Periodic solutions for a class of diffusive Nicholson’s blowflies model with Dirichlet boundary conditions
 Bingwen Liu^{1},
 Junxia Meng^{2}Email author and
 Weidong Jiao^{3}
https://doi.org/10.1186/1029242X2013449
© Liu et al.; licensee Springer. 2013
Received: 23 March 2013
Accepted: 29 August 2013
Published: 7 November 2013
Abstract
In this paper, we study the problem of periodic solutions for a class of diffusive Nicholson’s blowflies model with Dirichlet boundary conditions. By applying the Schauder fixed point theorem, the existence of nontrivial nonnegative periodic solutions of the considered model is established. Our results complement with some recent ones.
MSC:35B05, 35B10.
Keywords
1 Introduction
to describe the population of the Australian sheepblowfly Lucilia cuprina, where $u(t)$ is the population of the adult flies at the time t, p is the maximum per capita daily egg production rate, $1/a$ is the size at which the blowfly population reproduces at its maximum rate, δ is the per capita daily egg adult death rate, and τ is the maturation time. Since this equation explains Nicholson’s data of blowfly more accurately, the model and its modifications have been now refereed to as Nicholson’s blowflies model, we refer the reader to [2–6] and the references therein for some interesting results on this issue.
For the model above with either Dirichlet boundary conditions or Neumann boundary conditions, some results have been obtained. For example, So and Yang in [7] studied the asymptotic behavior of solutions of equation (1.2) under the Dirichlet boundary condition. In [8], Gourley considered the existence of travelling front solutions and their qualitative form for equation (1.2). The nonlinear stability of travelling wavefronts of equation (1.2) was investigated by Mei et al. in [9]. Yi and Zou [10] also established the global attractivity of the positive steady state of equation (1.2). Yi et al. in [11] established the threshold dynamics of equation (1.2) subject to the homogeneous Dirichlet boundary condition when the delayed reaction term is nonmonotone.
where $d(t)$, $\delta (t)$, $p(t)$, $\tau (t)$, $a(t)$ are positive, continuous and Tperiodic in t, $g(t,x)$ is the known function, which satisfies some structure conditions (see [19]), $g(t,x)$ is Tperiodic in the first argument, $\mathrm{\Omega}\subset {\mathbb{R}}^{\mathbf{n}}$ is a bounded domain with smooth boundary ∂ Ω, ${Q}_{T}=\mathrm{\Omega}\times [0,T]$, $\partial {Q}_{T}=\partial \mathrm{\Omega}\times [0,T]$, $x=({x}_{1},\dots ,{x}_{n})$ denotes the spatial variable vector in ${\mathbb{R}}^{n}$, and Δ is the Laplacian operator in ${\mathbb{R}}^{n}$.
The rest of this paper is organized as follows. In Section 2, we introduce some further notations and recall some useful results, which will be used in the later section. In Section 3, we give our main result and its proof.
2 Preliminaries
 (a)
Let us first introduce certain notations and definitions, constantly used throughout the present paper.
Let Ω be an ndimensional bounded domain in ${\mathbb{R}}^{n}$ with a boundary ∂ Ω and a closure $\overline{\mathrm{\Omega}}$. Assume throughout that ∂ Ω can be covered by a finite number of balls ${S}_{k}$ such that the portion $\partial \mathrm{\Omega}\cap {S}_{k}$ may be represented in the form ${x}_{i}=h({x}_{1},\dots ,{x}_{i1},{x}_{i+1},\dots ,{x}_{n})$ for some i, where h has the Hölder continuous (exponent α, $0<\alpha <1$) second derivatives.
where $P=(t,x)$, ${P}^{\prime}=({t}^{\prime},{x}^{\prime})$ and $d(P,{P}^{\prime})={[t{t}^{\prime}+{x{x}^{\prime}}^{2}]}^{1/2}$.
 (b)Let us now consider the linear boundary problem$\sum _{i,j=1}^{n}{a}_{ij}(t,x){u}_{{x}_{i}{x}_{j}}+\sum _{i=1}^{n}{b}_{i}(t,x){u}_{{x}_{i}}+c(t,x)u{u}_{t}=f(t,x)\phantom{\rule{1em}{0ex}}\text{in}{Q}_{T},$(2.1)
The following fundamental results concerning (2.1), (2.2), which are due to Fife [20] and Shmulev [21], are stated in a form suitable to our purpose.
Lemma 2.1 (See [20])
 (i)
$c(t,x)\le 0$ in ${\overline{Q}}_{T}$;
 (ii)
${\sum}_{i,j=1}^{n}{{a}_{ij}}_{\alpha}+{\sum}_{i=1}^{n}{{b}_{i}}_{\alpha}+{c}_{\alpha}\le M$, ${f}_{\alpha}<\mathrm{\infty}$ for some constant M;
 (iii)
$\phi (t,x)$ is the trace on ∂Q of a function $\mathrm{\Phi}(t,x)$ of class ${C}_{T}^{2+\alpha}({\overline{Q}}_{T})$.
where K depends only on ${a}_{0}$, M and ${Q}_{T}$.
Lemma 2.2 (See [21])
 (i)
${a}_{ij}(t,x)$, ${b}_{i}(t,x)$, $c(t,x)$ and $f(t,x)$ are continuous in ${\overline{Q}}_{T}$;
 (ii)
${\sum}_{i,j=1}^{n}{{a}_{ij}}_{\alpha}+{\sum}_{i=1}^{n}{{b}_{i}}_{0}+{c}_{0}\le {M}_{1}$ for some constant ${M}_{1}$;
 (iii)
${\sum}_{i,j=1}^{n}({sup}_{\partial {Q}_{T}}{a}_{ij}+{sup}_{(t,x),({t}^{\prime},{x}^{\prime})\in \partial {Q}_{T}}\frac{{a}_{ij}(t,x){a}_{ij}({t}^{\prime},{x}^{\prime})}{t{t}^{\prime}+x{x}^{\prime}})\le {M}_{2}$ for some constant ${M}_{2}$.
Lemma 2.3 (See [22])
For any $0<\alpha <\beta <1$, $0\le p\le q$, the bounded subsets of the space ${C}_{q+\beta}({Q}_{T})$ are precompact subsets of ${C}_{p+\alpha}({Q}_{T})$.
Lemma 2.4 (The Schauder fixed point theorem, see [22])
Let X be a closed convex subset of the Banach space Y, and let T be a continuous operator on X such that T X is contained in X, and T X is precompact. Then T has a fixed point, i.e., there exists a point $x\in X$ such that $\mathbf{T}x=x$.
3 Main result and its proof
The purpose of the present section is to prove the existence of a nonnegative periodic solution $u(t,x)$ of (1.4)(1.6).

