Principal spectral theory and asymptotic behavior of the spectral bound for partially degenerate nonlocal dispersal systems
-
Lei Zhang
Abstract
The purpose of this paper is to investigate the principal spectral theory and asymptotic behavior of the spectral bound for cooperative nonlocal dispersal systems, specifically focusing on the case where partial diffusion coefficients are zero, referred to as the partially degenerate case. We propose two sufficient conditions that ensure the existence of the principal eigenvalue in these partially degenerate systems. Additionally, we study the asymptotic behavior of the spectral bound for nonlocal dispersal operators with small and large diffusion coefficients, considering both non-degenerate and partially degenerate cases. Notably, we find a threshold-type result as the diffusion coefficients tend towards infinity in the partially degenerate case. Finally, we apply these findings to discuss the asymptotic behavior of the basic reproduction ratio in a viral diffusion model.
1 Introduction
In this paper, we investigate the principal eigenvalue theory and the asymptotic behavior of the spectral bound of the nonlocal dispersal systems with the form
Here d
i
≥ 0 is the diffusion coefficient; Ω is an open bounded domain in
which implies that the total particles are conserved in the dispersal process. This corresponds to the Neumann boundary condition (see, e.g., [1], [2]).
Recall that a square matrix is said to be cooperative if its off-diagonal elements are nonnegative, and nonnegative if all elements are nonnegative; a square matrix is said to be irreducible if it is not similar, via a permutation, to a block lower triangular matrix, and reducible if otherwise. We have the following assumptions.
For any
For any
For any
For each 1 ≤ i ≤ l, d i > 0.
For each 1 ≤ i ≤ l 1, d i > 0, and for each l 1 + 1 ≤ i ≤ l, d i = 0.
We call the system non-degenerate if all diffusion coefficients are positive, and partially degenerate if some diffusion coefficients are zero. Here (H4) and (H4′) corresponds the non-degenerate and partially degenerate cases, respectively. Some stream population models such as benthic-drift models and pulsed bio-reactor models with a hydraulic storage zone are described by partially degenerate reaction-diffusion systems (see, e.g., [3], [4], [5], [6], [7], [8], [9], [10]). Recently, partially degenerate nonlocal dispersal systems are beginning to be used in modeling such problems (see, e.g., [11], [12]).
In recent years, significant attention has been given to the research of nonlocal dispersal systems (see, e.g., [1], [13], [14], [15], [16], [17], [18]). It is of great importance to study the eigenvalue problem of the linear evolution systems with nonlocal dispersal. The main challenge to study this problem lies in the lack of compactness for the associated linear operators. Bates and Zhao [13], as well as Coville [19] established the principal eigenvalue theory for a scalar nonlocal dispersal equation by a generalized Krein–Rutman theorem, going back to Nussbaum [20] and Edmunds, Potter and Stuart [21]. Meanwhile, Shen and Zhang [22] also developed this eigenvalue problem using a powerful tool developed by Bürger [23], which investigated the eigenvalue problem of perturbations of positive semigroups. Huston, Shen and Vickers [24] and Coville [19] also proved the continuity of spectral bound and Lipschitz continuity of the principal eigenvalue with respect to parameters, respectively. More recently, Li, Coville and Wang [2] obtained the estimates for the principal eigenvalue and extended the existence theory. Berestycki, Coville and Vo [25] introduced the generalized principal eigenvalue.
Consider the following principal eigenvalue problem with nonlocal dispersal:
Here k(x) is a non-negative continuous function of
There is an open set Ω0 ⊂ Ω such that
The diffusion coefficient satisfies
It is natural to ask whether such results can be extended to cooperative systems. Shen and her collaborators [26], [27] gave a confirmed answer and extended the principal eigenvalue theory to the time-periodic case. Liang, Zhang and Zhao [28] developed the principal eigenvalue theory for nonlocal dispersal systems with time-delay in a time-periodic environment. Recently, Su, Wang and Zhang [29] studied the principal spectral theory and variational characterizations for cooperative systems with nonlocal and coupled diffusion without utilizing the generalized Krein–Rutman theorem. However, it seems that these methods and arguments may not be well adapted to the partially degenerate systems. Wang, Li and Sun [12] gave a sufficient condition for the existence of the principal eigenvalue problem for the linear evolution systems coupled with a nonlocal dispersal equation and an ordinary differential equation. It is worth pointing out that this eigenvalue problem has not been addressed in general cases. We thus aim to establish the principal eigenvalue theory of the partially degenerate nonlocal dispersal systems.
