Stability of traveling waves for partially degenerate nonlocal dispersal models in periodic habitats

Journal Pre-proof Stability of traveling waves for partially degenerate nonlocal dispersal models in periodic habitats

Xiongxiong Bao, Wan-Tong Li

PII: DOI: Reference:

S0893-9659(20)30082-3 https://doi.org/10.1016/j.aml.2020.106289 AML 106289

To appear in:

Applied Mathematics Letters

Received date : 11 December 2019 Revised date : 9 February 2020 Accepted date : 9 February 2020 Please cite this article as: X. Bao and W.-T. Li, Stability of traveling waves for partially degenerate nonlocal dispersal models in periodic habitats, Applied Mathematics Letters (2020), doi: https://doi.org/10.1016/j.aml.2020.106289. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Elsevier Ltd. All rights reserved.

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STABILITY OF TRAVELING WAVES FOR PARTIALLY DEGENERATE NONLOCAL DISPERSAL MODELS IN PERIODIC HABITATS XIONGXIONG BAO1 AND WAN-TONG LI2,∗

Abstract. Recently, the spreading speeds and periodic traveling wave solutions for a general class of partially degenerate nonlocal dispersal cooperative systems in time and space periodic habitats have been investigated. In this paper, we continue to study the stability and convergence rate for such time and space periodic traveling waves. We show that if the initial function perturbation is uniformly bounded with respect to a weighted maximum norm, the periodic traveling wave solution is globally exponentially stable when the wave speed is greater than the spreading speed. Keywords: Partially degenerate, Nonlocal dispersal, Time and space periodic habitat, Traveling wave solution, Stability. AMS subject classifications 35C07, 35K57, 45G15, 92D25.

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1. Introduction

In this paper, we investigate the stability and convergence rate of periodic traveling waves for the following partially degenerate nonlocal dispersal cooperative system in time and space periodic habitats:   ∂u1 = [(Ku (t, ·))(x) − u (t, x)] + F (t, x, u , u ), 1 1 1 1 2 ∂t (1.1)  ∂u2 = F2 (t, x, u1 , u2 ), ∂t

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where u1 = (u1 , ..., ui1 )> , u2 = (ui1 +1 , ..., uK )> and F1 (t, x, u1 , u2 ) = (F 1 , ..., F i1 )> , F2 (t, x, u) = R R (F i1 +1 , ..., F K )> for 1 ≤ i1 < K; (Ku1 (t, ·))(x) := ( RN k1 (y − x)u1 (t, y)dy, ..., RN ki1 (y − x)ui1 (t, y)dy) with that ki (·) is a C 1 nonnegative convolution kernel supported on a ball cenR tered at 0, RN ki (z)dz = 1 for i = 1, ..., i1 . Here F(t, x, u(t, x)) = (F 1 (t, x, u), ..., F K (t, x, u)) and (1.1) satisfy the following standard hypotheses, (H1) For each i = 1, . . . , K, F i (t, x, u) is C 1 in (t, x) ∈ R × RN and C 2 in u ∈ RK , and is periodic in (t, x) with period (T, P ) := (T, p1 , p2 , . . . , pN ). F(t, x, 0) = 0 and (1.1) admits a positive, unique and globally stable (T, P )-periodic solution u∗ (t, x). (H2) F(t, x, u) is cooperative in the sense that for any 1 ≤ i 6= j ≤ K, ∂F i (t, x, u) ≥ 0 ∂uj

for all u ∈ [0, u∗ (t, x)] and (t, x) ∈ R × RN .

Spreading speeds and traveling wave solution of partially degenerate cooperative system have attracted a lot of attention, for example, see [5,6,8,10,22,24,25] and references therein. Recently, nonlocal dispersal has been used to describe the movements and interactions of the organisms in a long range and there are many works on spreading speeds and traveling waves for nonlocal dispersal equations, see [2, 4, 7, 11, 14, 16, 20, 21, 23] and so on. Very recently, Bao and Li [1] have Date: February 9, 2020. School of Science, Chang’an University, Xi’an, Shaanxi 710064, P.R. China 2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, P. R. China ∗ Corresponding author ([email protected]).

