- Email: [email protected]
On the diffusive Nicholson’s blowflies equation with distributed delay
AlgebraLetters and its50Applications 466 (2015) 102–116 Applied Linear Mathematics (2015) 126–132
Contents lists at ScienceDirect Contents lists available at available ScienceDirect
LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
Inverse eigenvalue problem of Jacobi On the diffusive Nicholson’s blowflies equation withmatrix with mixed data distributed delay ∗ 1 Ying Keng Deng, Yixiang WuWei Department of Mathematics, University of LouisianaNanjing at Lafayette, Lafayette, LA 70504,and USA Department of Mathematics, University of Aeronautics Astronautics, Nanjing 210016, PR China
article
info a r t i c l e
abstract i n f o
a b s t r a c t
Article history: We study the global attractivity of the positive steady state of the diffusive NicholArticle history: Received 19 February 2015 In this paper, theFor inverse problem of reconstructing son’s equation with distributed delay. sucheigenvalue a problem, the monotone case has Received 2014 solved while the anon-monotone Received in revised form 16 June16 January been Jacobi matrix from its eigenvalues, leading principal one remains open, whichits is the consideration Accepted 20 September 2014 2015 submatrix and part of the eigenvalues of its submatrix of this paper. Accepted 16 June 2015 Available online 22 October 2014 is considered. The necessary and sufficient conditions for © 2015 Elsevier Ltd. All rights reserved. Submitted by Y. Wei Available online 2 July 2015
the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given. © 2014 Published by Elsevier Inc.
MSC: Keywords: 15A18 Reaction–diffusion equation 15A57 Distributed delay Sub- and supersolutions Keywords: Global attractivity Jacobi matrix Eigenvalue Inverse problem Submatrix
1. Introduction
In this paper, we solve an open problem on the global attractivity of the following diffusive Nicholson’s blowflies equation with distributed delay t t ut − d∆u = −τ u(x, t) + βτ f (t − s)u(x, s)ds exp − f (t − s)u(x, s)ds (1.1) −∞
−∞
for (x, t) ∈ Ω × (0, ∞), subject to no-flux boundary condition ∂u = 0, (x, t) ∈ ∂Ω × (0, ∞), ∂n where ∂/∂n is the outward normal derivative to ∂Ω , with initial condition 1
E-mail address: [email protected]. Tel.: +86 13914485239. u(x, t) = ϕ(x, t) ≥ 0, (x, t)
∈ Ω × (−∞, 0].
(1.2)
(1.3)
http://dx.doi.org/10.1016/j.laa.2014.09.031 Here, Ω is a bounded domain in RN with smooth boundary ∂Ω , the parameters β and τ are positive 0024-3795/© 2014 Published by Elsevier Inc. ∞ constants, and the nonnegative kernel function f satisfies 0 f (t)dt = 1. More specifically, if e < β ≤ e2 , we ∗ Corresponding author. E-mail address: [email protected] (Y. Wu).
http://dx.doi.org/10.1016/j.aml.2015.06.013 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
K. Deng, Y. Wu / Applied Mathematics Letters 50 (2015) 126–132
127
show that the unique positive steady state u∗ = ln β is globally attractive for any nonnegative, nontrivial bounded initial data. This model is proposed by Gourley and Ruan in [1] as a generalization of the diffusive Nicholson’s blowflies equation with discrete delay. It was shown that if β < 1 the zero steady state is globally asymptotically stable, if β ∈ (1, e2 ) the steady state u∗ is linearly stable, and if β ∈ (1, e] the steady state u∗ is globally attractive. However, the global attractivity of u∗ for the case β ∈ (e, e2 ] remains open, where the difficulty arises from the non-monotonicity of g(u) = βue−u . (If β ∈ (1, e], then u∗ ∈ (0, 1], and the authors of [1] took the advantage of the monotonicity of g in the interval (0, 1] to establish the global attractivity result). Nicholson’s blowflies equation was first used by Gurney et al. [2] to explain Nicholson’s experimental data [3], and this model has been intensively studied since then (see Ruan’s survey paper [4] for some earlier results). To mention a few studies, So et al. [5,6] considered the following diffusive Nicholson’s blowflies equation ut − d∆u = −τ u(x, t) + βτ u(x, t − 1)e−u(x,t−1) , and they proved the global attractivity of the positive steady state if β ∈ (1, e2 ] under Dirichlet boundary condition and if β ∈ (1, e] under Neumann boundary condition. The case β ∈ (e, e2 ] with Neumann boundary condition was later solved by Yi and Zou [7] through a dynamical systems argument. And a nonlocal version of this model was studied by Zhao [8] via a fluctuation method. If the domain is unbounded, the authors of [9] established a comparison principle for a diffusive nonlocal Nicholson’s blowflies equation and used it to prove the global attractivity result if β ∈ (1, e2 ], and their approach is also applicable to the bounded domain case. For a similar model without diffusion on R [10], Yi and Zou worked in the settings of compact open topology to determine the global dynamics. It is worth pointing out that all these studies concentrated on discrete delay, and their methods do not appear applicable to Eq. (1.1). 2. Global attractivity of the positive steady state In this section, we prove the global attractivity of the positive steady state u∗ = ln β. We adopt the method of successive improved sub- and supersolutions, which has been used by many authors [11–15] to study the global stability. Nevertheless, the problem under consideration here is more complicated due to the non-monotonicity. We first introduce Gourley and Ruan’s definition of coupled sub- and supersolutions of (1.1)–(1.3) (see [1]), which is motivated by Redlinger [16]. A pair of suitably smooth bounded functions v(x, t) and w(x, t) are called a pair of sub- and supersolutions for (1.1)–(1.3) on Ω × (0, ∞), respectively, if the following conditions are satisfied: 1. v(x, t) ≤ w(x, t) on Ω × [0, ∞); 2. The inequalities vt − d∆v ≤ −τ v + βτ f ∗ φ exp(−f ∗ φ), wt − d∆w ≥ −τ w + βτ f ∗ φ exp(−f ∗ φ) hold for all functions φ ∈ C((Ω × [0, ∞)) ∪ (Ω × (−∞, 0])) in Ω × (0, ∞), with v ≤ φ ≤ w, where f ∗ φ t denotes the convolution f ∗ φ = −∞ f (t − s)φ(x, s)ds; ∂v 3. ∂n = ∂w ∂n = 0 on ∂Ω × (0, ∞); 4. v(x, t) ≤ ϕ(x, t) ≤ w(x, t) in Ω × (−∞, 0]. The following lemma about the existence of the solution can be found in [1], which is originally due to Redlinger.
