Exponential Extinction of Nicholson's Blowflies System with Nonlinear Density-Dependent Mortality Terms

Abstract

This paper presents a new generalized Nicholson’s blowflies system with patch structure and nonlinear density-dependent mortality terms. Under appropriate conditions, we establish some criteria to guarantee the exponential extinction of this system. Moreover, we give two examples and numerical simulations to demonstrate our main results.

1. Introduction

To describe the population of the Australian sheep blowfly and agree well with the experimental date of Nicholson [1], Gurney et al. [2] proposed the following Nicholson's blowflies equation: 𝑁^′(𝑡)=−𝛿𝑁(𝑡)+𝑝𝑁(𝑡−𝜏)𝑒^{−𝑎𝑁(𝑡−𝜏)}. (1.1) Here, 𝑁(𝑡) is the size of the population at time 𝑡,𝑝 is the maximum per capita daily egg production, (1/𝑎) is the size at which the population reproduces at its maximum rate, 𝛿 is the per capita daily adult death rate, and 𝜏 is the generation time. There have been a large number of results on this model and its modifications (see, e.g., [3–8]). Recently, Berezansky et al. [9] pointed out that a new study indicates that a linear model of density-dependent mortality will be most accurate for populations at low densities and marine ecologists are currently in the process of constructing new fishery models with nonlinear density-dependent mortality rates. Consequently Berezansky et al. [9] presented the following Nicholson’s blowflies model with a nonlinear density-dependent mortality term 𝑁^′(𝑡)=−𝐷(𝑁(𝑡))+𝑃𝑁(𝑡−𝜏)𝑒^{−𝑎𝑁(𝑡−𝜏)}, (1.2) where 𝑃 is a positive constant and function 𝐷 might have one of the following forms: 𝐷(𝑁)=𝑎𝑁/(𝑁+𝑏) or 𝐷(𝑁)=𝑎−𝑏𝑒^−𝑁 with positive constants 𝑎,𝑏>0.

Wang [10] studied the existence of positive periodic solutions for the model (1.2) with 𝐷(𝑁)=𝑎−𝑏𝑒^−𝑁. Hou et al. [11] investigated the permanence and periodic solutions for the model (1.2) with 𝐷(𝑁)=𝑎𝑁/(𝑁+𝑏). Furthermore, Liu and Gong [12] considered the permanence for a Nicholson-type delay systems with nonlinear density-dependent mortality terms as follows: 𝑁^′₁(𝑡)=−𝐷₁₁𝑡,𝑁₁(𝑡)+𝐷₁₂𝑡,𝑁₂(𝑡)+𝑐₁(𝑡)𝑁₁𝑡−𝜏₁(𝑡)𝑒^{−𝛾1(𝑡)𝑁1(𝑡−𝜏1(𝑡))}𝑁^′₂(𝑡)=−𝐷₂₂𝑡,𝑁₂(𝑡)+𝐷₂₁𝑡,𝑁₁(𝑡)+𝑐₂(𝑡)𝑁₂𝑡−𝜏₂(𝑡)𝑒^{−𝛾2(𝑡)𝑁2(𝑡−𝜏2(𝑡))}, (1.3) where 𝐷_𝑖𝑗()=𝑎𝑡,𝑁^𝑖𝑗(𝑡)𝑁𝑏_𝑖𝑗(𝑡)+𝑁or𝐷_𝑖𝑗(𝑡,𝑁)=𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)𝑒^−𝑁, (1.4)𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖,𝛾_𝑖∶𝑅→(0,+∞) are all continuous functions bounded above and below by positive constants, and 𝜏_𝑗(𝑡)≥0 are bounded continuous functions, 𝑟_𝑖=sup_𝑡∈𝑅𝜏_𝑖(𝑡)>0, and 𝑖,𝑗=1,2.

On the other hand, since the biological species compete and cooperate with each other in real world, the growth models given by patch structure systems of delay differential equation have been provided by several authors to analyze the dynamics of multiple species (see, e.g., [13–16] and the reference therein). Moreover, the extinction phenomenon often appears in the biology, economy, and physics field and the main focus of Nicholson's blowflies model is on the scalar equation and results on patch structure of this model are gained rarely [14, 16], so it is worth studying the extinction of Nicholson's blowflies system with patch structure and nonlinear density-dependent mortality terms. Motivated by the above discussion, we shall derive the conditions to guarantee the extinction of the following Nicholson-type delay system with patch structure and nonlinear density-dependent mortality terms: 𝑁^′_𝑖(𝑡)=−𝐷_𝑖𝑖𝑡,𝑁_𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝐷_𝑖𝑗𝑡,𝑁_𝑗(𝑡+_𝑙𝑗=1)𝑐_𝑖𝑗(𝑡)𝑁_𝑖𝑡−𝜏_𝑖𝑗(𝑡)𝑒^{−𝛾𝑖𝑗(𝑡)𝑁𝑖(𝑡−𝜏𝑖𝑗(𝑡))}, (1.5) where 𝐷_𝑖𝑗()=𝑎𝑡,𝑁^𝑖𝑗(𝑡)𝑁𝑏_𝑖𝑗(𝑡)+𝑁or𝐷_𝑖𝑗(𝑡,𝑁)=𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)𝑒^−𝑁, (1.6)𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑘,𝛾_𝑖𝑘∶𝑅→(0,+∞) are all continuous functions bounded above and below by positive constants, and 𝜏_𝑖𝑘(𝑡)≥0 are bounded continuous functions, 𝑟_𝑖=max_1≤𝑗≤l{sup_𝑡∈𝑅𝜏_𝑖𝑗(𝑡)}>0, and 𝑖,𝑗=1,2,…,𝑛,𝑘=1,2,…,𝑙. Furthermore, in the case 𝐷_𝑖𝑗(𝑡,𝑁)=𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)𝑒^−𝑁, to guarantee the meaning of mortality terms we assume that 𝑎_𝑖𝑗(𝑡)>𝑏_𝑖𝑗(𝑡) for 𝑡∈𝑅 and 𝑖,𝑗=1,2,…,𝑛. The main purpose of this paper is to establish the conditions ensuring the exponential extinction of system (1.5).