Since the principal eigenvalue of (1.1) cannot be expressed explicitly or even by a variational formula, it is a challenging problem to quantitatively analyze the principal eigenvalue. Recently, Yang, Li, and Ruan [30] discussed the asymptotic behaviors of the principal eigenvalue of (1.2) for small and large diffusion coefficients:
One may ask how to characterize the above limiting profiles for cooperative systems. Such problems have been explored for patch models (see, e.g., [31], [32], [33]) and reaction-diffusion systems (see, e.g., [34], [35], [36], [37]). However, due to the lack of compactness for the associated linear operators, these questions are still open for nonlocal dispersal systems, especially for the partially degenerate case.
It is worth pointing out that the principal eigenvalue may not exist for nonlocal dispersal systems (see, e.g., [22, Section 6]). As is well known, the principal eigenvalue is a primary tool to obtain profound results in various differential equations, especially in elliptic and parabolic problems. For instance, the principal eigenvalue can be used to analyze the stability of equilibrium in the investigation of spatial dynamics (see, e.g., [38], [39], [40]) and characterize the spreading speed in the research of propagation dynamics (see, e.g., [41], [42]). When the principal eigenvalue does not exist, the spectral bound can serve as an alternative tool to analyze the stability of equilibrium and characterize the spreading speed for a scalar nonlocal dispersal equation (see, e.g., [43]). This motivates us to study the limiting profile of the spectral bound of the associated nonlocal dispersal operators with small and large diffusion coefficients for non-degenerate and partially degenerate cases.
The first purpose of this paper is to establish the principal eigenvalue theory of the partially degenerate nonlocal dispersal systems. For each 1 ≤ i ≤ l, write χ
i
(x) ≔ ∫Ω
k
i
(y, x)dy,
where
Definition 1.1.
The real number λ
∗ is called the principal eigenvalue of
For each
where M
11(x), M
12(x), M
21(x), M
22(x) are l
1 × l
1, l
1 × l
2, l
2 × l
1, l
2 × l
2 matrices, respectively. For any
Here I
2 is the identity l
2 × l
2 matrix. Define
Here
Theorem A.
(see Theorems 3.2 and 4.3) Assume that (H1)–(H3) and (H4′) hold. Then the following statements are valid.
If there is an open set Ω0 ⊂ Ω such that (η − H)−1 ∉ L 1(Ω0), then the principal eigenvalue of
Assume that
It is worth pointing out that the sufficient condition given in Theorem A(i) is consistent with that for the non-degenerate case in [26]. Moreover, this result generalize the sufficient condition given in [12]. In this paper, we employ a generalized Krein–Rutman theorem to obtain the above theorem. The main difficulty in carrying out this construction is to find a sufficient condition such that there is a “gap” between the spectral radius and the essential spectral radius of
The second purpose of this paper is to study the asymptotic behavior of the spectral bound of the associated nonlocal dispersal operators with small and large diffusion coefficients, both for the non-degenerate case and the partially degenerate case. Define a cooperative matrix
Theorem B.
(see Theorems 4.1, 4.2 and 4.4) Assume that (H1)–(H3) hold. Then the following statements are valid:
If, in addition, (H4) holds, then
If, in addition, (H4′) holds, then the following threshold results are valid:
If
If
Combining the structure of essential spectral points, perturbation theory, and the method of upper and lower approximation, we show that the spectral bound is continuous with respect to parameters (see Theorem A.1). As a straightforward consequence, the asymptotic behavior of the spectral bound as the diffusion coefficients approach zero is characterized (i.e., Theorem B(i)). To investigate the limiting profiles of the spectral bound with large diffusion coefficients, we distinguish between two cases. For the non-degenerate case (i.e., Theorem B(ii)), we employ ideas from [30] and [33]. To investigate the partially degenerate case (i.e., Theorem B(iii)), we combine perturbation techniques, comparison arguments, and methods used in the proof of the non-degenerate case and Theorem A.
The remaining part of the paper is organized as follows. In the next section, we provide the definition of the principal eigenvalue and present some preliminary results on positive operators and cooperative matrices. In Section 3, we establish the principal eigenvalue theory for partially degenerate linear evolution systems with nonlocal dispersal. In Section 4, we investigate the asymptotic behavior of the spectral bound of the associated nonlocal dispersal operators with small and large diffusion coefficients for both the non-degenerate case and the partially degenerate case. In Section 5, we employ these results to discuss the asymptotic behavior of the basic reproduction ratio for a infection model with cell-to-cell transmission and nonlocal viral dispersal. Additionally, we provide a proof of the continuity of the spectral bound of the associated nonlocal dispersal operators with respect to parameters in the Appendix.