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K. Deng, Y. Wu / Applied Mathematics Letters 50 (2015) 126–132
Lemma 2.1. Let v(x, t) and w(x, t) be a pair of sub- and supersolutions for (1.1)–(1.3). Suppose that the initial data ϕ ∈ C(Ω × (−∞, 0]) is bounded, nonnegative, uniformly H¨ older continuous with ϕ(x, 0) ∈ C 1 (Ω ). Then there exists a unique nonnegative classical solution u(x, t) of the problem (1.1)–(1.3) such that v(x, t) ≤ u(x, t) ≤ w(x, t) for (x, t) ∈ Ω × [0, ∞). The following result states that the solution of (1.1)–(1.3) is uniformly bounded, which can be found in [1]. Lemma 2.2. There exists a positive constant K = β/e such that for any ε > 0 u(x, t) ≤ K + ε
on Ω × [T, ∞),
for some T dependent on ϕ. Let g(u) = βue−u . It can be easily shown that g is increasing in (0, 1) and decreasing in (1, ∞). Moreover, the positive steady state satisfies 1 < u∗ < β/e, and there is a unique A ∈ (0, 1) such that g(A) = g(β/e). We can prove the following result. Lemma 2.3. Suppose that e < β ≤ e2 . Then the following statements hold. (i) g(A) = g(β/e) > 1. (ii) If g(λ) = µ and g(µ) = λ for 1 < λ ≤ µ ≤ β/e, then λ = µ = ln β. Proof. We only prove the second part. Suppose that g(λ) = µ and g(µ) = λ. Multiplying these two equations, we get λ + µ = 2 ln β. Substituting this into the first equation, we have βλe−λ + λ − 2 ln β = 0. Let m(x) = βxe−x +x−2 ln β. Then m′ (x) = βe−x −βxe−x +1 and m′′ (x) = βe−x (x−2). Since m′ (x) is decreasing on (1, 2) and increasing on (2, β/e) and m′ (2) = 1−β/e2 ≥ 0, m′ is nonnegative on (1, β/e). So m is increasing on (1, β/e). Thus λ = ln β is the unique solution of m(x) = 0 on [1, β/e]. Therefore λ = µ = ln β. We are now in a position to prove the main theorem on the global attractivity of the positive steady state u∗ = ln β of problem (1.1)–(1.3). Theorem 2.4. Suppose that e < β ≤ e2 and the initial data ϕ ∈ C(Ω × (−∞, 0]) is bounded, nontrivial, nonnegative, uniformly H¨ older continuous with ϕ(x, 0) ∈ C 1 (Ω ). Then the solution u(x, t) of problem (1.1)–(1.3) satisfies limt→∞ u(x, t) = u∗ uniformly for x ∈ Ω . Proof. We define u(t) = min u(x, t)
and u(t) = max u(x, t)
x∈Ω
for t > 0.
x∈Ω
Without loss of generality, we assume that ϕ(x, 0) is nontrivial. It follows that u(x, t) > 0 for (x, t) ∈ Ω × (0, ∞) since u is bounded below by the solution of the following problem: ut − ∆u = −τ u(x, t) (x, t) ∈ Ω × (0, ∞), ∂u (x, t) = 0 (x, t) ∈ ∂Ω × (0, ∞), ∂n u(x, 0) = ϕ(x, 0) x ∈ Ω. Consequently, u(t) > 0 for t ∈ (0, ∞). We then define I = [lim inf u(t), lim sup u(t)]. t→∞
t→∞
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K. Deng, Y. Wu / Applied Mathematics Letters 50 (2015) 126–132
It now suffices to show that I = {u∗ }. In view of Lemma 2.2, we have that I ⊆ [0, K] with K = β/e. We complete our proof in three steps. Step 1. We first improve the lower bound of I. We claim that I ⊆ (βδ1 e−δ1 , K] for some small δ1 > 0. To see this, let ε > 0 small be given. Since I ⊆ [0, K], there exists t1 > 1 such that u(x, t) ≤ u(t) ≤ K + ε for (x, t) ∈ Ω × [t1 − 1, ∞). And there exists t2 > t1 such that
∞
f (s)ds ≤ ε for t ≥ t2 . t−t1
Define δ1 = 1/2 min{min{u(t) : t1 − 1 ≤ t ≤ t2 }, A}. Since u(t) > 0, δ1 > 0. Let v 1 be given by 0 v 1 (t) = (δ1 − ε)(t − t1 + 1) δ − ε 1
t ≤ t1 − 1, t1 − 1 < t ≤ t1 , t1 < t ≤ t2 ,
v˙ 1 = −τ v 1 + τ β(δ1 − ε)(1 − ε) exp(−(δ1 − ε)(1 − ε))
(2.1) for t > t2 .