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BAO AND LI

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studied the spatial spreading speeds and traveling wave solution for partially degenerate nonlocal dispersal cooperative system (1.1) in time and space periodic media. Hence, it is a natural and important question to study the stability and convergent rate of space-time periodic traveling waves of (1.1). Although the stability of periodic traveling wave solution of scalar nonlocal dispersal equation in heterogenous media has been studied in [9,17–20], there are some difficulties to do this issue for cooperative system (1.1). The main difficulties come from the presence of both space and time periodic media and interaction between different components in a higher dimensional case, which lead to that the squeeze technique and methods in [18] do not work for this question. Fortunately, the method in a weighted space used in [13, 22] for the stability of traveling wave solution can overcome these difficulties. Thus in this paper we will show that when the wave speed is greater than the spreading speed, every solution of (1.1) converges to the periodic traveling wave solution (if exists) exponentially in time if the initial value is uniformly bounded in a weighted case. The existence and stability periodic traveling wave solution of (1.1) with critical speed are still challenging problems and remain open. Let S N −1 = {ξ ∈ RN | kξk = 1}. For any µ ∈ R, define the map Kξ,µ by setting Z  Z −µ(y−x)·ξ −µ(y−x)·ξ (Kξ,µ u1 (t, ·))(x) = e k1 (y − x)u1 (t, y)dy, ..., e ki1 (y − x)ui1 (t, y)dy , RN

RN

∂t

where

 ∂F i (t, x, 0) = ∂uj K×K

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where the kernel ki (·) (i = 1, ..., i1 ) is as in (1.1). Given any measurable and bounded function u0 (·), by general semigroup theory, there is a unique measurable solution u(t, x; u0 ) = (u1 (t, x; u0 ), ..., uK (t, x; u0 )) (see also [1]). Consider the linearization of (1.1) at 0,   ∂v1 = [(Kv (t, ·))(x) − v (t, x)] + A (t, x)v + A (t, x)v , 1 1 11 1 12 2 ∂t (1.2)  ∂v2 = A21 (t, x)v1 + A22 (t, x)v2 , A0 (t, x) =

! A11 (t, x) A12 (t, x) A21 (t, x) A22 (t, x)

∀(t, x) ∈ R × RN .

Then, for given µ ∈ R and ξ ∈ S N −1 , the eigenvalue problem associated to (1.2) is  ∂φ1   − ∂t + (Kξ,µ φ1 (t, ·))(x) − φ1 (t, x) + A11 (t, x)φ1 + A12 (t, x)φ2 = λφ1 (t, x), 2 (1.3) − ∂φ ∂t + A21 (t, x)φ1 + A22 (t, x)φ2 = λφ2 (t, x),    φi (· + T, ·) = φi (·, · + pl el ) = φi (·, ·), l = 1, 2, · · · , N and i = 1, 2.

Assume that

(H3) For any ξ ∈ S N −1 and µ ∈ R, the eigenvalue problem (1.3) admits a principal eigenvalue λ(ξ, µ, A0 ) with positive, time and space periodic eigenfunction (φ1 (t, x; µ, ξ), φ2 (t, x; µ, ξ)) and λ(A0 ) := λ(ξ, 0, A0 ) > 0. ∗

λ(ξ,µ,A0 ) ) By [1, Proposition 2.3], there is µ∗ ∈ (0, +∞) such that λ(µ . Let φ∗ (t, x) := µ∗ = inf µ>0 µ φ(t, x; µ∗ , ξ). Under the assumptions (H1)-(H3), if F(t, x, ρφ∗ (t, x)) ≤ ρFu (t, x, 0)φ∗ (t, x) for all ρ > 0 and any (t, x) ∈ R × RN , Bao and Li [1] has established that there is a bounded measurable function Φ : RN × R × RK → (RK )+ which generates a traveling wave solution U(t, x) = Φ(x − ctξ, t, ctξ) of (1.1) connecting u∗ (t, x) and 0 and propagating in the direction 0) of ξ with speed c for c > c∗ (ξ), where c∗ (ξ) = inf µ>0 λ(ξ,µ,A . Here c∗ (ξ) is the spreading speed µ of partially degenerate nonlocal dispersal system (1.1).

