ ).

Then there holds

cv(ξ ) ≥ d2 (J ∗ V (ξ ) − V (ξ )) + Since

 ξ −∞

(J ∗ V (η ) − V (η ))dη = =

 ξ  −∞

−∞ 1

0

 =

R

 ξ  

=

r (1 − 21 ) V ( ξ ). 1 − 1

R



R

R

(4.3)

J (x )(V (η − x ) − V (η ))dxdη

(−x )J (x )

(−x )J (x ) 

(−x )J (x )

1

0



1

V  (η − xθ )dθ dxdη

0

 ξ

V  (η − xθ )dηdxdθ

−∞

V (ξ − xθ )dθ dx,

J ∗ V (ξ ) − V (ξ ) is integrable on (−∞, ξ ] for any ξ ≤ ξˆ. Hence, from Eq. (4.3), we deduce that V(ξ ) is integrable on (−∞, ξ ] for any ξ ≤ ξˆ. Integrating two sides of inequality (4.3) from −∞ to ξ yields

r (1 − 21 ) 1 − 1

 ξ

−∞

V (η )dη ≤ cV (ξ ) + d2



R

xJ (x )



0

1

V (ξ − xθ )dθ dx.

Due to xV (ξ − θ x ) is non-increasing for θ ∈ [0, 1] and the symmetry of the kernel function J, we deduce

r (1 − 21 ) 1 − 1

 ξ

−∞

V ( η )d η ≤ c + d2





R

xJ (x )dx V (ξ ) = cV (ξ ).

22

H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

Since V(ξ ) is increasing with respect to ξ , there exists some τ > 0 such that

r (1 − 21 ) τ V (ξ − τ ) ≤ cV (ξ ). 1 − 1 ˆ Hence, we can choose a constant τ 0 > 0 sufficiently large  and some ν ∈ (0, 1), such that V (ξ − τ0 ) ≤ νV (ξ ) for each ξ ≤ ξ . Set W (x ) = V (x )e−μ0 x and μ0 = min τ1 ln( ν1 ), 0

λ0

, then

2

W (ξ − τ0 ) = V (ξ − τ0 )e−μ0 (ξ −τ0 ) ≤ νV (ξ )e−μ0 (ξ −τ0 ) = W (ξ ). Thus, there exists W0 > 0 such that W(ξ ) ≤ W0 for any ξ ≤ ξˆ, which implies that

V (ξ ) ≤ W0 eμ0 ξ for any

ξ ≤ ξˆ.

Since

cv (ξ ) ≤ d2 (J ∗ v(ξ ) − v(ξ )) + rv(ξ ), there exists W1 > 0 such that v(ξ ) ≤ W1 eμ0 ξ for any ξ ≤ ξˆ. Hence, we conclude that v(ξ )e−μ0 ξ and v (ξ )e−μ0 ξ are all bounded as ξ → −∞. Then we can deduce

sup{v(ξ )e−μ0 ξ } < ∞ and sup{v (ξ )e−μ0 ξ } < ∞. ξ ∈R

(4.4)

ξ ∈R

e−λξ v2 (ξ ) is uniformly bounded on R for 0 < λ ≤ μ0 . u (ξ ) Then, multiplying the second equation of (2.1) by e−λξ and integrating on R, we can obtain

Since 1 − β ≤ u(ξ ) ≤ 1 and 0 ≤ v(ξ ) ≤ 1,



d2

R

e−λξ J ∗ v(ξ )dξ − d2 

=r

R

R

and

 R

R

e−λξ v2 (ξ ) dξ . u (ξ )

In view of







e−λξ J ∗ v(ξ )dξ =

e−λξ v (ξ )dξ = λ

R

 R

e−λξ v(ξ )dξ − c

J (x )e−λx dx

d2

R

R

R

e−λξ v (ξ )dξ + r



R

e−λξ v(ξ )dξ

e−λξ v(ξ )dξ

e−λξ v(ξ )dξ ,

it follows

 





J (x )e−λx dx − d2 − cλ + r

 R

e−λξ v(ξ )dξ = r

 R

e−λξ v2 (ξ ) dξ , u (ξ )

that is,

(λ, c )

 R

e−λξ v(ξ )dξ = r

 R

e−λξ v2 (ξ ) dξ , u (ξ )

(4.5)

where (λ, c) is defined by Eq. (2.2). By the assumption 0 < c < c∗ , (λ, c) is always positive for all λ ∈ [0, λ0 ). All the two integrals in (4.5) are be analytically continued to the interval [0, λ0 ). Otherwise, by the theory of convergence region of two-sided Laplace transform (see[34,35]), the integral



R

e−λξ v(ξ )dξ

2 has a singularity at λ = μ∗ ∈ (0, λ0 ) and is analytic for all λ ∈ (0, μ∗ ). Notice vu((ξξ)) e−μ1 ξ is uniformly bounded for μ1 =

 v2 (ξ ) −λξ λ −μ∗ min 0 2 , μ0 , so the integral R e dξ is analytic for all μ < μ∗ + μ1 . It is a contradiction. Notice (4.5) can be u (ξ ) re-written as

 R



e−λξ (λ, c ) − r

 v (ξ ) v ( ξ )d ξ = 0. u (ξ )

(4.6)

This leads to a contradiction again in that (λ, c ) → +∞ as λ → λ0 − 0. So we conclude that if 0 < c < c∗ , there is no traveling waves satisfying the boundary conditions. 