And let w1 be given by t ≤ t1 − 1, M 1 w (t) = K + ε + (M − K − ε)(t1 − t) t1 − 1 < t ≤ t1 , K + ε t > t1 ,
(2.2)
where M is some large positive constant such that u ≤ M on Ω × R. Clearly, v 1 ≤ u ≤ w1 on Ω × (−∞, t2 ]. We then check that v 1 and w1 are a pair of sub- and supersolutions of (1.1)–(1.3), respectively. For t ≥ t2 and v 1 ≤ φ ≤ w1 , t t t f (t − s)φ(x, s)ds ≥ f (t − s)φ(x, s)ds ≥ (δ1 − ε) f (t − s)ds ≥ (δ1 − ε)(1 − ε). (2.3) −∞
t1
t1
Note that g is increasing on (0, 1) and decreasing on (1, K). Moreover, g is bounded by K and g(A) = g(K). And by the choice of δ1 , we have δ1 < A. So if ε is small, it follows from (2.3) that β(δ1 − ε)(1 − ε) exp(−(δ1 − ε)(1 − ε)) = g((δ1 − ε)(1 − ε)) ≤ g(f ∗ φ) = βf ∗ φ exp(−f ∗ φ) ≤ K + ε for all (x, t) ∈ Ω × [t2 , ∞) and v 1 ≤ φ ≤ w1 . Thus, it holds that vt1 − d∆v 1 ≤ −τ v 1 + βτ f ∗ φ exp(−f ∗ φ), wt1 − d∆w1 ≥ −τ w1 + βτ f ∗ φ exp(−f ∗ φ), for (x, t) ∈ Ω ×[t2 , ∞) and v 1 ≤ φ ≤ w1 . Hence, v 1 and w1 are a pair of sub- and supersolutions of (1.1)–(1.3) on Ω × (t2 , ∞), respectively. By Lemma 2.1, we have that v 1 ≤ u ≤ w1 . Since v 1 → β(δ1 − ε)(1 − ε) exp(−(δ1 − ε)(1 − ε)) and w1 → K + ϵ as t → ∞ and ε is arbitrary, we have I ⊆ [βδ1 e−δ1 , K].
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K. Deng, Y. Wu / Applied Mathematics Letters 50 (2015) 126–132
Step 2. We then claim that I ⊆ (1, K]. If δ2 := βδ1 e−δ1 > 1, we are done. Otherwise we continue to improve the lower bound of I and prove that I ⊆ [βδ2 e−δ2 , K]. Let ε > 0 small be given. Since I ⊆ [δ2 , K], there exists t3 > 1 such that δ2 − ε ≤ u(x, t) ≤ K + ε for (x, t) ∈ Ω × [t3 − 1, ∞). And there exists t4 > t3 such that
∞
f (s)ds ≤ ε for t ≥ t4 . t−t3
Let v 2 be given by 0 v 2 (t) = (δ2 − ε)(t − t3 + 1) δ − ε 2
t ≤ t3 − 1, t3 − 1 < t ≤ t3 , t3 < t ≤ t4 ,
v˙ 2 = −τ v 2 + τ β(δ2 − ε)(1 − ε) exp(−(δ2 − ε)(1 − ε))
(2.4) for t > t4 .
And let w2 be given by t ≤ t3 − 1, M 2 w (t) = K + ε + (M − K − ε)(t3 − t) t3 − 1 < t ≤ t3 , K + ε t > t3 ,
(2.5)
where M is the same as above. Clearly, v 2 ≤ u ≤ w2 on Ω × (−∞, t4 ], and one can check as in Step 1 that v 2 and w2 are a pair of sub- and supersolutions of problem (1.1)–(1.3) in Ω × (t4 , ∞), respectively. Since ε is arbitrary, we have that I ⊆ [βδ2 e−δ2 , K]. If βδ2 e−δ2 > 1, the claim is valid. Otherwise we can repeat the process to obtain a sequence {δk } satisfying δk = βδk−1 e−δk−1 . It is easy to see that there exists a δk > 1 for some k > 2. (Otherwise δk ≤ 1 for all k. Then the sequence {δk } is monotonically increasing because g is increasing on [0, 1]. So {δk } converges to ln β, which is impossible since ln β > 1.) As detailed above, we can construct sub- and supersolutions to improve the lower bound and show I ⊆ [βδk−1 e−δk−1 , K] = [δk , K]. Hence we have I ⊆ (1, K]. Step 3. We now show I = {u∗ }. Since I ⊆ (1, K], we can find λ1 > 1 such that I ⊆ [λ1 , K]. Since g(K) = g(β/e) > 1 by Lemma 2.3, we can make λ1 small such that λ1 < g(K). Let µ1 = K. We then define two sequences {µi } and {λi } by µi+1 = g(λi )
and λi+1 = g(µi )
for i > 1.
(2.6)
We claim that 1 < λ1 < λ2 < · · · < λi < u∗ < µi < · · · < µ2 < µ1 = K.
(2.7)
To see this, we first verify that 1 < u∗ < µ1 = K. (Note that u∗ = ln β and K = β/e with β ∈ (e, e2 ]. So 1 < u∗ is clear. Let h(x) = ln x − x/e. We have h′ (x) = 1/x − 1/e < 0 on (e, e2 ]. Hence h(β) < h(e) = 0, which implies ln β < β/e.) Since g is decreasing in (1, ∞) and 1 < u∗ < µ1 = K, we get g(K) = g(µ1 ) < g(u∗ ) = u∗ < g(1). Noticing g(1) = K = µ1 and g(K) = g(µ1 ) = λ2 by (2.6), this inequality leads to g(K) = λ2 < u∗ < µ1 = K. By the choices of λ1 , we have 1 < λ1 < g(K) = λ2 < u∗ < µ1 = K. Since g is decreasing on (1, ∞) and 1 < λ1 < u∗ , we have g(u∗ ) < g(λ1 ) < g(1). Since g(λ1 ) = µ2 by (2.6), this gives u∗ < µ2 < µ1 . Hence, we have 1 < λ1 < λ2 < u∗ < µ2 < µ1 . Continuing this process, we can prove (2.7).