For convenience, we introduce some notations. Throughout this paper, given a bounded continuous function g defined on 𝑅, let 𝑔⁺ and 𝑔⁻ be defined as 𝑔⁻=inf_𝑡∈𝑅𝑔(𝑡),𝑔⁺=sup_𝑡∈𝑅g(𝑡). (1.7)

Let 𝑅^𝑛(𝑅^𝑛₊) be the set of all (nonnegative) real vectors, we will use 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝑅^𝑛 to denote a column vector, in which the symbol (^𝑇) denotes the transpose of a vector. We let |𝑥| denote the absolute-value vector given by |𝑥|=(|𝑥₁|,…,|𝑥_𝑛|)^𝑇 and define ||𝑥||=max_{1≤𝑖≤𝑛}|𝑥_𝑖|. Denote 𝐶=∏^𝑛_𝑖=1𝐶([−𝑟_𝑖,0],𝑅)and𝐶₊=∏^𝑛_𝑖=1𝐶([−𝑟_𝑖,0],𝑅₊) as Banach spaces equipped with the supremum norm defined by ||𝜑||=sup_{−𝑟𝑖≤𝑡≤0}max_{1≤𝑖≤𝑛}|𝜑_𝑖(𝑡)| for all 𝜑(𝑡)=(𝜑₁(𝑡),…,𝜑_𝑛(𝑡))^𝑇∈𝐶 (or ∈𝐶₊). If 𝑥_𝑖(𝑡) is defined on [𝑡₀−𝑟_𝑖,𝜈) with 𝑡₀,𝜈∈𝑅 and 𝑖=1,…,𝑛, then we define 𝑥_𝑡∈𝐶 as 𝑥_𝑡=(𝑥¹_𝑡,…𝑥^𝑛_𝑡)^𝑇 where 𝑥^𝑖_𝑡(𝜃)=𝑥_𝑖(𝑡+𝜃) for all 𝜃∈[−𝑟_𝑖,0] and 𝑖=1,…,𝑛.

The initial conditions associated with system (1.5) are of the form: 𝑁_𝑡0=𝜑,𝜑=𝜑₁,…,𝜑_𝑛^𝑇∈𝐶₊,𝜑_𝑖(0)>0,𝑖=1,…,𝑛. (1.8) We write 𝑁_𝑡(𝑡₀,𝜑)(𝑁(𝑡;𝑡₀,𝜑)) for a solution of the initial value problem (1.5) and (1.8). Also, let [𝑡₀,𝜂(𝜑)) be the maximal right-interval of existence of 𝑁_𝑡(𝑡₀,𝜑).

Definition 1.1. The system (1.5) with initial conditions (1.8) is said to be exponentially extinct, if there are positive constants 𝑀 and 𝜅 such that |𝑁_𝑖(𝑡;𝑡₀,𝜑)|≤𝑀𝑒^{−𝜅(𝑡−𝑡0)},𝑖=1,2…,𝑛. Denote it as 𝑁_𝑖(𝑡;𝑡₀,𝜑)=𝑂(𝑒^{−𝜅(𝑡−𝑡0)}),𝑖=1,2,…,𝑛.
The remaining part of this paper is organized as follows. In Sections 2 and 3, we shall derive some sufficient conditions for checking the extinction of system (1.5). In Section 4, we shall give two examples and numerical simulations to illustrate our results obtained in the previous sections.

2. Extinction of Nicholson's Blowflies System with 𝐷_𝑖𝑗(𝑡,𝑁)=𝑎_𝑖𝑗(𝑡)𝑁/(𝑏_𝑖𝑗(𝑡)+𝑁)(𝑖,𝑗=1,2,…,𝑛)

Theorem 2.1. Suppose that there exists positive constant 𝐾₁ such that 𝑎⁻_𝑖𝑖𝑏⁺_𝑖𝑖+𝐾₁>^𝑛_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗𝑏⁻_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝛾⁻_𝑖𝑗𝑒𝐾₁,𝑖=1,2,…,𝑛. (2.1) Let 𝐸¹=𝜑∣𝜑∈𝐶₊,𝜑(0)>0,0≤𝜑_𝑖(𝑡)<𝐾₁,∀𝑡∈−𝑟_𝑖,0,𝑖=1,2,…,𝑛. (2.2) Moreover, assume 𝑁(𝑡;𝑡₀,𝜑) is the solution of (1.5) with 𝜑∈𝐸¹ and 𝐷_𝑖𝑗(𝑡,𝑁)=(𝑎_𝑖𝑗(𝑡)𝑁/(𝑏_𝑖𝑗(𝑡)+𝑁))(𝑖,𝑗=1,2,…,𝑛). Then, 0≤𝑁_𝑖𝑡;𝑡₀,𝜑<𝐾₁,∀𝑡∈𝑡₀𝜑𝜂𝜑,𝜂,𝑖=1,2,…,𝑛,=+∞. (2.3)