H. Cheng, R. Yuan / Applied Mathematics and Computation 338 (2018) 12–24

23

5. Discussion 1

The traveling waves obtained in this work describe that the predator free stationary state (0 ) invades the coexistence 1

state ( 1+1β ). The existence and non-existence of the traveling waves for model (1.3) indicate whether the coexistence state 1+β

of prey and predator exists or not. By the discussion of the above sections, we can know that whether the coexistence state of prey and predator exists or not is independent of the non-local spatial movement patterns of the predator individuals. On the contrary, the spreading speed depends on the aforementioned factors. From Lemma 2.1, it is easy to see that c∗ depends on the diffusion rate d2 and the growth rate r of the predator individuals. More specifically, in the case where

J (x ) =



1 4π ρ

e−x

2

/4ρ

with

ρ > 0,

we have

(λ, c ) = d2 eρλ − d2 − cλ + r, 2

∂ (λ, c ) 2 = d2 λ2 eρλ > 0. ∂ρ

Direct calculations give

dc∗ (r ) 1 = ∗ > 0, dr λ  dc∗ (d2 ) 1  ∗2 = ∗ eρλ − 1 > 0, dd2 λ dc∗ (ρ ) ∗2 = d2 λ∗ eρλ > 0, dρ which implies that the growth and movement of predator individuals can increase the spreading speed of the traveling waves. At the same time, under these assumptions of kernel function J, we can obtain that the critical velocity c∗ is strictly positive. But when J (x ) = J (−x ), the wave speed may be negative. Indeed, for asymmetric kernel, waves propagate ‘backward’ if predator individuals leave a location faster than they are replaced, but the waves do not propagate ‘backward’ for symmetric kernels. So it is difficult to obtain the existence of the traveling waves of (1.3) with asymmetric kernel. This remains an open problem for future study. Finally, we give the relationship between the local diffusion term and the nonlocal diffusion term. Especially, let J (x ) = δ (x ) + δ  (x ) where δ is the Dirac delta function, then (1.3) can be rewritten as

⎧ ∂ ⎪ ⎨ u(x, t ) = d1 u(x, t ) + u(x, t )(1 − u(x, t )) − β u(x, t )v(x, t ), ∂t   ∂ v(x, t ) ⎪ . ⎩ v(x, t ) = d2 v(x, t ) + rv(x, t ) 1 − ∂t u(x, t )

(5.1)

That is to say (5.1) is a local predator-prey model. It is a special case in [11]. So we can get when (u ) = β u (0 < β < 1), the almost planar transition waves in theorem 1.9 of [11] correspond to traveling waves. For c = c∗ , the existence or nonexistence of the traveling waves for the system (2.1) is not obtained. This is an open problem. Acknowledgments The authors would like to thank the anonymous referee for their careful reading and valuable comments. The work is supported by National Natural Science Foundation of China (11701341). References [1] M. Aziz-Alaoui, M.D. Okiye, Boundedness and global stability for a predator–prey model with modified leslie-gower and holling-type II schemes, Appl. Math. Lett. 16 (2003) 1069–1075. [2] P.W. Bates, P.C. Fife, X. Ren, X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal. 138 (1997) 105–136. [3] J. Carr, A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Am. Math. Soc. 132 (2004) 2433–2439. [4] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differ. Eq. 2 (1997) 125–160. [5] H. Cheng, R. Yuan, The spreading property for a prey-predator reaction–diffusion system with fractional diffusion, Frac. Calc. Appl. Anal. 18 (2015) 565–579. [6] C. Conley, R. Gardner, An application of the generalized morse index to travelling wave solutions of a competitive reaction–diffusion model, Indiana Univ. Math. J. 44 (1984) 319–343. [7] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl. 185 (2006) 461–485. [8] J. Coville, J. Dvila, S. Martinez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differ. Eq. 244 (2008) 3080–3118. [9] J. Coville, L. Dupaigne, Propagation speed of travelling fronts in non local reaction–diffusion equations, Nonlinear Anal. 60 (2005) 797–819. [10] J. Coville, L. Dupaigne, On a non-local equation arising in population dynamics, Proc. R. Soc. Edinb. Sect. A 137 (2007) 727–755. [11] A. Ducrot, Convergence to generalized transition waves for some holling-tanner prey–predator reaction–diffusion system, J. Math. Pures Appl. 100 (2013) 1–15.

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