Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip

A time-delayed reaction-diffusion system of mistletoes and birds with nonlocal effect in a two-dimensional strip is considered in this paper. By the background of model deriving, the bird diffuses with a Neumann boundary value condition, and the mistletoes does not diffuse and thus without boundary value condition. Making use of the theory of monotone semiflows and Kuratowski measure of non-compactness, we discuss the existence of spreading speed $c^\ast$. The value of $c^*$ is evaluated by using two auxiliary linear systems accompanied with approximate process.


1.
Introduction. Mistletoes are typical aerial stem-parasites plants (Kuijt [5]), whose seeds are mainly transmitted by fruit-eating birds highly specialized to consume their berries. Mistletoes produce flowers and fruits. Birds in some trees search and encounter mistletoe fruits, then handle the fruits and deposit their droppings with seeds after they ate the fruits. Once a mistletoe seed is deposited by a bird on an appropriate host plant, it sticks to a branch, germinates and forms a haustorium that taps into the xylem of the host to absorb water and minerals. Mistletoe was often regarded as a pest because it kills trees and devalues natural habitats, while it was recently recognized as an ecological keystone species, an organism that has a disproportionately pervasive influence over its community (Watson [17]).
In the mutually beneficial relationships between mistletoes and birds species that disperse mistletoe seeds, recently, Wang et al. [15] derived and proposed a reactiondiffusion system with proper initial and boundary value conditions: (D∇u 2 − γu 2 ∇u 1 ) · n(x) = 0, t > 0, x ∈ ∂Ω, u 1 (t, x) = u The purpose of this article is to study the dynamics of the model in a twodimensional strip because the species may be live in a two-dimensional strip rather than in a one-dimension space. Thus, in this article, we shall consider the reactiondiffusion model of mistletoes and birds with the chemotactic term γ = 0 in Ω = R × (−L, L), then the system (1) without initial value conditions turns into: k(x, y, z x , z y ) u 1 (t, z x , z y ) u 1 (t, z x , z y ) + ω u 2 (t, z x , z y )dz y dz x , t > 0, (x, y) ∈ R × (−L, L), where the parameters d m , α, d i , D, ω are positive constant, the time delay τ and parameter d are nonnegative, the boundary condition here expresses the requirement that the population size u i are confined inΩ which means that no migration occurs through the boundary ∂Ω and k(x, y, z x , z y ) is a kernel function with separated variables, that is, k(x, y, z x , z y ) = k(x − z x , y, z y ) = k 1 (x − z x )k 2 (y, z y ), depends only on the distance between x and z x , and satisfies the following assumptions throughout this paper.
(K1) k 1 is a nonnegative Lebesgue measurable function, k 1 (ξ) = k 1 (−ξ), ∀ ξ ∈ R, R k 1 (ξ)dξ = 1, and K 1 (ς) := R k 1 (ξ)e ςξ dξ < ∞, ∀ ς ∈ [0, ∞). (K2) k 2 (y, z y ) ∈ C([−L, L] × [−L, L], R + ), k 2 (y, z y ) = k 2 (z y , y), k 2 (y, z y ) > 0 for ∀y, z y ∈ [−L, L], and L −L k 2 (y, z y )dz y = 1 for every y ∈ [−L, L]. The main results for (2) in this article are about the spreading speed. Similar to [16], we use the theory of monotone semiflow developed by Liang and Zhao in [9]. However, there are some difficulties and challenging because of the region Ω has been changed into a two-dimensional strip now. Firstly, we must obtain the Green function by solving the following initial boundary value problem: Secondly, to estimate the upper and lower bounds of the spreading speed c * , we need more complex and elaborate techniques. The organization of the article is as follows. We give preliminaries in Section 2. In Section 3, we first translate (2) to an equivalent system (5), and then obtain the existence and uniqueness of solutions, the comparison principle and the strong positivity for system (5). In Section 4 and 5, by the usage of the theory of monotone semiflows developed in Liang & Zhao [9,27], we obtain the existence of the spreading speed c * . We give an estimation of c * in Section 6 in view of two linear systems accompanied with an approximate process. An appendix is added at the end to illustrate the eigenvalues and eigenfunctions for a specify eigenvalue problem. We hope that the results in this article could be significative for understanding about the spreading patterns and interaction rule among mistletoes and birds.
It is obviously that K and K 2 satisfy (2) K and K 2 are strongly positive, order preserved and compact.
Also see the statements after the assumption (K3) and the equation (49) in [15]. From [6,7], we know that there is no stable non-constant equilibrium of a cooperative system with homogeneous Neumann boundary value condition in a convex domain. Therefore, in this paper, we just consider constant equilibria of (2). It is easy to see that the system (2) has the following two boudnary equilibria: E 0 := (0, 0), E 1 := (0, 1). Note that the positive constant equilibria of (2) are determined by the following system: Substitute the second equation into the first one, then we have , then there is no other constant equilibrium. In the following, we further consider the situation ≥ 0, and obtain the following lemma by summarizing.
