Abstract and Applied Analysis
Volume 2012, Issue 1 302065
Research Article
Open Access

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

Wentao Wang, 

Corresponding Author

Wentao Wang

College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, China zjxu.edu.cn

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First published: 31 December 2012
https://doi.org/10.1155/2012/302065
Citations: 3
Academic Editor: Juntao Sun

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:
()
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
()
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:
()
where
()
aij, bij, ci, γi : R → (0, +∞) are all continuous functions bounded above and below by positive constants, and τj(t) ≥ 0 are bounded continuous functions, ri = sup t∈Rτi(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:
()
where
()
aij, bij, cik, γik : R → (0, +∞) are all continuous functions bounded above and below by positive constants, and τik(t) ≥ 0 are bounded continuous functions, ri = max 1≤j≤l{sup t∈Rτij(t)} > 0, and i, j = 1,2, …, n,   k = 1,2, …, l. Furthermore, in the case Dij(t, N) = aij(t) − bij(t)e−N, to guarantee the meaning of mortality terms we assume that aij(t) > bij(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
()

Let be the set of all (nonnegative) real vectors, we will use x = (x1, …, xn) T ∈ Rn 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 | , …, |xn | ) T and define | | x|| = max 1≤i≤n | xi|. Denote as Banach spaces equipped with the supremum norm defined by for all φ(t) = (φ1(t), …, φn(t)) T ∈ C (or ∈C+). If xi(t) is defined on [t0 − ri, ν) with t0, ν ∈ R and i = 1, …, n, then we define xt ∈ C as where for all θ ∈ [−ri, 0] and i = 1, …, n.

The initial conditions associated with system (1.5) are of the form:
()
We write Nt(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 Nt(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 . Denote it as .

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 Dij(t, N) = aij(t)N/(bij(t) + N)(i, j = 1,2, …, n)

Theorem 2.1. Suppose that there exists positive constant K1 such that

()
Let
()
Moreover, assume N(t; t0, φ) is the solution of (1.5) with φ ∈ E1 and Dij(t, N) = (aij(t)N/(bij(t) + N))(i, j = 1,2, …, n). Then,
()

Proof. Set N(t) = N(t; t0, φ)      for    all         t ∈ [t0, η(φ)). In view of φ ∈ C+, using Theorem 5.2.1 in [17, p. 81], we have Nt(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

()
Calculating the derivative of Ni(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
()
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

()
Then the solution N(t; t0, φ) of (1.5) with φ ∈ E1 and Dij(t, N) = (aij(t)N/(bij(t) + N))  (i, j = 1,2, …, n) is exponentially extinct as t → +∞.

Proof. Define continuous functions Γi(ω) by setting

()
Then, from (2.6), we obtain
()
The continuity of Γi(ω) implies that there exists λ > 0 such that
()
Let
()
Calculating the derivative of y(t) along the solution N(t) of system (1.5) with φ ∈ E1, we have
()
Let M1 denote an arbitrary positive number and set
()
We claim that
()
If this is not valid, there must exist t2 ∈ (t0, +∞) and i ∈ {1,2, …, n} such that
()
Then, from (2.3) and (2.11), we have
()
This contradiction implies that (2.13) holds. Thus,
()
This completes the proof.

3. Extinction of Nicholson′s Blowflies System with Dij(t, N) = aij(t) − bij(t)e−N(i, j = 1,2, …, n)

Theorem 3.1. Suppose that there exists positive constant K2 such that

()
()
Let
()
Moreover, assume N(t; t0, φ) is the solution of (1.5) with φ ∈ E2 and Dij(t, N) = aij(t) − bij(t)e−N  (i, j = 1,2, …, n). Then,
()
()

Proof. Set N(t) = N(t; t0, φ)  for    all  t ∈ [t0, η(φ)). Rewrite the system (1.5) as

()
where f(t, ϕ) = (f1(t, ϕ), f2(t, ϕ), …, fn(t, ϕ)) T and
()
In view of (3.2), whenever ϕ ∈ C satisfies ϕ ≥ 0,    ϕi(0) = 0 for some i and t ∈ R, then
()
Thus, using Theorem 5.2.1 in [17, p. 81], we have Nt(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
()
Calculating the derivative of Ni(t), together with (3.1) and the fact that sup u≥0 ue−u = 1/e, (1.5) and (3.9) imply that
()
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 and such that

()
()
Then the solution N(t; t0, φ) of (1.5) with φ ∈ E2 and Dij(t, N) = aij(t) − bij(t)e−N  (i, j = 1,2, …, n), is exponentially extinct as t → +∞.

Proof. Define continuous functions Γi(ω) by setting

()
Then, from (3.12), we obtain
()
The continuity of Γi(ω) implies that there exists such that
()
Let
()
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
()
Let M2 denote an arbitrary positive number and set
()
We claim that
()
If this is not valid, there must exist t4 ∈ (t0, +∞) and i ∈ {1,2, …, n} such that
()
Then, from (3.15) and (3.17), we have
()

This contradiction implies that (3.19) holds. Thus,

()
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:

()
Obviously, . Let K1 = e, then we have
()
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 Dij(t, N) = aij(t)N/(bij(t) + N)(i, j = 1,2, 3) and φ ∈ E1 = {φ∣φ ∈ C+,   φ(0) > 0  and  0 ≤ φi(t) < e,   for    all, 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.

Details are in the caption following the image
Figure 1
Open in figure viewerPowerPoint
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:

()
Obviously, . Let K2 = 1, then we have
()
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 Dij(t, N) = aij(t) − bij(t)e−N(i, j = 1,2, 3) and φ ∈ E2 = {φ∣φ ∈ C+,   φ(0) > 0  and  0 ≤ φi(t) < 1,   for    all,   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.

Details are in the caption following the image
Figure 2
Open in figure viewerPowerPoint
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).

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