Almost periodic solutions for a delayed Nicholson’s blowflies model with a nonlinear density-dependent mortality term
© Liu; licensee Springer. 2014
Received: 2 July 2013
Accepted: 7 February 2014
Published: 20 February 2014
A delayed Nicholson’s blowflies model with a nonlinear density-dependent mortality term is studied in this paper. Some sufficient conditions are obtained to guarantee the existence and global exponential stability of positive almost periodic solutions of this model. An example with numerical simulations is given to illustrate our main results.
MSC: 34C25, 34K13.
where the nonlinear density-dependent mortality function might have one of the following forms: or with positive constants .
where coefficients and delays are time-varying with or . Moreover, the dynamic behaviors on the existence of positive solutions, periodic solutions, persistence, permanence, oscillation and stability of Nicholson’s blowflies model (1.2) and its analogous equations have been studied extensively. We refer the reader to [3–9] and the references cited therein. On the other hand, the variation of the environment plays an important role in many biological and ecological dynamical systems. Fink  and He  pointed out that periodically varying environment and almost periodically varying environment are foundations for the theory of nature selection. Compared with periodic effects, almost periodic effects are more frequent. Hence, the effects of almost periodic environment on evolutionary theory have been the object of intensive analysis by numerous authors, and some of these results on Nicholson’s blowflies model without nonlinear density-dependent mortality term can be found in [12–14]. In particular, these results were obtained by using exponential dichotomy theory on almost periodic differential equations or functional differential equations with linear part. However, there is not any linear part in Nicholson’s blowflies model with a nonlinear density-dependent mortality term. Thus, many classical and traditional approaches fail to almost periodic problems on (1.2). Therefore, a new method must be sought to investigate the existence and stability of positive almost periodic solutions of (1.2).
where and are almost periodic functions, and .
Throughout this paper, let denote a nonnegative real number space, be the continuous functions space equipped with the usual supremum norm , and let . If is continuous and defined on with , then we define , where for all .
Then f is a locally Lipschitz map with respect to , which ensures the existence and uniqueness of the solution of (1.3) with admissible initial conditions (1.5).
We denote by an admissible solution of admissible initial value problem (1.3) and (1.5). Also, let be the maximal right-interval of the existence of .
The remainder of this paper is organized as follows. In Section 2, we give some definitions and lemmas, which tell us that some kinds of solutions to (1.3) are bounded. These results play an important role in Section 3 to establish the existence of almost periodic solutions of (1.3). Here we also study the local exponential stability of almost periodic solutions. The paper concludes with an example to illustrate the effectiveness of the obtained results by numerical simulation.
2 Preliminary results
In this section, we shall first recall some basic definitions, lemmas which are used in what follows.
A continuous function is said to be almost periodic on R if, for any , the set is relatively dense, i.e., for any , it is possible to find a real number with the property that, for any interval with length , there exists a number in this interval such that for all .
for all and .
This contradiction means that for all .
which is a contradiction and implies that is bounded on . From Theorem 2.3.1 in , we easily obtain .
This contradiction yields that (2.8) holds.
This ends the proof of Lemma 2.1. □
It is obvious that and is non-decreasing.
Now, we distinguish two cases to finish the proof.
In summary, there must exist such that holds for all . The proof of Lemma 2.2 is now complete. □
3 Main results
In this section, we establish sufficient conditions on the existence and global exponential stability of almost periodic solutions of (1.3).
for all j, t.
Therefore, is a solution of (1.3).
which implies that is an almost periodic solution of equation (1.3).
Finally, we prove that is globally exponentially stable.
This completes the proof of Theorem 3.1. □
In this section, we present an example and its numerical simulation to check the validity of results we obtained in the previous sections.
Remark 4.1 To the best of our knowledge, few authors have studied the problems of positive almost periodic solutions of Nicholson’s blowflies delayed systems with nonlinear density-dependent mortality terms. It is clear that the results in [11–13] and the references therein cannot be applicable to system (4.1) to prove the global exponential stability of a positive almost periodic solution. Moreover, one can find that the main results of  are restricted to considering the Nicholson’s blowflies delayed systems with the nonlinear density-dependent mortality term and give no opinions about . This implies that the results of the present paper are new and complement previously known results.
The author would like to express his sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. This work was supported by the construct program of the key discipline in Hunan province (Mechanical Design and Theory), the Scientific Research Fund of Hunan Provincial Natural Science Foundation of P.R. China (Grant No. 11JJ6006), the Natural Scientific Research Fund of Hunan Provincial Education Department of P.R. China (Grants Nos. 11C0916, 11C0915).
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