Lemma 2.1. The following statements are valid.
(i) Suppose < 0, then there are exact two constant equilibria of (2): E 0 = (0, 0), E 1 = (0, 1), that is, there is no positive constant equilibrium of (2). (ii) Suppose = 0. If d > 1, then there is a positive constant equilibria of (2), which is a double root of (4), otherwise, there is no positive constant equilibrium of (2).  The proof is similar to the proof in [16], we omit it here.
3. Existence, uniqueness and comparison principle. Throughout this article, we suppose that the following assumption holds.
We first give some notations. Let R + = [0, ∞) and Here, BC(R × [−L, L], R) denotes the space with all of bounded and uniformly continuous functions from R × [−L, L] to R, β = (β 1 , β 2 ) with β i ≥ 0. Note that any vector β ∈ R 2 is a constant functionβ in X . Obviously, X + is a positive cone in X and (X , X + ) is an ordered Banach space.
a compact open topology: φ n → φ (n → ∞) in X which is equivalent to φ n (x, y) → φ(x, y) (n → ∞) uniformly for (x, y) in any compact set on R × [−L, L]. Note that X is a Banach lattice.
Let (µ n , Ψ n (y)), n=0,1,... be the eigenvalues and their corresponding normalized eigenfuntions of operator − ∂ 2 ∂y 2 on [-L, L] with respect to homogeneous Neumann boundary condition. Let Γ(t, x, y, z x , z y ) be the Green function of initial-boundary value problem where (see more details in the Appendix). Therefore, for any given ψ ∈ X 2 , the solution of (6) can be written as for (x, y) ∈ R × (−L, L). Note that T 2 (t) is compact for t > 0 (see [25]). Let T 1 (t) := I be the identify operator defined on X 1 . Define a family of bounded linear operators T (t) = (T 1 (t), T 2 (t)), t ≥ 0 on X by Then, T (t) : X → X is a C 0 -semigroup with T (t)X + ⊆ X + for any t ≥ 0. For any φ = (φ 1 , φ 2 ) ∈ C + , define a map f = (f 1 , f 2 ): C + → X as follows: Therefore, (5) with the initial value φ ∈ C + can be written as an abstract system (7) has an equivalent integral form whose solution u(t, x, y; φ) of (8) is called a mild solution of (5). Here, φ = (φ 1 , φ 2 ) ∈ C + . Now, We define a pair of supersolution and subsolution for (5).
By the fact thatū(t, x, y), u(t, x, y) are a pair of supersolution and subsolution of (5), respectively, again applying Corollary 1 in Martin & Smith [10], we have which means that Similarly to the above arguments, we can obtain which together with the definition of supersolution and subsolution, and the fact that T (t)X + ⊂ X + for all t ≥ 0, leads to According to Proposition 3 in Martin & Smith [10] with We can also get a similar result for w 2 (t). Thus, we have that w(t) = (w 1 (t), w 2 (t)) 0 for t > 0, if w(0, x, y) > 0 for (x, y) ∈ R × [−L, L].
4. Monotone semiflow. Recall a family of operators {Π t } ∞ t≥0 which is called a semiflow on a metric space (Z, · Z ) with the norm · Z , if {Π t } ∞ t=0 satisfies the following properties: It is easy to see that property (c) holds if Π(·, v) is continuous on R + for each v ∈ Z, and Π(t, ·) is uniformly continuous for t in any bounded intervals in the sense that for any v 0 ∈ Z, bounded interval I and > 0, there exists δ = δ(v 0 , I, be the mild solution of (5) with initial condition u 0 = φ ∈ C E+ . Define a family of For C E+ is a positive invariant set for (5), it follows that Q t : C E+ → C E+ is the solution operator of (5) defined on C E+ .
Proof. From the integral form (8) of system (5) and the property of T (t), it is easy to see that Q t satisfies (a) and (b) of semiflow and is monotone. Now we will show that the property (c) of semiflow is satisfied. Firstly, for any given φ in C E+ , noting that from (5), ∂ ∂t u(t, x, y; φ) is bounded for (t, x, y) ∈ R + × R × [−L, L], that is, for any φ in C E+ , there exists M = M(φ) > 0, such that ∂ ∂t u(t, x, y; φ) ≤ M, which means that where · expresses the norm in R 2 . Therefore, Q t (φ) is continuous in t ∈ R + for each φ ∈ C E+ .

5.
Asymptotic speed of spread. In this section, we will apply the theorems in Liang & Zhao [9] and Zhao [27] to study the asymptotic speed of spread of system (5), and then state a corresponding result for (2). LetC = C([−τ, 0] × [−L, L], R 2 ) C with the maximum norm · C such that it is a Banach space. We can also regard every member ofC as a function in C.