2 Preliminaries
Let E be a Banach space with a cone K ⊂ E. We denote by Int(K) the interior of the cone K. We also use ≥(
We use r(A) and r e (A) to represent its spectral radius and essential spectral radius of A.
Definition 2.1.
Let E be a Banach space with a cone K ⊂ E. Assume that B is a bounded linear operator on E and there exists some c 0 such that B + c 0 I is a positive operator, where I is the identity map on E. The real number λ ∗ is called the principal eigenvalue of B if λ ∗ is an isolated eigenvalue of B with finite algebraic multiplicity corresponding to a positive eigenvector and λ ∗ ≥Reλ for all λ ∈ σ(B).
We remark that if λ ∗ is the principal eigenvalue of B, then λ ∗ = s(B).
Lemma 2.1.
Let E be a Banach space with a cone K ⊂ E. If B is a bounded positive linear operator on E and r(B) ∈ σ(B), then s(B) = r(B).
Proof.
It is obvious that r(B) ≥ s(B) by their definitions. Thanks to r(B) ∈ σ(B), we have r(B) ≤ s(B), and hence, s(B) = r(B).□
The following results can be derived directly by [47, Proposition 1] and the above lemma.
Remark 2.1.
Let E be a Banach space, and K ⊂ E be a normal cone with nonempty interior (for the definition of the normal cone, we refer to [48, Section 19]). If B is a bounded positive linear operator on E, then r(B) ∈ σ(B), and hence, s(B) = r(B).
Next, we review a generalized Krein–Rutman theorem (see, e.g., [20, Corollary 2.2]), which provide a primary tool to prove the existence of the principal eigenvalue for a non-compact bounded linear positive operator.
Lemma 2.2.
Let E be a Banach space with a total cone K ⊂ E (i.e.
Lemma 2.3.
Let E be a Banach space, and K ⊂ E be a cone with nonempty interior. Assume that B is a bounded positive linear operator on E and B n is strongly positive for some positive integer n. If r(B n ) is the principal eigenvalue of B n , then r(B) is the principal eigenvalue of B. Moreover, the following statements are valid:
There exists u ≫ 0 such that Bu = r(B)u.
r(B) is an algebraically simple eigenvalue of B.
u is the unique non-negative eigenvector of B up to scalar multiplications on E.
Proof.
In view of [44, Appendix A] or [48, Theorem 19.3], r(B n ) > 0 holds ture and the statements (i)–(iii) are valid with B replaced by B n . We assume that n ≥ 2 and let u ≫ 0 be an eigenvector of B n corresponding to r ≔ r(B n ). It then follows from Gelfand’s formula(see, e.g., [49, Theorem VI.6]) that λ n = r, where λ ≔ r(B) > 0.
Motivated by [42, Lemma 3.1], we first show that u is an eigenvector of B corresponding to λ. In view of B
n
[Bu] = B[B
n
u] = rBu, it is easy to see that Bu is an eigenvector of B
n
. Since r is an algebraically simple eigenvalue of B
n
, we have Bu = s
0
u ≥ 0 for some s
0 ≠ 0. Clearly, Bu > 0 and s
0 > 0, otherwise Bu = 0 yields that B
n
u = 0, which contradicts with r(B
n
) > 0. Thus,
(ii) Noting that r = λ n is an algebraically simple eigenvalue of B n , we first prove that λ is a geometrically simple eigenvalue of B. Indeed, if Bu′ = λu′ for some u′ ≠ 0, then B n u′ = λ n u′ implies that u′ is a scalar multiplication of u.
To show that λ is algebraically simple, we suppose that there exists a pair of v and t 0 such that (λI − B)v = t 0 u. By the fact that r = λ n is an algebraically simple eigenvalue of B n again, it then follows that t 0 = 0 and v is a scalar multiplication of u from
(iii) Suppose that there exists a pair of w > 0 and μ such that Bw = μw. It is easily seen that B n w = μ n w. Since u is the unique non-negative eigenvector of B n up to scalar multiplications on E, we then have μ n = r and w is a scalar multiplication of u.□
In order to facilitate the use of a generalized Krein–Rutman theorem, we provide the following lemma.
Lemma 2.4.