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STABILITY OF TRAVELING WAVES

3

In order to present the stability result of periodic traveling wave solution, we further consider the linearized eigenvalue problem of (1.1) at u∗ (t, x), that is

where

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(1.4)

 ∂ϕ1 ∗ ∗   − ∂t + (Kϕ1 (t, ·))(x) − ϕ1 (t, x) + A11 (t, x)ϕ1 + A12 (t, x)ϕ2 = λϕ1 (t, x), ∗ ∗ 2 − ∂ϕ ∂t + A21 (t, x)ϕ1 + A22 (t, x)ϕ2 = λϕ2 (t, x),    ϕi (· + T, ·) = ϕi (·, · + pl el ) = ϕi (·, ·), l = 1, 2, · · · , N

∗

A (t, x) =



∂F i (t, x, u∗ ) ∂uj



K×K

We assume that

=

! A∗11 (t, x) A∗12 (t, x) A∗21 (t, x) A∗22 (t, x)

and i = 1, 2,

∀(t, x) ∈ R × RN .

(H4) The eigenvalue problem (1.4) admits a principal eigenvalue λ(A∗ ) with positive, time and space periodic eigenfunction (ϕ1 (t, x), ϕ2 (t, x)) and λ(A∗ ) < 0. (H5) The reaction term F(t, x, u) is strictly subhomogeneous on [0, u∗ ] in the sense that F(t, x, νu) > νF(t, x, u) for any ν ∈ (0, 1), (t, x) ∈ R × RN .

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We remark that the principal eigenvalue theory of nonlocal dispersal operators plays an important role in the study of propagation phenomena of nonlocal dispersal system. There are quite a few results about the principal eigenvalue for nonlocal dispersal operators, for example, see [3, 4, 7, 12, 15, 26], etc. In particular, we refer to [1, 3, 12] for some criteria of the existence of principal eigenvalue for nonlocal dispersal eigenvalue problem (1.3) and (1.4). Then we have the following main results on the stability of periodic traveling waves of (1.1).

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Theorem 1.1. Assume that (H1)-(H5) hold. Suppose that U(t, x) = Φ(x − ctξ, t, ctξ) be a periodic traveling wave solution of (1.1) connecting u∗ and 0 and propagating in the direction ε of ξ with speed c > c∗ (ξ). Let µ ∈ (0, µ∗ ) such that c = λ(µ) µ and µ = µ + ε for small ε > 0 such ε ∗ that µ < min{2µ, µ }. For small δ > 0, we introduce the weight function  eµε (η−η0 ) , η ≥ η0 , ε ω (η) := 1, η < η0 , where η0 ∈ R is chose such that |Ui (t, x)−u∗i (t, x)| < δ for all x·ξ−ct < η0 and i = 1, ..., K. Then there is a real number ε0 > 0 such that for any given initial value u0 (x) with 0 ≤ u0 (x) ≤ u∗ (0, x) and [u0 (x) − U(0, x)]ω ε (x · ξ) ∈ L∞ (RN , RK ),

we have

sup |ui (t, x; u0 ) − Ui (t, x)| ≤ Ce−ε0 t , x∈R

∀i = 1, ..., K, t ≥ 0

for some constant C > 0. Theorem 1.1 shows that when c > c∗ (ξ), if the initial perturbation is uniformly bounded in a weighted space, solution u(t, x; u0 ) of (1.1) converges exponentially to traveling wave solution U(t, x). The result also provides the exponential convergence rate. For more example on this result, we refer to see [13].

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2. Proof of Theorem 1.1

(2.1)

ε1

K X i=1

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In this section, we will prove the stability and convergence rate of time periodic traveling wave solutions and prove Theorem 1.1. Note that ϕ(t, x) = (ϕ1 (t, x), ϕ2 (t, x)) and λ(A∗ ) < 0 are the principal eigenfunction and eigenvalue of (1.4). Choose 0 < ε < −λ(A∗ ) and ε1 > 0 small such that ϕi (t, x) < ε min {ϕi (t, x)}, 1≤i≤K

∀(t, x) ∈ R × RN , ∀i = 1, ..., K.

Choose δ > 0 small such that for any function V = (V1 , ..., VK ) and V = (V 1 , ..., V K ) satisfying |Vi (t, x) − u∗i (t, x)| < δ, |V i (t, x) − u∗i (t, x)| < δ and Vi (t, x) ≤ V i (t, x) for all (t, x) ∈ R × RN and i = 1, ..., K, we have  K  X ∂F i i i ∗ (2.2) |F (t, x, V) − F (t, x, V)| < (t, x, u ) + ε1 (V i (t, x) − Vi (t, x)) ∂uj j=1

for all (t, x) ∈ R ×

RN

and i = 1, ..., K.