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K. Deng, Y. Wu / Applied Mathematics Letters 50 (2015) 126–132
Now we assume that I ⊆ [λi , µi ] has been proved for some i ≥ 1. Let ε > 0 small be given. Since I ⊆ [λi , µi ], there exists t5 > 1 such that for (x, t) ∈ Ω × [t5 − 1, ∞).
λi − ε ≤ u(x, t) ≤ µi + ε And there exists t6 > t5 such that
∞
f (s)ds ≤ ε for t ≥ t6 . t−t5
Let v i be given by 0 i v (t) = (λi − ε)(t − t5 + 1) λ − ε i
t ≤ t5 − 1, t5 − 1 < t ≤ t5 , t5 < t ≤ t6 ,
(2.8)
v˙ i = −τ v i + τ β(εM + ε + µi ) exp(−(εM + ε + µi ))
for t > t6 .
And let wi be given by t ≤ t5 − 1, M i w (t) = µi + ε + (M − µi − ε)(t5 − t) t5 − 1 < t ≤ t5 , µ + ε t 5 < t ≤ t6 , i w˙ i = −τ wi + τ β(λi − ε)(1 − ε) exp(−(λi − ε)(1 − ε))
(2.9)
for t > t6 ,
where M is the same as above. It then follows that v i ≤ u ≤ wi on Ω × (−∞, t6 ]. For t ≥ t6 and v i ≤ φ ≤ wi , t t5 t f ∗φ= f (t − s)φ(x, s)ds = f (t − s)φ(x, s)ds + f (t − s)φ(x, s)ds −∞
−∞
t5
≤ εM + ε + µi ,
(2.10)
and
t
f ∗φ=
t
f (t − s)φ(x, s)ds ≥ −∞
f (t − s)φ(x, s)ds ≥ (λi − ε)(1 − ε).
(2.11)
t5
Since g is decreasing on (1, ∞) and λi > 1, it follows from (2.10)–(2.11) that β(εM + ε + µi ) exp(−(εM + ε + µi )) ≤ βf ∗ φ exp(−f ∗ φ) ≤ β(λi − ε)(1 − ε) exp(−(λi − ε)(1 − ε)) for all (x, t) ∈ Ω × [t6 , ∞) and v i ≤ φ ≤ wi , when ε is small. Thus, it holds that vti − d∆v i ≤ −τ v i + βτ f ∗ φ exp(−f ∗ φ), wti − d∆wi ≥ −τ wi + βτ f ∗ φ exp(−f ∗ φ) for (x, t) ∈ Ω × [t6 , ∞) and v i ≤ φ ≤ wi . So v i and wi are a pair of sub- and supersolutions of problem (1.1)–(1.3) in Ω × (t6 , ∞). Since v i → g(εM + ε + µi ) and wi → g((λi − ε)(1 − ε)), we have I ⊆ [g(εM + ε + µi ), g((λi − ε)(1 − ε))]. Since ε is arbitrary, we have I ⊆ [g(µi ), g(λi )] = [λi+1 , µi+1 ]. Hence by induction, it holds that I ⊆ [λn , µn ] for all n. Because of the monotonicity of {λi } and {µi }, there exist λ and µ with 1 < λ ≤ µ < K such that λi → λ
and µi → µ.
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K. Deng, Y. Wu / Applied Mathematics Letters 50 (2015) 126–132
Therefore, we have I ⊆ [λ, µ]. Moreover, by the definition of {λi } and {µi }, we find that λ = g(µ) and µ = g(λ). It then follows from Lemma 2.3 that λ = µ = u∗ . Hence, I = {u∗ }, and the proof is completed. Acknowledgments The authors would like to thank the referees for their helpful comments. References [1] S.A. Gourley, S. Ruan, Dynamics of the diffusive Nicholson’s blowflies equation with distributed delay, Proc. Roy. Soc. Edinburgh 130A (2000) 1275–1291. [2] W.S.C. Gurney, S.P. Blythe, R.M. Nisbet, Nicholson’s blowflies revisited, Nature 287 (1980) 17–21. [3] A.J. Nicholson, An outline of the dynamics of animal population, Aust. J. Zool. 2 (1954) 9–65. [4] S. Ruan, Spatial–temporal dynamics in nonlocal epidemiological models, in: Mathematics for Life Science and Medicine, Springer, Berlin, Heidelberg, 2007, pp. 97–122. [5] J.W.-H. So, Y. Yang, Dirichlet problem for the diffusive Nicholson’s blowflies equation, J. Differential Equations 150 (1998) 317–348. [6] J.W.-H. So, J.S. Yu, Global attractivity and uniform persistence in Nicholson’s blowflies, Differ. Equ. Dyn. Syst. 2 (1994) 11–18. [7] T. Yi, X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case, J. Differential Equations 245 (2008) 3376–3388. [8] X. Zhao, Global attractivity in a class of nonmonotone reaction–diffusion equations with time delay, Can. Appl. Math. Q. 17 (2009) 271–281. [9] Z. Wang, W. Li, Dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity, Proc. Roy. Soc. Edinburgh 140A (2010) 1081–1109. [10] T. Yi, X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differential Equations 251 (2011) 2598–2611. [11] K. Deng, Y. Wu, Asymptotic behavior for a reaction–diffusion population model with delay, Discrete Contin. Dyn. Syst. Ser. B 20 (2015) 385–395. [12] S.A. Gourley, N.F. Briton, On a modified Volterra population equation with diffusion, Nonlinear Anal. 21 (1993) 389–395. [13] S.A. Gourley, J.W.-H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol. 44 (2002) 49–78. [14] R. Laister, Global asymptotic behavior in some functional parabolic equations, Nonlinear Anal. 50 (2002) 347–361. [15] R. Redlinger, On Volterra’s population equation with diffusion, SIAM J. Math. Anal. 16 (1985) 135–142. [16] R. Redlinger, Existence theorems for semilinear parabolic systems with functionals, Nonlinear Anal. 8 (1984) 667–682.
