Global exponential stability of positive periodic solutions for a delayed Nicholsonʼs blowflies model

Abstract This paper is concerned with a non-autonomous delayed Nicholsonʼs blowflies model. Under proper conditions, we employ a novel argument to establish a criterion on the global exponential stability of positive periodic solutions. This answers an open problem proposed by Berezansky et al. (2010) [2] . We also provide numerical simulations to support the theoretical result.

[1]  Junxia Meng,et al.  The positive almost periodic solution for Nicholson-type delay systems with linear harvesting terms☆ , 2012 .

[2]  Viktor Tkachenko,et al.  A Global Stability Criterion for Scalar Functional Differential Equations , 2003, SIAM J. Math. Anal..

[3]  Qiyuan Zhou The positive periodic solution for Nicholson-type delay system with linear harvesting terms , 2013 .

[4]  Kai Wang,et al.  Periodic solutions, permanence and global attractivity of a delayed impulsive prey–predator system with mutual interference ☆ , 2013 .

[5]  M. Hirsch,et al.  4. Monotone Dynamical Systems , 2005 .

[6]  S. P. Blythe,et al.  Nicholson's blowflies revisited , 1980, Nature.

[7]  Hal L. Smith,et al.  An introduction to delay differential equations with applications to the life sciences / Hal Smith , 2010 .

[8]  Wei Chen,et al.  Positive almost periodic solution for a class of Nicholson's blowflies model with multiple time-varying delays , 2011, J. Comput. Appl. Math..

[9]  Chaoxiong Du,et al.  Existence of positive periodic solutions for a generalized Nicholson's blowflies model , 2008 .

[10]  Wentao Wang,et al.  Existence and exponential stability of positive almost periodic solution for Nicholson-type delay systems , 2011 .

[11]  Bingwen Liu Global dynamic behaviors for a delayed Nicholson's blowflies model with a linear harvesting term , 2013 .

[12]  Hassan A. El-Morshedy,et al.  Global attractivity in a population model with nonlinear death rate and distributed delays , 2014 .

[13]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[14]  On positive periodic solution for the delay Nicholson’s blowflies model with a harvesting term , 2012 .

[15]  Jianhong Wu,et al.  Global dynamics of Nicholsonʼs blowflies equation revisited: Onset and termination of nonlinear oscillations , 2013 .

[16]  Jehad O. Alzabut,et al.  Almost periodic solutions for an impulsive delay Nicholson's blowflies model , 2010, J. Comput. Appl. Math..

[17]  A. Nicholson An outline of the dynamics of animal populations. , 1954 .

[18]  Teresa Faria,et al.  Asymptotic stability for delayed logistic type equations , 2006, Math. Comput. Model..

[19]  Yang Kuang,et al.  Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology , 1992 .

[20]  Juan J. Nieto,et al.  A new approach for positive almost periodic solutions to a class of Nicholson's blowflies model , 2013, J. Comput. Appl. Math..

[22]  G. Karakostas,et al.  Stable steady state of some population models , 1992 .

[23]  G. Ladas,et al.  Linearized oscillations in population dynamics. , 1987, Bulletin of mathematical biology.

[24]  Lian Duan,et al.  Permanence and periodic solutions for a class of delay Nicholson’s blowflies models , 2013 .

[25]  Elena Braverman,et al.  Persistence and extinction in spatial models with a carrying capacity driven diffusion and harvesting , 2013 .

[26]  Wentao Wang Positive periodic solutions of delayed Nicholson’s blowflies models with a nonlinear density-dependent mortality term , 2012 .

[27]  Elena Braverman,et al.  Nicholson's blowflies differential equations revisited: Main results and open problems , 2010 .

[28]  Lijuan Wang Almost periodic solution for Nicholson’s blowflies model with patch structure and linear harvesting terms , 2013 .

[29]  K. Cooke,et al.  Interaction of maturation delay and nonlinear birth in population and epidemic models , 1999 .

[30]  Xingfu Zou,et al.  Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case☆ , 2008 .

[31]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.