Proof. Let U± (t, x) is the solution of (1.1) with initial value U+ (0, x) = max{u0 (x), U(0, x)},

U− (0, x) = min{u0 (x), U(0, x)},

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respectively. The comparison principle implies that

0 ≤ U− (t, x) ≤ min{u(t, x; u0 ), U(t, x)} ≤ max{u(t, x; u0 ), U(t, x)} ≤ U+ (t, x) ≤ u∗ (t, x).

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We first show that U+ (t, x) converges to U(t, x) exponentially in time. Let V(t, x) := U+ (t, x)− U(t, x). In view of 0 ≤ V(0, x) ≤ ku0 (x) − U(0, x)k, we obtain that V(0, x)ω ε (x · ξ) is uniformly bounded on RN . Case (I): x · ξ − ct ≥ η0 . By (H5), for any i = 1, ..., K, we have i

i

F (t, x, U1 + V1 , ..., UK + VK ) − F (t, x, U1 , ..., UK ) ≤

Define

V i (t, x, M ) = eM t Vi (t, x),

i

∂F ∂uj

j=1

∂uj

(t, x, 0)Vj (t, x).

∀i = 1, ..., K,

where M is a positive constant to be determined later. Let  (M − 1)u + eM t F i (t, x, e−M t u), i i F (t, x, u) = M ui + eM t F i (t, x, e−M t u), Note that, for any 1 ≤ i 6= j ≤ K,

K X ∂F i

i = 1, . . . , i1 , i = i1 + 1, . . . , K.

(t, x, u) ≥ 0 for u ∈ [0, u∗ (t, x)] and (t, x) ∈ R × RN .

Choose M > 0 larger enough such that  i −1 + M + ∂F i (t, x, e−M t u) ≥ 0, i = 1, . . . , i , ∂F 1 ∂ui (t, x, u) := i ∂F −M t M + ∂ui (t, x, e u) ≥ 0, i = i1 + 1, . . . , K ∂ui

for u ∈ [0, u∗ (t, x)] and (t, x) ∈ R+ × RN . Then   Z t Z i K X ∂F  V i (t, x; M ) ≤ V i (0, x; M ) + ki (y − x)V i (τ, x; M )dy + (τ, x, 0)V i (τ, x; M ) dτ ∂uj 0 RN j=1

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STABILITY OF TRAVELING WAVES

for i ∈ {1, ..., i1 } and

for i ∈ {i1 + 1, ..., K}. Define

0 j=1

∂uj

(τ, x, 0)V i (τ, x; M )dτ

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V i (t, x; M ) ≤ V i (0, x; M ) +

Z tX i K ∂F

5

ε (x·ξ−η )+λε t 0

Vi0 (t, x) = C1 φεi (t, x)e−µ

,

i = 1, ..., K,

where µε = µ + ε, (φε1 (t, x), ..., φεK (t, x)) is the eigenfunction of (1.3) associated to the eigenvalue λε := λ(ξ, µε , A0 ). Since V(0, x)ω ε (x · ξ) is uniformly bounded on RN , we choose C1 > 0 0 0 sufficiently large such that V(0, x) ≤ V0 (0, x) for any x ∈ RN . Let V i (t, x; M ) = eM t V i (t, x) for 0 any i = 1, ..., K. Then V i (t, x; M ) satisfies   Z t Z i K X ∂F 0 0 0 0  V i (t, x; M ) = V i (0, x; M ) + ki (y − x)V i (τ, x; M )dy + (τ, x, 0)V i (τ, x; M ) dτ ∂u N j 0 R j=1

for i ∈ {1, ..., i1 } and V

0 i (t, x; M )

=V

0 i (0, x; M )

Z tX i K ∂F 0 + (τ, x, 0)V i (τ, x; M )dτ ∂uj 0 j=1

λ(ξ,µε ,A0 ) µε 0

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for i ∈ {i1 + 1, ..., K}. Note that for any ε > 0, c >

and x · ξ − ct ≥ η0 . By the

comparison principle (see [1]), we obtain V(t, x; M ) ≤ V (t, x; M ), which implies that

