Multiple periodic solutions for discrete Nicholson’s blowflies type system. (English) Zbl 1474.39010

Ding, Hui-Sheng; Dix, Julio G.

Abstr. Appl. Anal. 2014, Article ID 659152, 6 p. (2014).

Summary: This paper is concerned with the existence of multiple periodic solutions for discrete Nicholson’s blowflies type system. By using the Leggett-Williams fixed point theorem, we obtain the existence of three nonnegative periodic solutions for discrete Nicholson’s blowflies type system. In order to show that, we first establish the existence of three nonnegative periodic solutions for the \(n\)-dimensional functional difference system \(y \left(k + 1\right) = A \left(k\right) y \left(k\right) + f \left(k, y \left(k - \tau\right)\right), k \in \mathbb Z\), where \(A \left(k\right)\) is not assumed to be diagonal as in some earlier results. In addition, a concrete example is also given to illustrate our results.

MSC:

39A12	Discrete version of topics in analysis
92D25	Population dynamics (general)

References:

[1]	Nicholson, A. J., An outline of the dynamics of animal populations, Australian Journal of Zoology, 2, 9-65 (1954)
[2]	Gurney, W. S.; Blythe, S. P.; Nisbet, R. M., Nicholson’s blowflies revisited, Nature, 287, 17-21 (1980)
[3]	Liu, B., The existence and uniqueness of positive periodic solutions of Nicholson-type delay systems, Nonlinear Analysis: Real World Applications, 12, 6, 3145-3151 (2011) · Zbl 1231.34119
[4]	Liu, B.; Gong, S., Permanence for Nicholson-type delay systems with nonlinear density-dependent mortality terms, Nonlinear Analysis: Real World Applications, 12, 4, 1931-1937 (2011) · Zbl 1232.34109
[5]	Berezansky, L.; Idels, L.; Troib, L., Global dynamics of Nicholson-type delay systems with applications, Nonlinear Analysis: Real World Applications, 12, 1, 436-445 (2011) · Zbl 1208.34120
[6]	Chen, W.; Wang, L., Positive periodic solutions of Nicholson-type delay systems with nonlinear density-dependent mortality terms, Abstract and Applied Analysis, 2012 (2012) · Zbl 1260.34149
[7]	Zhou, Q., The positive periodic solution for Nicholson-type delay system with linear harvesting terms, Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, 37, 8, 5581-5590 (2013) · Zbl 1274.34209
[8]	Saker, S. H.; Agarwal, S., Oscillation and global attractivity in a periodic Nicholson’s blowflies model, Mathematical and Computer Modelling, 35, 7-8, 719-731 (2002) · Zbl 1012.34067
[9]	Saker, S. H., Oscillation of continuous and discrete diffusive delay Nicholson’s blowflies models, Applied Mathematics and Computation, 167, 1, 179-197 (2005) · Zbl 1075.92051
[10]	So, J. W.-H.; Yu, J. S., On the stability and uniform persistence of a discrete model of Nicholson’s blowflies, Journal of Mathematical Analysis and Applications, 193, 1, 233-244 (1995) · Zbl 0834.39009
[11]	Alzabut, J.; Bolat, Y.; Abdeljawad, T., Almost periodic dynamics of a discrete Nicholson’s blowflies model involving a linear harvesting term, Advances in Difference Equations, 2012, article 158 (2012) · Zbl 1377.39025
[12]	Alzabut, J. O., Existence and exponential convergence of almost periodic solutions for a discrete Nicholson’s blowflies model with a nonlinear harvesting term, Mathematical Sciences Letters, 2, 201-207 (2013)
[13]	Padhi, S.; Qian, C.; Srivastava, S., Multiple periodic solutions for a first order nonlinear functional differential equation with applications to population dynamics, Communications in Applied Analysis, 12, 3, 341-351 (2008) · Zbl 1187.34092
[14]	Padhi, S.; Srivastava, S.; Dix, J. G., Existence of three nonnegative periodic solutions for functional differential equations and applications to hematopoiesis, Panamerican Mathematical Journal, 19, 1, 27-36 (2009) · Zbl 1187.34119
[15]	Padhi, S.; Pati, S., Positive periodic solutions for a nonlinear functional differential equation, Georgian Academy of Sciences A. Razmadze Mathematical Institute: Memoirs on Differential Equations and Mathematical Physics, 51, 109-118 (2010) · Zbl 1220.34090
[16]	Alzabut, J. O., Almost periodic solutions for an impulsive delay Nicholson’s blowflies model, Journal of Computational and Applied Mathematics, 234, 1, 233-239 (2010) · Zbl 1196.34095
[17]	Ding, H.-S.; Nieto, J. J., A new approach for positive almost periodic solutions to a class of Nicholson’s blowflies model, Journal of Computational and Applied Mathematics, 253, 249-254 (2013) · Zbl 1288.92017
[18]	Leggett, R. W.; Williams, L. R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Mathematics Journal, 28, 4, 673-688 (1979) · Zbl 0421.47033
[19]	Dix, J. G.; Padhi, S.; Pati, S., Multiple positive periodic solutions for a nonlinear first order functional difference equation, Journal of Difference Equations and Applications, 16, 9, 1037-1046 (2010) · Zbl 1204.39013
[20]	Raffoul, Y. N., Positive periodic solutions of nonlinear functional difference equations, Electronic Journal of Differential Equations, 55, 1-8 (2002) · Zbl 1007.39005
[21]	Raffoul, Y. N.; Tisdell, C. C., Positive periodic solutions of functional discrete systems and population models, Advances in Difference Equations, 3, 369-380 (2005) · Zbl 1111.39008
[22]	Islam, M. N.; Raffoul, Y. N., Periodic solutions of neutral nonlinear system of differential equations with functional delay, Journal of Mathematical Analysis and Applications, 331, 2, 1175-1186 (2007) · Zbl 1118.34057
[23]	Jiang, D.; Wei, J.; Zhang, B., Positive periodic solutions of functional differential equations and population models, Electronic Journal of Differential Equations, 71, 1-13 (2002)
[24]	Zeng, Z.; Bi, L.; Fan, M., Existence of multiple positive periodic solutions for functional differential equations, Journal of Mathematical Analysis and Applications, 325, 2, 1378-1389 (2007) · Zbl 1110.34043
[25]	Zhang, W.; Zhu, D.; Bi, P., Existence of periodic solutions of a scalar functional differential equation via a fixed point theorem, Mathematical and Computer Modelling, 46, 5-6, 718-729 (2007) · Zbl 1145.34041