Proof. Set 𝑁(𝑡)=𝑁(𝑡;𝑡₀,𝜑)forall𝑡∈[𝑡₀,𝜂(𝜑)). In view of 𝜑∈𝐶₊, using Theorem 5.2.1 in [17, p. 81], we have 𝑁_𝑡(𝑡₀,𝜑)∈𝐶₊ for all 𝑡∈[𝑡₀,𝜂(𝜑)). Assume, by way of contradiction, that (2.3) does not hold. Then, there exist 𝑡₁∈[𝑡₀,𝜂(𝜑)) and 𝑖∈{1,2,…,𝑛} such that 𝑁_𝑖𝑡₁=𝐾₁,0≤𝑁_𝑗(𝑡)<𝐾₁∀𝑡∈𝑡₀−𝑟_𝑗,𝑡₁,𝑗=1,2,…,𝑛. (2.4) Calculating the derivative of 𝑁_𝑖(𝑡), together with (2.1) and the fact that sup_u≥0𝑢𝑒^−𝑢=1/𝑒 and 𝑎(𝑡)𝑁/(𝑏(𝑡)+𝑁)≤𝑎(𝑡)𝑁/𝑏(𝑡) for all 𝑡∈𝑅,𝑁≥0, (1.5) and (2.4) imply that 0≤𝑁^′_𝑖𝑡₁=−𝐷_𝑖𝑖𝑡₁,𝑁_𝑖𝑡₁+^𝑛_{𝑗=1,𝑗≠𝑖}𝐷_𝑖𝑗𝑡₁,𝑁_𝑗𝑡₁+^𝑙_𝑗=1𝑐_𝑖𝑗𝑡₁𝑁_𝑖𝑡₁−𝜏_𝑖𝑗𝑡₁𝑒^{−𝛾𝑖𝑗(𝑡1)𝑁𝑖(𝑡1−𝜏𝑖𝑗(𝑡1))}𝑎≤−^𝑖𝑖𝑡₁𝑁_𝑖𝑡₁𝑏_𝑖𝑖𝑡₁+𝑁_𝑖𝑡₁+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑡₁𝑁_𝑗𝑡₁𝑏_𝑖𝑗𝑡₁+^𝑙_𝑗=1𝑐_𝑖𝑗𝑡₁𝛾_𝑖𝑗𝑡₁1𝑒≤−𝑎⁻_𝑖𝑖𝑏⁺_𝑖𝑖+𝐾₁+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗𝑏⁻_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝛾⁻_𝑖𝑗𝑒𝐾₁𝐾₁<0, (2.5) which is a contradiction and implies that (2.3) holds. From Theorem 2.3.1 in [18], we easily obtain 𝜂(𝜑)=+∞. This ends the proof of Theorem 2.1.

Theorem 2.2. Suppose that there exists positive constant 𝐾₁ satisfying (2.1) and 𝑎⁻_𝑖𝑖𝑏⁺_𝑖𝑖+𝐾₁>^𝑛_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗𝑏⁻_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗,𝑖=1,2,…,𝑛. (2.6) Then the solution 𝑁(𝑡;𝑡₀,𝜑) of (1.5) with 𝜑∈𝐸¹ and 𝐷_𝑖𝑗(𝑡,𝑁)=(𝑎_𝑖𝑗(𝑡)𝑁/(𝑏_𝑖𝑗(𝑡)+𝑁))(𝑖,𝑗=1,2,…,𝑛) is exponentially extinct as 𝑡→+∞.