Let E be a Banach space, and K ⊂ E be a cone with nonempty interior. Assume that B is a bounded linear operator with s(B) > s
e
(B) and B + c
0
I is positive for some c
0 > 0. Then s(B) is the principal eigenvalue of B. If, in addition,
There exists u ≫ 0 such that Bu = s(B)u.
s(B) is an algebraically simple eigenvalue.
u is the unique non-negative eigenvector of B up to scalar multiplications on E.
There exists δ > 0 such that Reλ < s(B) − δ for any λ ∈ σ(B) \{s(B)}.
Proof.
It is easy to see that
Since B is a bounded linear operator, it then follows that all spectral points of B are bounded. Thus, s(B) > s
e
(B) implies that r(B + c
1
I) > r
e
(B + c
1
I) for some large constant c
1 with c
1 > c
0 due to (2.1). Lemma 2.2 yields that r(B + c
1
I) is the principal eigenvalue of B + c
1
I. By Lemma 2.1, we have r(B + c
1
I) = s(B + c
1
I) = s(B) + c
1, and hence, s(B) is the principal eigenvalue of B. Since
In view of s(B) > s
e
(B), to prove (iv), it is sufficient to verify that there is no spectral point μ ≠ s(B) such that Reμ ≥ s(B). Otherwise,
Lemma 2.5.
Let E be a Banach space, K ⊂ E be a cone with nonempty interior, and B be a bounded positive linear operator on E. If Bu = 0 for some u ∈ Int(K), then B is zero operator, that is, Bv = 0 for all v ∈ E.
Proof.
In view of u ∈ Int(K), there exists r > 0 such that u ≥ rv for all v ∈ E with ‖v‖ = 1. Since B is positive, we have 0 = Bu ≥ rBv, ∀v ∈ E with ‖v‖ = 1. Thus, Bv ≤ 0 and −Bv = B(−v) ≤ 0 imply that Bv = 0 for all v ∈ E.□
Lemma 2.6.
Let E be a Banach space, and K ⊂ E be a cone with nonempty interior. Assume that A is a bounded strongly positive linear operator and B is nonzero positive operator on E. We further assume that r(A) and r(A + B) are the principal eigenvalue of A and A + B, respectively. Then s(A + B) = r(A + B) > s(A) = r(A) > 0.
Proof.
Lemma 2.1 yields that s(A + B) = r(A + B) and s(A) = r(A). According to [50, Theorem 1.1] and [48, Theorem 19.3], it is easy to see that r(A + B) ≥ r(A) > 0. Suppose, by contradiction, that λ ≔ r(A + B) = r(A) > 0. By the definition of the principal eigenvalue, there exists u ≫ 0 and v ≫ 0 such that (A + B)u = λu and Av = λv. Choose t 0 > 0 such that u − t 0 v ≥ 0 but u − t 0 v ∉ Int(K). If u − t 0 v = 0, then
implies that Bu = 0, and hence, B is zero operator by Lemma 2.5, which is a contradiction. If u − t
0
v > 0, since A is strongly positive, then (2.2) yields that
In the remaining of the section, we present some results of cooperative and irreducible matrices, which provide some primary tools for the analysis later in this paper.
Lemma 2.7.
If A is an l × l non-negative and irreducible matrix whose diagonal elements are all positive, then A p is strongly positive for all p ≥ l − 1.
Proof.
Choose ϵ
0 small such that A − ϵ
0
I is non-negative. Since A can be written as A − ϵ
0
I + ϵ
0
I, so the conclusion holds for p = l − 1 by [51, (8.3.5)]. An easy computation yields that
We recall a result about the spectral bound of an irreducible cooperative matrix (see, e.g., [52]). For completeness, here we provide an elementary proof.
Lemma 2.8.
Assume that A is a cooperative and irreducible l × l matrix, B is a non-negative l × l non-zero matrix, then s(A + B) > s(A).
Proof.
Without loss of generality, we assume that A is a non-negative matrix whose diagonal elements are all positive, for otherwise we can replace A by A + μI with some large positive μ. Here I is the identity l × l matrix. Lemma 2.1 yields that s(A) = r(A) and s(A + B) = r(A + B). It suffices to show that r(A + B) > r(A).
According to Lemma 2.7, A l−1 and A l are strongly positive. We notice that BA l−1 is nonzero, so is (A + B) l − A l . From Lemma 2.6, it follows that [r(A + B)] l = r((A + B) l ) > r(A l ) = [r(A)] l . This completes the proof.□
We now present the last result of this section.
Lemma 2.9.