Contents lists at ScienceDirect Contents lists available at available ScienceDirect
LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
Inverse eigenvalue problem of Jacobi On the diffusive Nicholson’s blowflies equation withmatrix with mixed data distributed delay ∗ 1 Ying Keng Deng, Yixiang WuWei Department of Mathematics, University of LouisianaNanjing at Lafayette, Lafayette, LA 70504,and USA Department of Mathematics, University of Aeronautics Astronautics, Nanjing 210016, PR China
article
info a r t i c l e
abstract i n f o
a b s t r a c t
Article history: We study the global attractivity of the positive steady state of the diffusive NicholArticle history: Received 19 February 2015 In this paper, theFor inverse problem of reconstructing son’s equation with distributed delay. sucheigenvalue a problem, the monotone case has Received 2014 solved while the anon-monotone Received in revised form 16 June16 January been Jacobi matrix from its eigenvalues, leading principal one remains open, whichits is the consideration Accepted 20 September 2014 2015 submatrix and part of the eigenvalues of its submatrix of this paper. Accepted 16 June 2015 Available online 22 October 2014 is considered. The necessary and sufficient conditions for © 2015 Elsevier Ltd. All rights reserved. Submitted by Y. Wei Available online 2 July 2015
the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given. © 2014 Published by Elsevier Inc.
MSC: Keywords: 15A18 Reaction–diffusion equation 15A57 Distributed delay Sub- and supersolutions Keywords: Global attractivity Jacobi matrix Eigenvalue Inverse problem Submatrix
1. Introduction
In this paper, we solve an open problem on the global attractivity of the following diffusive Nicholson’s blowflies equation with distributed delay t t ut − d∆u = −τ u(x, t) + βτ f (t − s)u(x, s)ds exp − f (t − s)u(x, s)ds (1.1) −∞
−∞
for (x, t) ∈ Ω × (0, ∞), subject to no-flux boundary condition ∂u = 0, (x, t) ∈ ∂Ω × (0, ∞), ∂n where ∂/∂n is the outward normal derivative to ∂Ω , with initial condition 1
E-mail address: [email protected]. Tel.: +86 13914485239. u(x, t) = ϕ(x, t) ≥ 0, (x, t)
∈ Ω × (−∞, 0].
(1.2)
(1.3)
http://dx.doi.org/10.1016/j.laa.2014.09.031 Here, Ω is a bounded domain in RN with smooth boundary ∂Ω , the parameters β and τ are positive 0024-3795/© 2014 Published by Elsevier Inc. ∞ constants, and the nonnegative kernel function f satisfies 0 f (t)dt = 1. More specifically, if e < β ≤ e2 , we ∗ Corresponding author. E-mail address: [email protected] (Y. Wu).
http://dx.doi.org/10.1016/j.aml.2015.06.013 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
K. Deng, Y. Wu / Applied Mathematics Letters 50 (2015) 126–132
127
show that the unique positive steady state u∗ = ln β is globally attractive for any nonnegative, nontrivial bounded initial data. This model is proposed by Gourley and Ruan in [1] as a generalization of the diffusive Nicholson’s blowflies equation with discrete delay. It was shown that if β < 1 the zero steady state is globally asymptotically stable, if β ∈ (1, e2 ) the steady state u∗ is linearly stable, and if β ∈ (1, e] the steady state u∗ is globally attractive. However, the global attractivity of u∗ for the case β ∈ (e, e2 ] remains open, where the difficulty arises from the non-monotonicity of g(u) = βue−u . (If β ∈ (1, e], then u∗ ∈ (0, 1], and the authors of [1] took the advantage of the monotonicity of g in the interval (0, 1] to establish the global attractivity result). Nicholson’s blowflies equation was first used by Gurney et al. [2] to explain Nicholson’s experimental data [3], and this model has been intensively studied since then (see Ruan’s survey paper [4] for some earlier results). To mention a few studies, So et al. [5,6] considered the following diffusive Nicholson’s blowflies equation ut − d∆u = −τ u(x, t) + βτ u(x, t − 1)e−u(x,t−1) , and they proved the global attractivity of the positive steady state if β ∈ (1, e2 ] under Dirichlet boundary condition and if β ∈ (1, e] under Neumann boundary condition. The case β ∈ (e, e2 ] with Neumann boundary condition was later solved by Yi and Zou [7] through a dynamical systems argument. And a nonlocal version of this model was studied by Zhao [8] via a fluctuation method. If the domain is unbounded, the authors of [9] established a comparison principle for a diffusive nonlocal Nicholson’s blowflies equation and used it to prove the global attractivity result if β ∈ (1, e2 ], and their approach is also applicable to the bounded domain case. For a similar model without diffusion on R [10], Yi and Zou worked in the settings of compact open topology to determine the global dynamics. It is worth pointing out that all these studies concentrated on discrete delay, and their methods do not appear applicable to Eq. (1.1). 2. Global attractivity of the positive steady state In this section, we prove the global attractivity of the positive steady state u∗ = ln β. We adopt the method of successive improved sub- and supersolutions, which has been used by many authors [11–15] to study the global stability. Nevertheless, the problem under consideration here is more complicated due to the non-monotonicity. We first introduce Gourley and Ruan’s definition of coupled sub- and supersolutions of (1.1)–(1.3) (see [1]), which is motivated by Redlinger [16]. A pair of suitably smooth bounded functions v(x, t) and w(x, t) are called a pair of sub- and supersolutions for (1.1)–(1.3) on Ω × (0, ∞), respectively, if the following conditions are satisfied: 1. v(x, t) ≤ w(x, t) on Ω × [0, ∞); 2. The inequalities vt − d∆v ≤ −τ v + βτ f ∗ φ exp(−f ∗ φ), wt − d∆w ≥ −τ w + βτ f ∗ φ exp(−f ∗ φ) hold for all functions φ ∈ C((Ω × [0, ∞)) ∪ (Ω × (−∞, 0])) in Ω × (0, ∞), with v ≤ φ ≤ w, where f ∗ φ t denotes the convolution f ∗ φ = −∞ f (t − s)φ(x, s)ds; ∂v 3. ∂n = ∂w ∂n = 0 on ∂Ω × (0, ∞); 4. v(x, t) ≤ ϕ(x, t) ≤ w(x, t) in Ω × (−∞, 0]. The following lemma about the existence of the solution can be found in [1], which is originally due to Redlinger.