(2.3) Vi (t, x) ≤ Vi0 (t, x) = C1 φεi (t, x)e−µ

ε (x·ξ−ct−η ) 0

ε −λε )t

e−(cµ

ε −λε )t

e1 e−(cµ ≤C

,

∀i = 1, ..., K,

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e > 0. for any (t, x) ∈ R × RN and some constant C Case (II): x · ξ − ct < η0 . Note that U(t, x) is very close to u∗ (t, x) for all x·ξ−ct < η. Since U(t, x) ≤ U(t, x)+V(t, x) ≤ u∗ (t, x), we obtain from (2.2) that  K  X ∂F i i + i ∗ F (t, x, U ) − F (t, x, U) < (t, x, u ) + ε1 Vj (t, x), ∀i = 1, ..., K. ∂uj j=1

Thus V(t, x) satisfies  Z K  X ∂F i ∂Vi ∗ (t, x) ≤ ki (y − x)Vi (t, x) − Vi (t, x) + (t, x, u ) + ε1 Vj (t, x) ∂t ∂uj RN for i ∈ {1, ..., i1 } and

K

X ∂Vi (t, x) ≤ ∂t j=1



j=1

 ∂F i ∗ (t, x, u ) + ε1 Vj (t, x), ∂uj

i ∈ {i1 + 1, ..., K}

for (t, x) ∈ R × RN with x · ξ − ct < η0 . Choose ε0 := min{cµε − λε , −λ(A∗ ) − ε}. Define Vei (t, x) := C2 ϕi (t, x)e−ε0 t ,

i = 1, ..., K,

where (ϕ1 (t, x), ..., ϕK (t, x)) and λ(A∗ ) < 0 are the principal eigenfunction and eigenvalue of linearized eigenvalue problem (1.4), C2 is large enough such that Vi (0, x) ≤ Vei (0, x) for x ∈ RN and i = 1, ..., K. Due to ε0 ≤ −λ(A∗ ) − ε and (2.1), we have ∂ Vei ∂ϕi (t, x) =C2 (t, x)e−ε0 t − C2 ε0 e−ε0 t ϕi (t, x) ∂t ∂t

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BAO AND LI

≥(KVei (t, ·))(x) − Vei (t, x) + =(KVei (t, ·))(x) − Vei (t, x) +

K X ∂F i j=1

∂uj

(t, x, u∗ )Vej (t, x) − ε0 Vei (t, x) − λ(A∗ )Vei (t, x)

 i  K X ∂F ∂F i (t, x, u∗ )Vej (t, x) + (t, x, u∗ ) + ε Vei (t, x) ∂uj ∂ui

repro of

=(KVei (t, ·))(x) − Vei (t, x) +

j=1,i6=j

K  X ∂F i j=1

∂uj



(t, x, u ) + ε1 Vej (t, x) ∗

for i ∈ {1, ..., i1 }. Similarly, we can obtain  K  X ∂ Vei ∂F i ∗ (t, x, u ) + ε1 Vej (t, x) (t, x) ≥ ∂t ∂uj j=1

e x) and for some for i ∈ {i1 + 1, ..., K}. Then by comparison principle, we have V(t, x) ≤ V(t, e2 > 0, constant C (2.4)

e2 e−ε0 t , Vi (t, x) ≤ Vei (t, x) = C2 ϕi (t, x)e−ε0 t ≤ C

∀i = 1, ..., K.

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Together with (2.3) and (2.4), there is a positive constant C such that Vi (t, x) ≤ Ce−ε0 t for t ≥ 0, x ∈ RN and ∀i = 1, ..., K. Define W(t, x) := U(t, x) − U− (t, x). Consider W(t, x) = (W1 (t, x), ..., W2 (t, x)). By symmetry and the similar method for V(t, x), we can have Wi (t, x) ≤ Ce−εt for t ≥ 0, x ∈ R and i = 1, ..., K. That is, U− (t, x) also converges to U(t, x) exponentially in time. Due to |ui (t, x; u0 ) − Ui (t, x)| ≤ max{|Ui± (t, x) − Ui (t, x)|}, thus there is supx∈R |ui (t, x; u0 ) − Ui (t, x)| ≤ Ce−ε0 t for i = 1, ..., K and t ≥ 0. This completes the proof. 