Assume that
Here A 11, A 12, A 21, A 22 are l 1 × l 1, l 1 × l 2, l 2 × l 1, l 2 × l 2 matrices. Let I, I 1 and I 2 be l × l, l 1 × l 1 and l 2 × l 2 identity matrices, respectively. Then the following statements are valid:
s(A) > s(A 22).
Write A μ ≔ A − C, where
Proof.
(i) Without loss of generality, we assume that A is a non-negative matrix whose diagonal elements are all positive, for otherwise we can replace A by A + cI with some large positive c. Let B = diag(A 11, A 22). It is easy to see that
Suppose, by contradiction, that λ ≔ s(A) = s(A 22). Lemma 2.1 yields that s(A) = r(A).
According to Lemma 2.7, A
l−1 and A
l
are strongly positive. By the Perron–Frobenius theorem (see, e.g., [53, Theorem 4.3.1]), there exist
Choose t
0 > 0 such that
(ii) We first prove that
Therefore,
Suppose that the inequality holds, that is,
It then follows that H ∈ σ(W) from
Lemma 2.8 yields a contradiction that H ≤ s(W) < s(A) = H, which leads to
(iii) Let λ(
μ
) ≔ s(A
μ
) be the principal eigenvalue of A
μ
with a strongly positive eigenvector
For each i = 1, …, l 1, we divide the i-th equation of (2.3) by μ i to obtain
Lemma 2.8 yields that λ( μ ) is decreasing with respect to μ i ∈ (0, + ∞), i = 1, …, l 1. Notice that λ( μ ) ≥ s(A 22) for all μ with μ i > 0, i = 1, …, l 1, we have
exists and λ
∗ ≥ s(A
22). We normalize
x
(
μ
) by
This implies that λ ∗ is an eigenvalue of A 22, and hence, λ ∗ ≤ s(A 22). Therefore, λ ∗ = s(A 22).
(iv) For a given λ > s(A 22), write
Thus,
We notice that A is irreducible, so is N − M. Moreover, for each j = l
1 + 1, …, l, the j-th column of N is nonzero. According to [54, Lemma 3.4] and [55, Theorem 2.2.7],
3 Principal eigenvalue for the partially degenerate case
In this section, we establish the principal eigenvalue theory for partially degenerate nonlocal dispersal systems. Let
where
Lemma 3.1.
Assume that (H1)–(H3) and (H4′) hold. Then the following statements are valid:
For any
For any
For any
Let
where
We remark that
In this section, we consider the following eigenvalue problem
Recall
Lemma 3.2.
Assume that (H1)–(H2) hold and K
i
(x, x) > 0 for all
Since
Lemma 3.3.
Assume that (H1) hold. Then the following statements are valid:
Lemma 3.4.
Assume that (H1)–(H3) and (H4′) hold. If A(x) are non-negative matrices whose diagonal elements are positive for all
Proof.
By the arguments similar to those in [22, Proposition 2.2],
and hence,
Theorem 3.1.
Assume that (H1)–(H3) and (H4′) hold. If there exists some
u
∗ ≫ 0 and
λ ∗ is an algebraically simple eigenvalue.
u ∗ is the unique non-negative eigenvector up to scalar multiplications on X.
Proof
Choose
Let I be an l × l identity matrix and
The remaining parts can be derived by Lemma 2.4.□
Define
Here
Define four linear operators
Let
Thus, (3.1) can be rewritten as
Lemma 3.5.
Assume that (H1)–(H2) hold. Then the following statements are valid:
s(A(x)) > s(A 22(x)) for all
η > η 22.
Proof.
In view of
For any
Here I
2 is the identity l
2 × l
2 matrix, and F
η
(x) is well-defined for all
For any λ > η
22, let
Here
Lemma 3.6.
Assume that (H1)–(H3) and (H4′) hold. Then the following statements are valid:
s(F H(x)(x)) = H(x),
h(x) ≤ H(x),
Proof.
According to [46, Theorem 7.27] and [28, Proposition 2.7], we obtain (i) and (ii). Part (iii) follows from Lemma 2.9(ii).
(iv) By [50, Theorem 1.1], for any given
(v) It is easy to see that
This implies that
(vi) For any λ > η
22, define
Since p is jointly continuous in
(vii) In view of [50, Theorem 1.1],
Case 1:
Thanks to (i) and Lemma 2.4,
Case 2:
For any given ϵ > 0, there exists δ 1 > 0 small enough with λ 1 > δ 1 + η 22 such that
due to (vi). According to [56, Theorem IV.3.1 and Remark IV.3.2], there exists δ 2 > 0 small enough with λ 1 > δ 2 + η 22 such that
It then follows that
if |λ − λ 1| ≤ δ, where δ = min(δ 1, δ 2). Combining the above two inequalities, we conclude that
This completes the proof.□
Next, we show the relation between (3.1) and (3.4).