Proof. Define continuous functions Γ_i(𝜔) by setting Γ_i(𝜔)𝑎=𝜔−^−𝑖𝑖𝑏⁺_𝑖𝑖+𝐾₁+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗𝑏⁻_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝑒^𝜔𝑟𝑖,𝑖=1,2,…,𝑛. (2.7) Then, from (2.6), we obtain Γ_𝑖(0)𝑎=−^−𝑖𝑖𝑏⁺_𝑖𝑖+𝐾₁+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗𝑏⁻_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗<0,𝑖=1,2,…,𝑛. (2.8) The continuity of Γ_𝑖(𝜔) implies that there exists 𝜆>0 such that Γ_𝑖(𝜆)𝑎=𝜆−^−𝑖𝑖𝑏⁺_𝑖𝑖+𝐾₁+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗𝑏⁻_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝑒^𝜆𝑟𝑖<0,𝑖=1,2,…,𝑛. (2.9) Let 𝑦_𝑖(𝑡)=𝑁_𝑖(𝑡)𝑒^{𝜆(𝑡−𝑡0)},𝑖=1,2,…,𝑛. (2.10) Calculating the derivative of 𝑦(𝑡) along the solution 𝑁(𝑡) of system (1.5) with 𝜑∈𝐸¹, we have 𝑦^′_𝑖(𝑡)=𝜆𝑦_𝑖(𝑡)+𝑒^{𝜆(𝑡−𝑡0)}𝑁^′_𝑖(𝑡)=𝜆𝑦_𝑖(𝑡)−𝑎_𝑖𝑖(𝑡)𝑦_𝑖(𝑡)𝑏_𝑖𝑖(𝑡)+𝑁_𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)𝑦_𝑗(𝑡)𝑏_𝑖𝑗(𝑡)+𝑁_𝑗(𝑡)+^𝑙_𝑗=1𝑐_𝑖𝑗(𝑡)𝑒^{𝜆𝜏𝑖𝑗(𝑡)}𝑦_𝑖𝑡−𝜏_𝑖𝑗(𝑡)𝑒^{−𝛾𝑖𝑗(𝑡)𝑁𝑖(𝑡−𝜏𝑖𝑗(𝑡))},𝑖=1,2,…,𝑛. (2.11) Let 𝑀₁ denote an arbitrary positive number and set 𝑀₁>𝑦_𝑖(𝑡),∀𝑡∈𝑡₀−𝑟_𝑖,𝑡₀,𝑖=1,2,…,𝑛. (2.12) We claim that 𝑦_𝑖(𝑡)<𝑀₁,∀𝑡∈𝑡₀,+∞,𝑖=1,2,…,𝑛. (2.13) If this is not valid, there must exist 𝑡₂∈(𝑡₀,+∞) and 𝑖∈{1,2,…,𝑛} such that 𝑦_𝑖𝑡₂=𝑀₁,𝑦_𝑗(𝑡)<𝑀₁,∀𝑡<𝑡₂,𝑗=1,2,…,𝑛. (2.14) Then, from (2.3) and (2.11), we have 0≤𝑦^′_𝑖𝑡₂=𝜆𝑦_𝑖𝑡₂−𝑎_𝑖𝑖𝑡₂𝑦_𝑖𝑡₂𝑏_𝑖𝑖𝑡₂+𝑁_𝑖𝑡₂+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑡₂𝑦_𝑗𝑡₂𝑏_𝑖𝑗𝑡₂+𝑁_𝑗𝑡₂+^𝑙_𝑗=1𝑐_𝑖𝑗𝑡₂𝑒^{𝜆𝜏𝑖𝑗(𝑡2)}𝑦_𝑖𝑡₂−𝜏_𝑖𝑗𝑡₂𝑒^{−𝛾𝑖𝑗(𝑡2)𝑁𝑖(𝑡2−𝜏𝑖𝑗(𝑡2))}≤𝜆𝑀₁−𝑎_𝑖𝑖𝑡₂𝑀₁𝑏_𝑖𝑖𝑡₂+𝐾₁+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑡₂𝑀₁𝑏_𝑖𝑗𝑡₂+^𝑙_𝑗=1𝑐_𝑖𝑗𝑡₂𝑒^𝜆𝑟𝑖𝑀₁≤𝑎𝜆−^−𝑖𝑖𝑏⁺_𝑖𝑖+𝐾₁+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗𝑏⁻_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝑒^𝜆𝑟𝑖𝑀₁<0. (2.15) This contradiction implies that (2.13) holds. Thus, 𝑁_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑒^{−𝜆(𝑡−𝑡0)}≤𝑀₁𝑒^{−𝜆(𝑡−𝑡0)}∀𝑡∈𝑡₀−𝑟_𝑖,+∞,𝑖=1,2,…,𝑛. (2.16) This completes the proof.

3. Extinction of Nicholson's Blowflies System with 𝐷_𝑖𝑗(𝑡,𝑁)=𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)𝑒^−𝑁(𝑖,𝑗=1,2,…,𝑛)

Theorem 3.1. Suppose that there exists positive constant 𝐾₂ such that 𝑎⁻_𝑖𝑖>^𝑛_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑏⁻_𝑖𝑗𝑒^−𝐾2+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝛾⁻_𝑖𝑗𝑒,𝑖=1,2,…,𝑛,−𝑎_𝑖𝑖(𝑡)+𝑏_𝑖𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)≥0,𝑖=1,2,…,𝑛. (3.1)(3.2) Let 𝐸²=𝜑∣𝜑∈𝐶₊,𝜑(0)>0,0≤𝜑_𝑖(𝑡)<𝐾₂,∀𝑡∈−𝑟_𝑖,0,𝑖=1,2,…,𝑛. (3.3) Moreover, assume 𝑁(𝑡;𝑡₀,𝜑) is the solution of (1.5) with 𝜑∈𝐸² and 𝐷_𝑖𝑗(𝑡,𝑁)=𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)𝑒^−𝑁(𝑖,𝑗=1,2,…,𝑛). Then, 0≤𝑁_𝑖𝑡;𝑡₀,𝜑<𝐾₂,∀𝑡∈𝑡₀𝜑𝜂𝜑,𝜂,𝑖=1,2,…,𝑛,=+∞. (3.4)(3.5)