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Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions which improved the presentation of this manuscript. Xiongxiong Bao was partially supported by NSF of China (11701041) and the Fundamental Research Funds for the Central Universities (300102129201, CHD) and Wan-Tong Li was partially supported by NSF of China (11731005, 11671180). References

[1] X. Bao, W.T. Li, Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats, Nonlinear Anal. Real World Appl. 51 (2020) 102975. [2] X. Bao, W.T. Li, Z.C. Wang, Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system, Commun. Pure Appl. Anal. 19 (2020) 253-277. [3] X. Bao, W. Shen, Criteria for the existence of principal eigenvalue of time periodic cooperative linear system with nonlocal dispersal, Proc. Amer. Math. Soc. 145 (2017) 2881-2894. [4] X. Bao, W. Shen, Z. Shen, Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems, Commun. Pure Appl. Anal. 18 (2019) 361-396. [5] V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomath. 97, Springer-Verlag, Heidelberg, 1993 [6] V. Capasso, R.E. Wilson, Analysis of reaction-diffusion system modeling man-environment-man epidemics, SIAM J. Appl. Math. 57 (1997) 327-346. [7] J. Coville, J. D´ avila, S. Mart´inez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. I. Poincar´ e-AN 30 (2013) 179-223.

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[8] J. Fang, X.Q. Zhao, Monotone wave fronts for partially degenerate reaction-diffusion system, J. Dynam. Differential Equations 21 (2009) 663-680. [9] F. Hamel, L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc. 13 (2011) 345-390. [10] B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion system, J. Differential Equations 252 (2012) 4842-4861. [11] W.T. Li, L, Zhang, G.B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst. 35 (2015) 1531-1560. [12] X. Liang, L. Zhang, X.Q. Zhao, The principal eigenvalue for degenerate periodic reaction-diffusion systems, SIAM J. Math. Anal. 49 (2017) 3603-3636. [13] Z. Ouyang, C. Ou, Global stability and convergence rate of traveling waves for a nonlocal model in periodic media, Discrete Contin. Dyn. Syst. Ser. B 17 (2012) 993-1007. [14] S. Pan, W.T. Li, G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffsion systems and applications, Z. Angew. Math. Phys. 60 (2009) 377-392. [15] N. Rawal, W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Differential Equations 24 (2012) 927-954. [16] N. Rawal, W. Shen, A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst. 35 (2015) 1609-1640. [17] W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent of monostable equations, J. Dynam. Differential Equations 23 (2011) 1-44. [18] W. Shen, Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence, Nonlinearity 30 (2017) 3466-3491. [19] W. Shen, Z. Shen, Transition fronts in nonlocal Fisher-KPP equations in heterogeneous media, Commun. Pure Appl. Anal. 15 (2016) 1193-1213. [20] W. Shen, A. Zhang, Traveling wave solutions of monostable equations with nonlocal dispersal in space periodic habitats, Comm. Appl. Nonlinear Anal. 19 (2012) 73-101. [21] W. Shen, A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitates, J. Differential Equations 249 (2010) 747-795. [22] X. Wang, X.Q. Zhao, Pulsating waves of a paratially degenerate reaction-diffusion system in a periodic habitats, J. Differential Equations 259 (2015) 7238-7259. [23] J.B. Wang, W.T. Li, J.W. Sun, Global dynamics and spreading speeds for a partially degenerate system with non-local dispersal in periodic habitats, Proc. Royal Soc. Edinburgh A 148 (2018) 849-880. [24] C. Wu, D. Xiao, X.Q. Zhao, Spreading speeds of a partially degenerate reaction diffusion system in a periodic habitats, J. Differential Equations 255 (2013) 3983-4011. [25] S.L. Wu, Y.J. Sun, S.Y. Liu, Traveling fonts and entire solutions in partially degenerate reaction-diffusion system with monostable nonlinearity, Discret. Contin. Dyn. Syst. 33 (2013) 921-946. [26] F.Y. Yang, W.T. Li, J.W. Sun, Principals for some nonlocal eigenvalue problems and applications, Discret. Contin. Dyn. Syst. 36 (2016) 4027-4049.

*Credit Author Statement

Journal Pre-proof 1 We ensure that we have written entirely original works, and if we have used the work and/or words of others, that this has been appropriately cited or quoted. Also, our manuscript “Stability of Traveling waves for Partially Degenerate Nonlocal

repro of

Dispersal Models in Periodic Habitats (AML-D-19-0265) has not been previously published, is not currently submitted for review to any other journal, and will not be submitted elsewhere before a decision is made by this journal. Also, Sincerely yours, Wan-Tong Li

PhD, Professor, Cuiying Professor,

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Lanzhou University