Proposition 3.1.
Assume that (H1)–(H3) and (H4′) hold. The following statements are equivalent.
There exists λ ∗ > η and u ∗ ≫ 0 such that
There exists λ ∗ > η such that
There exists a pair of λ ∗ > η and
There exists a pair of λ ∗ > η and
Proof.
(i) ⇒ (ii). It has been proved in Theorem 3.1.
(ii) ⇒ (i). Lemma 2.4 implies that there exists
u
∗ ≫ 0 such that
(iii) ⇒ (iv). Define
(iv) ⇒ (iii). We notice that λ
∗ > η,
(iv) ⇒ (v). This can be derived by Lemma 2.4 directly.
(v) ⇒ (i)(vi). Let
(vi) ⇒ (v). Obviously.
(i) ⇒ (v). We use
(vi) ⇒ (iv). It is a straightforward result of Theorem 3.1.□
In the remaining of this section, we give a sufficient condition for the existence of the principal eigenvalue, that is, provide a criterion to derive one of the equivalent statements in Proposition 3.1. We need the following lemma, which is a straightforward result of Lemma 3.2.
Lemma 3.7.
Assume that (H1)–(H3) and (H4′) hold. If there is an open set Ω0 ⊂ Ω such that (η − h)−1 ∉ L
1(Ω0), then
The subsequent lemma presents a sufficient condition for the existence of the principal eigenvalue when l 1 = 1.
Lemma 3.8.
Assume that l 1 = 1 and (H1)–(H3) and (H4′) hold. Then there exists C > 0 independent of x such that
Moreover, if (η − H)−1 ∉ L 1(Ω0) for some open set Ω0 ⊂ Ω, then (η − h)−1 ∉ L 1(Ω0).
Proof.
We notice that F η (x) = h(x) and F H(x)(x) = H(x) when l 1 = 1, so
Let
We conclude that
Remark 3.1.
Assume that l
1 = 1, (H1)–(H3) and (H4′) hold, and there exists an open set Ω0 ⊂ Ω such that (η − H)−1 ∉ L
1(Ω0). According to Proposition 3.1 and Lemmas 3.7 and 3.8, we have
We are now in a position to prove the main result of this section, that is, to provide a sufficient condition for the existence of the principal eigenvalue.
Theorem 3.2.
Assume that (H1)–(H3) and (H4′) hold. If there is an open set Ω0 ⊂ Ω such that (η − H)−1 ∉ L
1(Ω0), then
Proof.
Define
where
Define
This completes the proof.□
Remark 3.2.
The conclusions of Theorem 3.1 and 3.2 and Proposition 3.1 are still valid when assumptions (H1)–(H3) and (H4′) are replaced by (i)–(iii) of Lemma 3.1.
4 Asymptotic behavior
In this section, we investigate the asymptotic behavior of the spectral bound as the diffusion coefficients go to zero and infinity for the non-degenerate case and partially degenerate case. We use the same notations X, X
+, X
1, X
1,+, X
2 and X
2,+ as in Section 3. Let
For any given
where
Thus,
Theorem 4.1.
Assume that (H1) holds. Then
4.1 Non-degenerate case with large diffusion
In this subsection, we investigate the asymptotic behavior of the spectral bound of
Lemma 4.1.
Assume that (H3) holds. Then for each 1 ≤ i ≤ l, there exists a unique strongly positive continuous function p
i
on
that is,
Proof.
For any given 1 ≤ i ≤ l, by the Krein–Rutman theorem (see, e.g., [48, Theorem 19.3]), there exists a positive number r and a strongly positive continuous function p on
This implies that
Integrating the above equation over Ω, we obtain r = 1. By Lemma 2.3(iii), there is no other positive continuous function satisfies (4.2).□
In the rest of the section, we use the notation p i as in (4.1).
Lemma 4.2.
Assume that (H1)–(H4) hold. We have the estimate |s(P( d ))| ≤ C, where the constant C is independent of d .
Proof.
Choose
Let
where
Define
According to [50, Theorem 1.1], it is easy to see that
It suffices to prove that
In view of (H3), χ
i
(x) = ∫
Ω
k
i
(y, x)dy > 0,
Lemma 4.3.