Define the reflection operator R on C by R Proof. It is easy to see that Q t satisfies (A1) -(A3) with β = E + for t > 0. Let Q t := Q t |C be the restriction of Q t toC, thenQ t is a semiflow generated from the initial boundary value problem: Note that the above problem has the same equilibria as (5) and E 0 is unstable. ForQ t is a strongly monotone semiflow in [0, E + ]C, appealing to Dancer-Hess connecting orbit lemma (see Chapter 2 of Zhao [28]), the semiflowQ t admits a strongly monotone full orbit connecting E 0 and E + . Thus, (A4) holds for t > 0.
Proof. We note that (5) has another integral form whereT (t) = (T 1 (t), T 2 (t)) T defined on X with Here f 2 is defined in Section 3 andf 1 is defined as Indeed,T (t) : X → X is a C 0 -semigroup withT (t)X + ⊆ X + for any t ≥ 0.
We define a linear operator L(t)[φ] := (T 1 (t)φ 1 (0)(·, ·), 0) for ∀φ ∈ C + and a nonlinear map Note that Q t = L(t) + S(t) for t ≥ 0 and for each φ ∈ C E+ , we have then it follows that L(t) ≤ e −dmt for ∀t > 0. For the expression of (11) and the compactness of T 2 (t), we have that S(t) is compact for each t > τ . Therefore, for any subset A ⊂ C E+ and x ∈ R, it follows that Thus, Q t satisfies (A5) for t > τ .
Let u(t, x, y) = w(t, y)e −ςx , then we obtain where K 1 (ς) := R k 1 (ξ)e ςξ dξ. Furthermore, we claim that Note that the integral operator 0. Thus, it follows that K 2 possesses a sequence of eigenvalues j such that j ∈ R, and the only possible limit point of { j } is zero (see [2,15]). What's more, as K 2 is strongly positive, from Krein-Rutman theorem (see [11]), the principal eigenvalue 0 = 1 and Φ 0 (y) ≡ 1. Here, we denote Φ j (y) as the corresponding eigenfunction of eigenvalue j for K 2 for j ≥ 0. Substitute w(t, y) = e λt (Φ j (y), Ψ n (y)) T with j, n = 0, 1, ... into (14), where Ψ n is defined in Appendix. Recall that K 2 [Φ j ](y) = j Φ j (y) and Ψ n (y) = −µ n Ψ n (y), then we have It is obvious that the above matrix is in Frobenius form (see the definition of Frobenius form of matrix in Weinberger, Lewis & Li [19, page 179]). For the use of convenience, we introduce the following lemma from Hale [4] or Smith [11].
Let ε → 0 in (19), we obtain the linearized system of (5) at E 0 : (21) That is, c * is the spreading speed of the system (21).
Summarizing the above discussion, we obtain the estimation of c * . Theorem 6.3. The following estimation for the spreading speed c * of system (5) holds.
Herec * is the spreading speed of the system (13), and c * is the spreading speed of the system (21).

Remark 3.
Let the solution operator of (21) is M t . We are not able to have the the result: for all u ∈ C E+ , t ≥ 0. We can only obtain the result between the solution operators of (5) and (13): Therefore, The conclusion in Theorem 6.3 could not lead to the linear determinacy as mentioned in [19].
In this article, we commit ourself to obtain the threshold asymptotic patterns related to spreading speed of model (2). We want to mention here that the model (2) is an unusual system, which has several characteristics. Firstly, the system is defined on a strip domain, and it is a partially degenerate system. Secondly, the boundary value of which is defined only for the second species. Thirdly, the model is both spatial nonlocal and with time delay. Moreover, as stated in Remark 3, the solution operator of the system could not compare with the solution operator of its linearized system at zero equilibrium. Therefore, the linear determinacy is still an open problem.
We also mention here, that if we take k 1 (x) to be the Dirac-delta function, we believe that the discussions for spreading speed can be proceeded. 8. Appendix. Consider the following initial boundary value problem with ψ ∈ X 2 : Using the separation of variables: w(t, x, y) = V (t, x)Y (y), and V (t, x), Y (y) = 0, then we have , t > 0, (x, y) ∈ R × (−L, L).
For the left-hand side of the above is the function of (t, x) and the right-hand side is the function of y, there must exist a constant µ such that Y (y) + µY (y) = 0, y ∈ (−L, L), Y (y) = 0, y ∈ {−L, L}.
Thus, there are a series of solutions of (22): t > 0, (x, y) ∈ R × [−L, L], B k is any constant, k = 0, 1, ... Therefore, the solution of (22) can be written as We have from w(0, x, y) = ψ(x, y) = ∞ k=1 V k (0, x)B k Ψ k (y) that is a classical solution of (22). Therefore, we can conclude that the Green function of (22) is Let ψ(x, y) ≡ 1 for (x, y) ∈ R×[−L, L], then we have from (22) that w(t, x, y; ψ) ≡ 1 is the solution of (22) with regard to the initial function ψ. By the expression of w(t, x, y) above, we see