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Lemma 2.1. Let v(x, t) and w(x, t) be a pair of sub- and supersolutions for (1.1)–(1.3). Suppose that the initial data ϕ ∈ C(Ω × (−∞, 0]) is bounded, nonnegative, uniformly H¨ older continuous with ϕ(x, 0) ∈ C 1 (Ω ). Then there exists a unique nonnegative classical solution u(x, t) of the problem (1.1)–(1.3) such that v(x, t) ≤ u(x, t) ≤ w(x, t) for (x, t) ∈ Ω × [0, ∞). The following result states that the solution of (1.1)–(1.3) is uniformly bounded, which can be found in [1]. Lemma 2.2. There exists a positive constant K = β/e such that for any ε > 0 u(x, t) ≤ K + ε
on Ω × [T, ∞),
for some T dependent on ϕ. Let g(u) = βue−u . It can be easily shown that g is increasing in (0, 1) and decreasing in (1, ∞). Moreover, the positive steady state satisfies 1 < u∗ < β/e, and there is a unique A ∈ (0, 1) such that g(A) = g(β/e). We can prove the following result. Lemma 2.3. Suppose that e < β ≤ e2 . Then the following statements hold. (i) g(A) = g(β/e) > 1. (ii) If g(λ) = µ and g(µ) = λ for 1 < λ ≤ µ ≤ β/e, then λ = µ = ln β. Proof. We only prove the second part. Suppose that g(λ) = µ and g(µ) = λ. Multiplying these two equations, we get λ + µ = 2 ln β. Substituting this into the first equation, we have βλe−λ + λ − 2 ln β = 0. Let m(x) = βxe−x +x−2 ln β. Then m′ (x) = βe−x −βxe−x +1 and m′′ (x) = βe−x (x−2). Since m′ (x) is decreasing on (1, 2) and increasing on (2, β/e) and m′ (2) = 1−β/e2 ≥ 0, m′ is nonnegative on (1, β/e). So m is increasing on (1, β/e). Thus λ = ln β is the unique solution of m(x) = 0 on [1, β/e]. Therefore λ = µ = ln β. We are now in a position to prove the main theorem on the global attractivity of the positive steady state u∗ = ln β of problem (1.1)–(1.3). Theorem 2.4. Suppose that e < β ≤ e2 and the initial data ϕ ∈ C(Ω × (−∞, 0]) is bounded, nontrivial, nonnegative, uniformly H¨ older continuous with ϕ(x, 0) ∈ C 1 (Ω ). Then the solution u(x, t) of problem (1.1)–(1.3) satisfies limt→∞ u(x, t) = u∗ uniformly for x ∈ Ω . Proof. We define u(t) = min u(x, t)
and u(t) = max u(x, t)
x∈Ω
for t > 0.
x∈Ω
Without loss of generality, we assume that ϕ(x, 0) is nontrivial. It follows that u(x, t) > 0 for (x, t) ∈ Ω × (0, ∞) since u is bounded below by the solution of the following problem: ut − ∆u = −τ u(x, t) (x, t) ∈ Ω × (0, ∞), ∂u (x, t) = 0 (x, t) ∈ ∂Ω × (0, ∞), ∂n u(x, 0) = ϕ(x, 0) x ∈ Ω. Consequently, u(t) > 0 for t ∈ (0, ∞). We then define I = [lim inf u(t), lim sup u(t)]. t→∞
t→∞
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It now suffices to show that I = {u∗ }. In view of Lemma 2.2, we have that I ⊆ [0, K] with K = β/e. We complete our proof in three steps. Step 1. We first improve the lower bound of I. We claim that I ⊆ (βδ1 e−δ1 , K] for some small δ1 > 0. To see this, let ε > 0 small be given. Since I ⊆ [0, K], there exists t1 > 1 such that u(x, t) ≤ u(t) ≤ K + ε for (x, t) ∈ Ω × [t1 − 1, ∞). And there exists t2 > t1 such that
∞
f (s)ds ≤ ε for t ≥ t2 . t−t1
Define δ1 = 1/2 min{min{u(t) : t1 − 1 ≤ t ≤ t2 }, A}. Since u(t) > 0, δ1 > 0. Let v 1 be given by 0 v 1 (t) = (δ1 − ε)(t − t1 + 1) δ − ε 1
t ≤ t1 − 1, t1 − 1 < t ≤ t1 , t1 < t ≤ t2 ,
v˙ 1 = −τ v 1 + τ β(δ1 − ε)(1 − ε) exp(−(δ1 − ε)(1 − ε))
(2.1) for t > t2 .