Proof. Set 𝑁(𝑡)=𝑁(𝑡;𝑡₀,𝜑)forall𝑡∈[𝑡₀,𝜂(𝜑)). Rewrite the system (1.5) as𝑁^′(𝑡)=𝑓𝑡,𝑁_𝑡, (3.6) where 𝑓(𝑡,𝜙)=(𝑓₁(𝑡,𝜙),𝑓₂(𝑡,𝜙),…,𝑓_𝑛(𝑡,𝜙))^𝑇 and 𝑓_𝑖𝑡,𝜙=−𝑎_𝑖𝑖(𝑡)+𝑏_𝑖𝑖(𝑡)𝑒^{−𝜙𝑖(0)}+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)𝑒^{−𝜙𝑗(0)}+^𝑙_𝑗=1𝑐_𝑖𝑗(𝑡)𝜙_𝑖−𝜏_𝑖𝑗(𝑡)𝑒^{−𝛾𝑖𝑗(𝑡)𝜙𝑖(−𝜏𝑖𝑗(𝑡))},𝑖=1,2,…,𝑛,𝜙∈𝐶. (3.7) In view of (3.2), whenever 𝜙∈C satisfies 𝜙≥0,𝜙_𝑖(0)=0 for some 𝑖 and 𝑡∈𝑅, then 𝑓_𝑖𝑡,𝜙=−𝑎_𝑖𝑖(𝑡)+𝑏_𝑖𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)𝑒^{−𝜙𝑗(0)}+^𝑙_𝑗=1𝑐_𝑖𝑗(𝑡)𝜙_𝑖−𝜏_𝑖𝑗(𝑡)𝑒^{−𝛾𝑖𝑗(𝑡)𝜙𝑖(−𝜏𝑖𝑗(𝑡))}≥−𝑎_𝑖𝑖(𝑡)+𝑏_𝑖𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)≥0. (3.8) Thus, using Theorem 5.2.1 in [17, p. 81], we have 𝑁_𝑡(𝑡₀,𝜑)∈𝐶₊ for all 𝑡∈[𝑡₀,𝜂(𝜑)) and 𝜑∈𝐸²⊂𝐶₊. Assume, by way of contradiction, that (3.4) does not hold. Then, there exist 𝑡₃∈[𝑡₀,𝜂(𝜑)) and 𝑖∈{1,2,…,𝑛} such that 𝑁_𝑖𝑡₃=𝐾₂,0≤𝑁_𝑗(𝑡)<𝐾₂∀𝑡∈𝑡₀−𝑟_𝑗,𝑡₃,𝑗=1,2,…,𝑛. (3.9) Calculating the derivative of 𝑁_𝑖(𝑡), together with (3.1) and the fact that sup_𝑢≥0𝑢𝑒^−𝑢=1/𝑒, (1.5) and (3.9) imply that 0≤𝑁^′_𝑖𝑡₃=−𝐷_𝑖𝑖𝑡₃,𝑁_𝑖𝑡₃+^𝑛_{𝑗=1,𝑗≠𝑖}𝐷_𝑖𝑗𝑡₃,𝑁_𝑗𝑡₃+^𝑙_𝑗=1𝑐_𝑖𝑗𝑡₃𝑁_𝑖𝑡₃−𝜏_𝑖𝑗𝑡₃×𝑒^{−𝛾𝑖𝑗(𝑡3)𝑁𝑖(𝑡3−𝜏𝑖𝑗(𝑡3))}≤−𝑎_𝑖𝑖𝑡₃+𝑏_𝑖𝑖𝑡₃𝑒^−𝐾2+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑡₃−𝑏_𝑖𝑗𝑡₃𝑒^−𝐾2+^𝑙_𝑗=1𝑐_𝑖𝑗𝑡₃𝛾_𝑖𝑗𝑡₃1𝑒≤−𝑎⁻_𝑖𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑏⁻_𝑖𝑗𝑒^−𝐾2+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝛾⁻_𝑖𝑗𝑒<0, (3.10) which is a contradiction and implies that (3.4) holds. From Theorem 2.3.1 in [18], we easily obtain 𝜂(𝜑)=+∞. This ends the proof of Theorem 3.1.

Theorem 3.2. Let (3.1) and (3.2) hold. Moreover, suppose that there exist two positive constants 𝜆 and M such that −𝑎_𝑖𝑖(𝑡)+𝑏_𝑖𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)≤𝑀𝑒⁻𝜆_{(𝑡−𝑡0)}𝑏,𝑡∈𝑅,𝑖=1,2,…,𝑛,_−𝑖𝑖𝐾>1+²2𝑏⁺_𝑖𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑏⁺_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗,𝑖=1,2,…,𝑛. (3.11)(3.12) Then the solution 𝑁(𝑡;𝑡₀,𝜑) of (1.5) with 𝜑∈𝐸² and 𝐷_𝑖𝑗(𝑡,𝑁)=𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)𝑒^−𝑁(𝑖,𝑗=1,2,…,𝑛), is exponentially extinct as t→+∞.