Assume that (H1)–(H4) hold. If min1≤i≤l
d
i
→ +∞, then
Proof.
By the analysis in Lemmas 3.3 and 3.5, it is not hard to see that
Recall
Theorem 4.2.
Assume that (H1)–(H4) hold. If min1≤i≤l d i → +∞, then
Proof.
By Lemma 4.3, there exists
We divide the i-th equation of (4.3) by
Noting that Lemma 4.2 leads to
we then have
uniformly on
Then (4.4) is equivalent to
uniformly on
pointwise on
pointwise on
which is bounded. By the dominated convergence theorem, for any 1 ≤ i ≤ l, we have
Since
uniformly on
uniformly on
and w
i
is continuous on
Letting n k → +∞, we have
that is,
4.2 Partially degenerate case with large diffusion
In this subsection, we study the asymptotic behavior of the spectral bound of
and
Lemma 4.4.
Assume that (H1) and (H2) hold. Then the following statements are valid:
If
Proof.
(i) Since p
j
is a strongly positive continuous function on
(ii) We notice that if
In view of [50, Theorem 1.1], it is easy to see that
(iii) Define a function
Obviously, f(γ) is decreasing and continuous with respect to γ ∈ (η
22, + ∞). In view of
Lemma 4.5.
Assume that (H1)–(H3) and (H4′) hold. Then the following statements are valid:
If
Proof.
Write
where
Let
We next introduce a family of linear operators
and present their properties.
Lemma 4.6.
Assume that (H1)–(H3) and (H4′) hold. Then the following statements are valid:
For any given γ > η 22, there exists
If
Proof.
Part (i) follows from Lemma 4.3 and Theorem 4.2. Part (ii) has been shown in Lemma 3.6. Part (iii) can be derived by the arguments similar to those in Lemma 4.4(iii).□
Lemma 4.7.
Assume that (H1)–(H3) and (H4′) hold and
Moreover,
Proof.
For any given ϵ
0 > 0 with γ* − 2ϵ
0 > η
22, in view of Lemma 4.3, we can choose
According to Lemma 4.6(i), there exists some
Now we assume that
Lemma 4.6(iii) yields that
by Lemma 4.6(ii). Proposition 3.1(iv) implies that
Summarizing Lemmas 4.3 and 4.7, we obtain another sufficient condition for the existence of the principal eigenvalue.
Theorem 4.3.
Assume that (H1)–(H3) hold. If, in addition, (H4) holds or (H4′) and
By repeating the arguments in the proof of Lemma 4.2, we have the following results.
Lemma 4.8.
Assume that (H1)–(H3) and (H4′) hold. We obtain the estimate s(P( d )) ≤ C, where the constant C is independent of d .
Finally, we are in a position to prove the main result of this subsection.
Theorem 4.4.
Assume that (H1)–(H3) and (H4′) hold. Then the following statements are valid:
If
If
Proof.
(i) Choose ϵ
0 > 0 such that γ* − 2ϵ
0 > η
22. By Lemma 4.7, there exists
For any 1 ≤ i ≤ l
1, we divide the i-th row of (4.9) by
Notice that Lemmas 4.7 and 4.8 yield that
Moreover, there exists a subsequence n
k
such that
for some constant v
i
. Write
it then follows that
as n
k
→ +∞. Therefore,
Letting n k → +∞, we have
This implies that
(ii) We now consider the case
Choose α
0 > 0 such that
let
and define
For any α > α
0, we have
Next, we prove the claim and fix α > α
0. Write
In addition,
Consequently, f(η
22 + α − α
0) ≤ 0, f(γ*(α)) = 0 and
which derive our claim.
For any given ϵ > 0, choose α
1 > α
0 > 0 with
In view of ∫
Ω
p
i
(x)dx = 1, ∀i = 1, …, l
1, using (i) with
Moreover, [50, Theorem 1.1] implies that
if
5 An application
In this section, we employ the above result to analyze the asymptotic behavior of the basic reproduction ratio for a infection model with cell-to-cell transmission and nonlocal viral dispersal. The authors in [58], [59], [60] investigated a viral infection model using a partially degenerate reaction-diffusion system. To account for the rapid movement of viruses, in this paper, we assume that the diffusion process exhibits nonlocal dispersion. Thus the modified model can be expressed as follows:
for all
System (5.1) has a unique infected-free steady state (0, S*(x), 0). For convenience, write
We linearize system (5.1) at the infected-free steady state (0, S*(x), 0) to obtain the following cooperative linear system:
for all
Let
According to [61, Section 3], for any d ≥ 0, the next generation operator is −F[B(d)]−1 and the basic reproduction ratio is
Then we can present the main result of this section:
Theorem 5.1.