And let w1 be given by t ≤ t1 − 1, M 1 w (t) = K + ε + (M − K − ε)(t1 − t) t1 − 1 < t ≤ t1 , K + ε t > t1 ,
(2.2)
where M is some large positive constant such that u ≤ M on Ω × R. Clearly, v 1 ≤ u ≤ w1 on Ω × (−∞, t2 ]. We then check that v 1 and w1 are a pair of sub- and supersolutions of (1.1)–(1.3), respectively. For t ≥ t2 and v 1 ≤ φ ≤ w1 , t t t f (t − s)φ(x, s)ds ≥ f (t − s)φ(x, s)ds ≥ (δ1 − ε) f (t − s)ds ≥ (δ1 − ε)(1 − ε). (2.3) −∞
t1
t1
Note that g is increasing on (0, 1) and decreasing on (1, K). Moreover, g is bounded by K and g(A) = g(K). And by the choice of δ1 , we have δ1 < A. So if ε is small, it follows from (2.3) that β(δ1 − ε)(1 − ε) exp(−(δ1 − ε)(1 − ε)) = g((δ1 − ε)(1 − ε)) ≤ g(f ∗ φ) = βf ∗ φ exp(−f ∗ φ) ≤ K + ε for all (x, t) ∈ Ω × [t2 , ∞) and v 1 ≤ φ ≤ w1 . Thus, it holds that vt1 − d∆v 1 ≤ −τ v 1 + βτ f ∗ φ exp(−f ∗ φ), wt1 − d∆w1 ≥ −τ w1 + βτ f ∗ φ exp(−f ∗ φ), for (x, t) ∈ Ω ×[t2 , ∞) and v 1 ≤ φ ≤ w1 . Hence, v 1 and w1 are a pair of sub- and supersolutions of (1.1)–(1.3) on Ω × (t2 , ∞), respectively. By Lemma 2.1, we have that v 1 ≤ u ≤ w1 . Since v 1 → β(δ1 − ε)(1 − ε) exp(−(δ1 − ε)(1 − ε)) and w1 → K + ϵ as t → ∞ and ε is arbitrary, we have I ⊆ [βδ1 e−δ1 , K].
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Step 2. We then claim that I ⊆ (1, K]. If δ2 := βδ1 e−δ1 > 1, we are done. Otherwise we continue to improve the lower bound of I and prove that I ⊆ [βδ2 e−δ2 , K]. Let ε > 0 small be given. Since I ⊆ [δ2 , K], there exists t3 > 1 such that δ2 − ε ≤ u(x, t) ≤ K + ε for (x, t) ∈ Ω × [t3 − 1, ∞). And there exists t4 > t3 such that
∞
f (s)ds ≤ ε for t ≥ t4 . t−t3
Let v 2 be given by 0 v 2 (t) = (δ2 − ε)(t − t3 + 1) δ − ε 2
t ≤ t3 − 1, t3 − 1 < t ≤ t3 , t3 < t ≤ t4 ,
v˙ 2 = −τ v 2 + τ β(δ2 − ε)(1 − ε) exp(−(δ2 − ε)(1 − ε))
(2.4) for t > t4 .
And let w2 be given by t ≤ t3 − 1, M 2 w (t) = K + ε + (M − K − ε)(t3 − t) t3 − 1 < t ≤ t3 , K + ε t > t3 ,
(2.5)
where M is the same as above. Clearly, v 2 ≤ u ≤ w2 on Ω × (−∞, t4 ], and one can check as in Step 1 that v 2 and w2 are a pair of sub- and supersolutions of problem (1.1)–(1.3) in Ω × (t4 , ∞), respectively. Since ε is arbitrary, we have that I ⊆ [βδ2 e−δ2 , K]. If βδ2 e−δ2 > 1, the claim is valid. Otherwise we can repeat the process to obtain a sequence {δk } satisfying δk = βδk−1 e−δk−1 . It is easy to see that there exists a δk > 1 for some k > 2. (Otherwise δk ≤ 1 for all k. Then the sequence {δk } is monotonically increasing because g is increasing on [0, 1]. So {δk } converges to ln β, which is impossible since ln β > 1.) As detailed above, we can construct sub- and supersolutions to improve the lower bound and show I ⊆ [βδk−1 e−δk−1 , K] = [δk , K]. Hence we have I ⊆ (1, K]. Step 3. We now show I = {u∗ }. Since I ⊆ (1, K], we can find λ1 > 1 such that I ⊆ [λ1 , K]. Since g(K) = g(β/e) > 1 by Lemma 2.3, we can make λ1 small such that λ1 < g(K). Let µ1 = K. We then define two sequences {µi } and {λi } by µi+1 = g(λi )
and λi+1 = g(µi )
for i > 1.
(2.6)
We claim that 1 < λ1 < λ2 < · · · < λi < u∗ < µi < · · · < µ2 < µ1 = K.
(2.7)
To see this, we first verify that 1 < u∗ < µ1 = K. (Note that u∗ = ln β and K = β/e with β ∈ (e, e2 ]. So 1 < u∗ is clear. Let h(x) = ln x − x/e. We have h′ (x) = 1/x − 1/e < 0 on (e, e2 ]. Hence h(β) < h(e) = 0, which implies ln β < β/e.) Since g is decreasing in (1, ∞) and 1 < u∗ < µ1 = K, we get g(K) = g(µ1 ) < g(u∗ ) = u∗ < g(1). Noticing g(1) = K = µ1 and g(K) = g(µ1 ) = λ2 by (2.6), this inequality leads to g(K) = λ2 < u∗ < µ1 = K. By the choices of λ1 , we have 1 < λ1 < g(K) = λ2 < u∗ < µ1 = K. Since g is decreasing on (1, ∞) and 1 < λ1 < u∗ , we have g(u∗ ) < g(λ1 ) < g(1). Since g(λ1 ) = µ2 by (2.6), this gives u∗ < µ2 < µ1 . Hence, we have 1 < λ1 < λ2 < u∗ < µ2 < µ1 . Continuing this process, we can prove (2.7).