(H_{1}) $d(t)$, $\delta (t)$ are Hölder continuous (exponent α, $0<\alpha <1$) in $[0,T]$, and there exist positive constants ${a}_{0}$, ${a}_{1}$ such that$d(t)\ge {a}_{0}>0,\phantom{\rule{2em}{0ex}}a(t)\ge {a}_{1}>0.$

(H_{2}) $0\le g(t,x)\in {C}^{\alpha}({\overline{Q}}_{T})$, $g(t,x)\not\equiv 0$, $\psi (t,x,u)\in {C}^{\alpha}({\overline{Q}}_{T})$, $u<\mathrm{\infty}$, i.e., $g(t,x)$, $\psi (t,x,u)$ satisfy the local Hölder condition in $(t,x)$ with the exponent α, $\psi (t,x,u)$ satisfies a local Hölder condition in u, uniformly with respect to t.

(H_{3}) there exists a function $\mathrm{\Phi}(t,x)\in {C}_{T}^{2+\alpha}({\overline{Q}}_{T})$ such that $\mathrm{\Phi}(t,x){}_{\partial \mathrm{\Omega}}$=0.
We shall now prove an existence theorem by employing the Schauder fixed point theorem.
Theorem 3.1 Suppose that the assumptions (H_{1})(H_{3}) are satisfied, then there exists a nonnegative Tperiodic solution $u(t,x)$ of problem (3.1)(3.3). Furthermore, $u(t,x)$ belongs to ${C}_{T}^{1+\nu}({\overline{Q}}_{T})$ for any $0<\nu <1$ and to ${C}_{T}^{2+\gamma}({\overline{Q}}_{T})$ for some $0<\gamma <1$.
where ${v}_{+}=max\{0,v(t\tau (t),x)\}$, $(x,t)\in {Q}_{T}$, obviously, ${v}_{+}=max\{0,v(t\tau (t),x)\}\ge 0$.
On the one hand, we prove the existence of the periodic solution of problem (3.4)(3.6) under the conditions of Theorem 3.1, we split the proof into two cases: (i) ${v}_{+}=0$; (ii) ${v}_{+}>0$.
Case (i). If ${v}_{+}=0$, the coefficients are periodic and Hölder continuous on ${\overline{Q}}_{T}$, a unique periodic solution u exists according to Lemma 2.1.
Obviously, V is a closed convex set in the Banach space ${C}_{T}^{1+\nu}({\overline{Q}}_{T})$, from (3.8), we can see that the mapping T maps V into itself.
in view of Lemma 2.3, we know that the bounded subsets of the space ${C}_{T}^{1+{\nu}^{\prime}}({\overline{Q}}_{T})$ are precompact subsets of ${C}_{T}^{1+\nu}({\overline{Q}}_{T})$, so we conclude that T maps V into a compact subset of V.
where $x,y\ge 0$, $0<\theta <1$.
then the continuity of T in the $(1+\nu )$norms is proved.
By Lemma 2.4, it follows that T has a fixed point ${u}^{\ast}\in {C}_{T}^{1+\nu}({Q}_{T})$, ${u}^{\ast}$ is then a periodic solution of (3.4)(3.6), and it belongs to ${C}_{T}^{1+\nu}({\overline{Q}}_{T})$, ν is arbitrary. Moreover, ${u}^{\ast}$ also belongs to ${C}_{T}^{2+\gamma}({\overline{Q}}_{T})$ for some $0<\gamma <1$, following directly from Lemma 2.1.
On the other hand, we claim that the Tperiodic solution u of problem (3.4)(3.6) is nonnegative and nontrival.
since $g(t,x)\not\equiv 0$, we see that Tperiodic solution is nontrivial. The proof of Theorem 3.1 is completed. □
Remark 3.1 In [2–6, 13, 15], the authors studied the dynamical behavior of delayed Nicholson’s blowflies equation, however, the effect of diffusive term was seldom considered in these references. In this sense, the main result obtained in the present paper extended the mentioned results.
Remark 3.2 It is worth pointing out that the coefficients of equation (1.4) are timevarying, which is more difficult to research than the constant case. To the best of our knowledge, there are very few works concerning the studied problem, which implies that the results of this paper are more general, and they effectually complement the previously known results.
Declarations
Acknowledgements
The authors would like to express the sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. This work was supported by the National Natural Science Foundation of China (Grant No. 11201184), the Construct Program of the Key Discipline in Hunan Province (Mechanical Design and Theory), the Science Fund for Distinguished Young Scholars of Zhejiang Province of China (Grant No. R1100002), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (Grant Nos. LY12A01018, Y6110436), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R. China (Grant No. Z201122436).
Authors’ Affiliations
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