Then the following statements are valid:
If
Proof.
(i) For any μ > 0, d ≥ 0, define
(ii) For any μ > 0, θ > 0, define
For any μ > 0 and γ > η 22(μ), define
We remark that
In the case where
By Theorem 4.4(i) with
We next prove that
In the case where
Thanks to Theorem 4.4(ii) with
It suffices to prove that
We finish this section with a brief discussion. The asymptotic behavior of the basic reproduction ratio for the general cooperative non-degenerate nonlocal dispersal system can be explored using Theorems 4.1 and 4.2, similar to the approach in [62], [63]. However, for the general partially degenerate case, further investigation is required, which we leave as a future endeavor.
Acknowledgments
I would like to thank Profs. Xiao-Qiang Zhao and Xing Liang for helpful discussions during the preparation of this work. I am also grateful to the anonymous referees for their careful reading and helpful comments, which led to an improvement of my original manuscript.
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
-
Use of Large Language Models, AI and Machine Learning Tools: I use Chatgpt to improve language.
-
Conflict of interest: There is no conflict of interest.
-
Research funding: This research is supported by the National Natural Science Foundation of China (12471168, 12171119) and the Fundamental Research Funds for the Central Universities (GK202304029, GK202306003, GK202402004).
-
Data availability: Not applicable.
Appendix A. Continuity of the spectral bound
The purpose of this section is to establish the continuity of the spectral bound with respect to parameters whether it degenerates or not. We use the same notations X, X
+, A,
where
Define a bounded linear operator
Here A′(x) and
Lemma A.1.
Assume that (H1) holds. For any ϵ > 0, there exists δ > 0 such that
If
If
We are now in a position to prove the main result of this section.
Theorem A.1.
Assume that (H1) holds. For any ϵ > 0, there exists δ > 0 such that
A′(x) is cooperative for all
Proof.
For any given number ϵ > 0, we proceed our proof into three steps.
Step 1: Prove the conclusion when
According to [50, Theorem 1.1], we have
By Lemma A.1, there exists δ
1 > 0 such that
In the case of
In the case of
then
and
The conclusion follows by choosing δ = min(δ 1, δ 2).
Step 2: Prove the conclusion when
According to [50, Theorem 1.1], we have
then
and
The conclusion follows by choosing δ = δ 3.
Step 3: Finish the proof in general cases.
Let
Clearly,
and
Let
where
Define
According to [50, Theorem 1.1], it is easy to see
Choosing δ = min(δ 1, δ 2, δ 3), we have
and
This implies that
and
Consequently,
which completes the proof.□
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Articles in the same Issue
- Frontmatter
- Research Articles
- Decay estimates for defocusing energy-critical Hartree equation
- A surprising property of nonlocal operators: the deregularising effect of nonlocal elements in convolution differential equations
- Long-time asymptotic behavior for the Hermitian symmetric space derivative nonlinear Schrödinger equation
- The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy
- Regularity of center-outward distribution functions in non-convex domains
- Existence and multiplicity of solutions for fractional p-Laplacian equation involving critical concave-convex nonlinearities
- Periodic solutions for a coupled system of wave equations with x-dependent coefficients
- Remarks on analytical solutions to compressible Navier–Stokes equations with free boundaries
- Homogenization of Smoluchowski-type equations with transmission boundary conditions
- Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity
- Principal spectral theory and asymptotic behavior of the spectral bound for partially degenerate nonlocal dispersal systems
Articles in the same Issue
- Frontmatter
- Research Articles
- Decay estimates for defocusing energy-critical Hartree equation
- A surprising property of nonlocal operators: the deregularising effect of nonlocal elements in convolution differential equations
- Long-time asymptotic behavior for the Hermitian symmetric space derivative nonlinear Schrödinger equation
- The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy
- Regularity of center-outward distribution functions in non-convex domains
- Existence and multiplicity of solutions for fractional p-Laplacian equation involving critical concave-convex nonlinearities
- Periodic solutions for a coupled system of wave equations with x-dependent coefficients
- Remarks on analytical solutions to compressible Navier–Stokes equations with free boundaries
- Homogenization of Smoluchowski-type equations with transmission boundary conditions
- Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity
- Principal spectral theory and asymptotic behavior of the spectral bound for partially degenerate nonlocal dispersal systems