Proof. Define continuous functions Γ_i(𝜔) by setting Γ_𝑖(𝜔)=𝜔−𝑏⁻_𝑖𝑖𝐾+1+²2𝑏⁺_𝑖𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑏⁺_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝑒^𝜔𝑟𝑖,𝑖=1,2,…,𝑛. (3.13) Then, from (3.12), we obtain Γ_𝑖(0)=−𝑏⁻_𝑖𝑖𝐾+1+²2𝑏⁺_𝑖𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑏⁺_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗<0,𝑖=1,2,…,𝑛. (3.14) The continuity of Γ_i(𝜔) implies that there exists 0<𝜇<𝜆 such that Γ_𝑖𝜇=𝜇−𝑏⁻_𝑖𝑖𝐾+1+²2𝑏⁺_𝑖𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑏⁺_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝑒^𝜇𝑟𝑖<0,𝑖=1,2,…,𝑛. (3.15) Let 𝑥_𝑖(𝑡)=𝑁_𝑖(𝑡)𝑒^{𝜇(𝑡−𝑡0)},𝑖=1,2,…,𝑛. (3.16) Calculating the derivative of 𝑥(𝑡) along the solution 𝑁(𝑡) of system (1.5) with 𝜑∈𝐸², in view of (3.4) and (3.11), we have 𝑥^′_𝑖(𝑡)=𝜇𝑥_𝑖(𝑡)+𝑒^{𝜇(𝑡−𝑡0)}𝑁^′_𝑖(𝑡)=𝜇𝑥_𝑖(𝑡)+𝑒^{𝜇(𝑡−𝑡0)}−𝑎_𝑖𝑖(𝑡)+𝑏_𝑖𝑖(𝑡)𝑒^{−𝑁𝑖(𝑡)}+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)𝑒^{−𝑁𝑗(𝑡)}+^𝑙_𝑗=1𝑐_𝑖𝑗(𝑡)𝑒^{𝜇𝜏𝑖𝑗(𝑡)}𝑥_𝑖𝑡−𝜏_𝑖𝑗(𝑡)𝑒^{−𝛾𝑖𝑗(𝑡)𝑁𝑖(𝑡−𝜏𝑖𝑗(𝑡))}≤𝜇𝑥_𝑖(𝑡)+𝑒^{𝜇(𝑡−𝑡0)}−𝑎_𝑖𝑖(𝑡)+𝑏_𝑖𝑖(𝑡)1−𝑁_𝑖(𝑡)+12𝑁²_𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)1−𝑁_𝑗(𝑡)+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝑒^𝜇𝑟𝑖𝑥_𝑖𝑡−𝜏_𝑖𝑗(𝑡)=𝜇𝑥_𝑖(𝑡)+𝑒^{𝜇(𝑡−𝑡0)}−𝑎_𝑖𝑖(𝑡)+𝑏_𝑖𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)−𝑏_𝑖𝑖(𝑡)𝑥_𝑖(𝑡)+12𝑏_𝑖𝑖(𝑡)𝑁_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝑏_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝑒^𝜇𝑟𝑖𝑥_𝑖𝑡−𝜏_𝑖𝑗(𝑡)≤𝜇𝑥_𝑖(𝑡)+𝑀𝑒^(𝜇−𝜆_{)(𝑡−𝑡0)}−𝑏⁻_𝑖𝑖𝑥_𝑖(𝑡)+𝐾₂2𝑏⁺_𝑖𝑖𝑥_𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝑏⁺_𝑖𝑗𝑥_𝑗(𝑡)+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝑒^𝜇𝑟𝑖𝑥_𝑖𝑡−𝜏_𝑖𝑗(𝑡). (3.17) Let 𝑀₂ denote an arbitrary positive number and set 𝑀₂>max𝑥_𝑖(𝑡),𝑀∀𝑡∈𝑡₀−𝑟_𝑖,𝑡₀,𝑖=1,2,…,𝑛. (3.18) We claim that 𝑥_𝑖(𝑡)<𝑀₂,∀𝑡∈𝑡₀,+∞,𝑖=1,2,…,𝑛. (3.19) If this is not valid, there must exist 𝑡₄∈(𝑡₀,+∞) and 𝑖∈{1,2,…,𝑛} such that 𝑥_𝑖𝑡₄=𝑀₂,𝑥_𝑗(𝑡)<𝑀₂,∀𝑡<𝑡₄,𝑗=1,2,…,𝑛. (3.20) Then, from (3.15) and (3.17), we have 0≤𝑥^′_𝑖𝑡₄≤𝜇𝑥_𝑖𝑡₄+𝑀𝑒^(𝜇−𝜆_{)(𝑡4−𝑡0)}−𝑏⁻_𝑖𝑖𝑥_𝑖𝑡₄+𝐾₂2𝑏⁺_𝑖𝑖𝑥_𝑖𝑡₄+^𝑛_{𝑗=1,𝑗≠𝑖}𝑏⁺_𝑖𝑗𝑥_𝑗𝑡₄+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝑒^𝜇𝑟𝑖𝑥_𝑖𝑡₄−𝜏_𝑖𝑗𝑡₄≤𝜇+1−𝑏⁻_𝑖𝑖+𝐾₂2𝑏⁺_𝑖𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑏⁺_𝑖𝑗+^𝑙_𝑗=1𝑐⁺_𝑖𝑗𝑒^𝜇𝑟𝑖𝑀₂<0. (3.21)
This contradiction implies that (3.19) holds. Thus, 𝑁_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑒^{−𝜇(𝑡−𝑡0)}≤𝑀₂𝑒^{−𝜇(𝑡−𝑡0)}∀𝑡∈𝑡₀−𝑟_𝑖,+∞,𝑖=1,2,…,𝑛. (3.22) This completes the proof.