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Now we assume that I ⊆ [λi , µi ] has been proved for some i ≥ 1. Let ε > 0 small be given. Since I ⊆ [λi , µi ], there exists t5 > 1 such that for (x, t) ∈ Ω × [t5 − 1, ∞).
λi − ε ≤ u(x, t) ≤ µi + ε And there exists t6 > t5 such that
∞
f (s)ds ≤ ε for t ≥ t6 . t−t5
Let v i be given by 0 i v (t) = (λi − ε)(t − t5 + 1) λ − ε i
t ≤ t5 − 1, t5 − 1 < t ≤ t5 , t5 < t ≤ t6 ,
(2.8)
v˙ i = −τ v i + τ β(εM + ε + µi ) exp(−(εM + ε + µi ))
for t > t6 .
And let wi be given by t ≤ t5 − 1, M i w (t) = µi + ε + (M − µi − ε)(t5 − t) t5 − 1 < t ≤ t5 , µ + ε t 5 < t ≤ t6 , i w˙ i = −τ wi + τ β(λi − ε)(1 − ε) exp(−(λi − ε)(1 − ε))
(2.9)
for t > t6 ,
where M is the same as above. It then follows that v i ≤ u ≤ wi on Ω × (−∞, t6 ]. For t ≥ t6 and v i ≤ φ ≤ wi , t t5 t f ∗φ= f (t − s)φ(x, s)ds = f (t − s)φ(x, s)ds + f (t − s)φ(x, s)ds −∞
−∞
t5
≤ εM + ε + µi ,
(2.10)
and
t
f ∗φ=
t
f (t − s)φ(x, s)ds ≥ −∞
f (t − s)φ(x, s)ds ≥ (λi − ε)(1 − ε).
(2.11)
t5
Since g is decreasing on (1, ∞) and λi > 1, it follows from (2.10)–(2.11) that β(εM + ε + µi ) exp(−(εM + ε + µi )) ≤ βf ∗ φ exp(−f ∗ φ) ≤ β(λi − ε)(1 − ε) exp(−(λi − ε)(1 − ε)) for all (x, t) ∈ Ω × [t6 , ∞) and v i ≤ φ ≤ wi , when ε is small. Thus, it holds that vti − d∆v i ≤ −τ v i + βτ f ∗ φ exp(−f ∗ φ), wti − d∆wi ≥ −τ wi + βτ f ∗ φ exp(−f ∗ φ) for (x, t) ∈ Ω × [t6 , ∞) and v i ≤ φ ≤ wi . So v i and wi are a pair of sub- and supersolutions of problem (1.1)–(1.3) in Ω × (t6 , ∞). Since v i → g(εM + ε + µi ) and wi → g((λi − ε)(1 − ε)), we have I ⊆ [g(εM + ε + µi ), g((λi − ε)(1 − ε))]. Since ε is arbitrary, we have I ⊆ [g(µi ), g(λi )] = [λi+1 , µi+1 ]. Hence by induction, it holds that I ⊆ [λn , µn ] for all n. Because of the monotonicity of {λi } and {µi }, there exist λ and µ with 1 < λ ≤ µ < K such that λi → λ
and µi → µ.
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Therefore, we have I ⊆ [λ, µ]. Moreover, by the definition of {λi } and {µi }, we find that λ = g(µ) and µ = g(λ). It then follows from Lemma 2.3 that λ = µ = u∗ . Hence, I = {u∗ }, and the proof is completed. Acknowledgments The authors would like to thank the referees for their helpful comments. References [1] S.A. Gourley, S. Ruan, Dynamics of the diffusive Nicholson’s blowflies equation with distributed delay, Proc. Roy. Soc. Edinburgh 130A (2000) 1275–1291. [2] W.S.C. Gurney, S.P. Blythe, R.M. Nisbet, Nicholson’s blowflies revisited, Nature 287 (1980) 17–21. [3] A.J. Nicholson, An outline of the dynamics of animal population, Aust. J. Zool. 2 (1954) 9–65. [4] S. Ruan, Spatial–temporal dynamics in nonlocal epidemiological models, in: Mathematics for Life Science and Medicine, Springer, Berlin, Heidelberg, 2007, pp. 97–122. [5] J.W.-H. So, Y. Yang, Dirichlet problem for the diffusive Nicholson’s blowflies equation, J. Differential Equations 150 (1998) 317–348. [6] J.W.-H. So, J.S. Yu, Global attractivity and uniform persistence in Nicholson’s blowflies, Differ. Equ. Dyn. Syst. 2 (1994) 11–18. [7] T. Yi, X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case, J. Differential Equations 245 (2008) 3376–3388. [8] X. Zhao, Global attractivity in a class of nonmonotone reaction–diffusion equations with time delay, Can. Appl. Math. Q. 17 (2009) 271–281. [9] Z. Wang, W. Li, Dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity, Proc. Roy. Soc. Edinburgh 140A (2010) 1081–1109. [10] T. Yi, X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differential Equations 251 (2011) 2598–2611. [11] K. Deng, Y. Wu, Asymptotic behavior for a reaction–diffusion population model with delay, Discrete Contin. Dyn. Syst. Ser. B 20 (2015) 385–395. [12] S.A. Gourley, N.F. Briton, On a modified Volterra population equation with diffusion, Nonlinear Anal. 21 (1993) 389–395. [13] S.A. Gourley, J.W.-H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol. 44 (2002) 49–78. [14] R. Laister, Global asymptotic behavior in some functional parabolic equations, Nonlinear Anal. 50 (2002) 347–361. [15] R. Redlinger, On Volterra’s population equation with diffusion, SIAM J. Math. Anal. 16 (1985) 135–142. [16] R. Redlinger, Existence theorems for semilinear parabolic systems with functionals, Nonlinear Anal. 8 (1984) 667–682.