4. Numerical Examples

In this section, we give two examples and numerical simulations to demonstrate the results obtained in previous sections.

Example 4.1. Consider the following Nicholson's blowflies system with patch structure and nonlinear density-dependent mortality terms: 𝑁^′₁(𝑡)(|𝑁=−25+cos3𝑡|)₁(𝑡)5+|sin2𝑡|+𝑁₁(𝑡)+(1+|sin2𝑡|)𝑁₂(𝑡)3+|cos3𝑡|+𝑁₂(𝑡)+(1+|cos2𝑡|)𝑁₃(𝑡)3+|sin3𝑡|+𝑁₃(𝑡)+141+cos²𝑡𝑁₁(𝑡−2|sin𝑡|)𝑒^{−4𝑁1(𝑡−2|sin𝑡|)}+141+sin²𝑡𝑁₁(𝑡−2|cos𝑡|)𝑒^{−4𝑁1(𝑡−2|cos𝑡|)}𝑁^′₂(𝑡)(|𝑁=−25+sin3𝑡|)₂(𝑡)5+|cos2𝑡|+𝑁₂(𝑡)+(1+|cos2𝑡|)𝑁₁(𝑡)3+|sin3𝑡|+𝑁₁(𝑡)+(1+|sin2𝑡|)𝑁₃(𝑡)3+|cos3𝑡|+𝑁₃(𝑡)+141+sin²𝑡𝑁₂(𝑡−2|cos𝑡|)𝑒^{−4𝑁2(𝑡−2|cos𝑡|)}+141+cos²𝑡𝑁₂(𝑡−2|sin𝑡|)𝑒^{−4𝑁2(𝑡−2|sin𝑡|)}𝑁^′₃(𝑡)(|𝑁=−25+sin5𝑡|)₃(𝑡)5+|cos6𝑡|+𝑁₃(𝑡)+(1+|cos3𝑡|)𝑁₁(𝑡)3+|sin2𝑡|+𝑁₁(𝑡)+(1+|sin3𝑡|)𝑁₂(𝑡)3+|cos2𝑡|+𝑁₂(𝑡)+141+sin²2𝑡𝑁₃(𝑡−2|cos2𝑡|)𝑒^{−4𝑁3(𝑡−2|cos2𝑡|)}+141+cos²2𝑡𝑁₃(𝑡−2|sin2𝑡|)𝑒^{−4𝑁3(𝑡−2|sin2𝑡|)}. (4.1) Obviously, 𝑎⁻_𝑖𝑖=25,𝑏⁺_𝑖𝑖=6,(𝑖=1,2,3),𝑎⁺_𝑖𝑗=2,𝑏⁻_𝑖𝑗=3,(𝑖,𝑗=1,2,3,𝑖≠𝑗),𝑐⁺_𝑖𝑗=1/2,𝛾⁻_𝑖𝑗=4,(𝑖=1,2,3,𝑗=1,2). Let 𝐾₁=𝑒, then we have 25=𝑎6+e𝑏⁺_𝑖𝑖+^−𝑖𝑖𝐾₁>³_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗𝑏⁻_𝑖𝑗+²_𝑗=1𝑐⁺_𝑖𝑗𝛾⁻_𝑖𝑗𝑒𝐾₁=43+14𝑒²,25=𝑎6+𝑒𝑏⁺_𝑖𝑖+^−𝑖𝑖𝐾₁>³_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗𝑏⁻_𝑖𝑗+²_𝑗=1𝑐⁺_𝑖𝑗=73. (4.2) Then (4.2) imply that the system (4.1) satisfies (2.1) and (2.6). Hence, from Theorems 2.1 and 2.2, the solution 𝑁(𝑡) of system (4.1) with 𝐷_𝑖𝑗(𝑡,𝑁)=𝑎_𝑖𝑗(𝑡)𝑁/(𝑏_𝑖𝑗(𝑡)+𝑁)(𝑖,𝑗=1,2,3) and 𝜑∈𝐸¹={𝜑∣𝜑∈C₊,𝜑(0)>0and0≤𝜑_𝑖(𝑡)<𝑒,forall,𝑡∈[−2,0],𝑖=1,2,3} is exponentially extinct as 𝑡→+∞ and 𝑁(𝑡)=𝑁(𝑡,0,𝜑)=𝑂(𝑒^−𝜅𝑡),𝜅≈0.0001. The fact is verified by the numerical simulation in Figure 1.

Figure 1 

Numerical solution 𝑁(𝑡)=(𝑁1(𝑡),𝑁2(𝑡),𝑁3(𝑡))𝑇 of system (4.1) for initial value 𝜑(𝑡)≡(0.5,1.7,2.6)𝑇.

Example 4.2. Consider the following Nicholson's blowflies system with patch structure and nonlinear density-dependent mortality terms:𝑁^′₁(𝑡)=−(12+|sin𝑡|)+(11+|cos𝑡|)𝑒^{−𝑁1(𝑡)}+11+2|−1sin𝑡|2+12|cos𝑡|𝑒^{−𝑁2(𝑡)}+11+2||−1sin𝑡2+12|cos𝑡|𝑒^{−𝑁3(𝑡)}+141+cos²𝑡𝑁₁(𝑡−2|sin𝑡|)𝑒^{−𝑁1(𝑡−2|sin𝑡|)}+141+sin²𝑡𝑁₁(𝑡−2|cos𝑡|)𝑒^{−𝑁1(𝑡−2|cos𝑡|)}𝑁^′₂(𝑡)=−(12+|cos𝑡|)+(11+|sin𝑡|)𝑒^{−𝑁2(𝑡)}+11+2||−1cos𝑡2+12|sin𝑡|𝑒^{−𝑁1(𝑡)}+11+2|−1cos𝑡|2+12|sin𝑡|𝑒^{−𝑁3(𝑡)}+141+sin²𝑡𝑁₂(𝑡−2|cos𝑡|)𝑒^{−𝑁2(𝑡−2|cos𝑡|)}+141+cos²𝑡𝑁₂(𝑡−2|sin𝑡|)𝑒^{−𝑁2(𝑡−2|sin𝑡|)}𝑁^′₃(𝑡)=−(12+|sin2𝑡|)+(11+|cos2𝑡|)𝑒^{−𝑁3(𝑡)}+11+2||−1sin2𝑡2+12|cos2𝑡|𝑒^{−𝑁1(𝑡)}+11+2|−1sin2𝑡|2+12|cos2𝑡|𝑒^{−𝑁2(𝑡)}+141+cos²2𝑡𝑁₃(𝑡−2|sin2𝑡|)𝑒^{−𝑁3(𝑡−2|sin2𝑡|)}+141+sin²2𝑡𝑁₃(𝑡−2|cos2𝑡|)𝑒^{−𝑁3(𝑡−2|cos2𝑡|)}. (4.3) Obviously, 𝑎⁻_𝑖𝑖=12,𝑏⁻_𝑖𝑖=11,𝑏⁺_𝑖𝑖=12,(𝑖=1,2,3),𝑎⁺_𝑖𝑗=3/2,𝑏⁻_𝑖𝑗=1/2,𝑏⁺_𝑖𝑗=1,(𝑖,𝑗=1,2,3,𝑖≠𝑗),𝑐⁺_𝑖𝑗=1/2,𝛾⁻_𝑖𝑗=1,(𝑖=1,2,3,𝑗=1,2). Let 𝐾₂=1, then we have 12=𝑎⁻_𝑖𝑖>³_{𝑗=1,𝑗≠𝑖}𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑏⁻_𝑖𝑗𝑒^−𝐾2+²_𝑗=1𝑐⁺_𝑖𝑗𝛾⁻_𝑖𝑗𝑒=3+12𝑒,𝑖=1,2,3,−𝑎_𝑖𝑖(𝑡)+𝑏_𝑖𝑖(𝑡)+³_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)=0,𝑖=1,2,3,11=𝑏⁻_𝑖𝑖𝐾>1+²2𝑏⁺_𝑖𝑖+³_{𝑗=1,𝑗≠𝑖}𝑏⁺_𝑖𝑗+²_𝑗=1𝑐⁺_𝑖𝑗=10,𝑖=1,2,3. (4.4) Then (4.4) imply that the system (4.3) satisfies (3.1), (3.2), (3.11), and (3.12). Hence, from Theorems 3.1 and 3.2, the solution 𝑁(𝑡) of system (4.1) with 𝐷_𝑖𝑗(𝑡,𝑁)=𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)𝑒^−𝑁(𝑖,𝑗=1,2,3) and 𝜑∈𝐸²={𝜑∣𝜑∈𝐶₊,𝜑(0)>0and0≤𝜑_𝑖(𝑡)<1,forall,𝑡∈[−2,0],𝑖=1,2,3} is exponentially extinct as 𝑡→+∞ and 𝑁(𝑡)=𝑁(𝑡,0,𝜑)=𝑂(𝑒^−𝜅𝑡),𝜅≈0.0001. The fact is verified by the numerical simulation in Figure 2.

Figure 2 

Numerical solution 𝑁(𝑡)=(𝑁1(𝑡),𝑁2(𝑡),𝑁3(𝑡))𝑇 of system (4.3) for initial value 𝜑(𝑡)≡(0.1,0.5,0.9)𝑇.

Remark 4.3. To the best of our knowledge, few authors have considered the problems of the extinction of Nicholson's blowflies model with patch structure and nonlinear density-dependent mortality terms. Wang [10] and Hou et al. [11] have researched the permanence and periodic solution for scalar Nicholson's blowflies equation with a nonlinear density-dependent mortality term. Liu and Gong [12] have considered the permanence for Nicholson-type delay systems with nonlinear density-dependent mortality terms and Takeuchi et al. [13] have investigated the global stability of population model with patch structure. Faria [14], Liu [15], and Berzansky et al. [16] have, respectively, studied the local and global stability of positive equilibrium for constant coefficients of Nicholson's blowflies model with patch structure. It is clear that all the results in [10–16] and the references therein cannot be applicable to prove the extinction of (4.1) and (4.3). This implies that the results of this paper are new.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant no. 11201184), the Natural Scientific Research Fund of Zhejiang Province of China (Grant nos. Y6110436, LY12A01018), and Innovation Program of Shanghai Municipal Education Commission (Grant no. 13YZ127).