DYNAMICS OF A TWO-PATCH NICHOLSON'S BLOWFLIES MODEL WITH RANDOM DISPERSAL

Houfu Liu; Yuanyuan Cong; Ying Su

doi:10.11948/20210268

2022 Volume 12 Issue 2

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Houfu Liu, Yuanyuan Cong, Ying Su. DYNAMICS OF A TWO-PATCH NICHOLSON'S BLOWFLIES MODEL WITH RANDOM DISPERSAL[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 692-711. doi: 10.11948/20210268

Citation:

Houfu Liu, Yuanyuan Cong, Ying Su. DYNAMICS OF A TWO-PATCH NICHOLSON'S BLOWFLIES MODEL WITH RANDOM DISPERSAL[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 692-711. doi: 10.11948/20210268

DYNAMICS OF A TWO-PATCH NICHOLSON'S BLOWFLIES MODEL WITH RANDOM DISPERSAL

1.
Department of Mathematics, Harbin Institute of Technology, No 92 West Dazhi street, Harbin 150001, China

Corresponding author: Email: ysu@hit.edu.cn(Y. Su)
Fund Project: The authors were supported by the National Natural Science Foundation of China (11971129, 11771108)

Abstract

Abstract

The global dynamics of the Nicholson's blowfly reaction-diffusion model with zero Dirichlet boundary condition is less understood. In this paper, we provide a discrete version of diffusive Nichlson's blowflies equation with zero Dirichlet boundary condition. Local and global stability of the equilibria are obtained by some comparison arguments, fluctuation method and the theory of exponential ordering. Hopf bifurcation at the positive equilibrium and the global existence of the periodic solutions are studied by local and global Hopf bifurcation theory.
- 1Nicholson's blowflies model /
- Dirichlet boundary condition /
- stability /
- Hopf bifurcations
MSC: 34K18, 92D25, 35B10, 35K57

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References

[1]	K. A. G. Azevedo and L. A. C. Laderira, Hopf bifurcation for a class of partial differnetial equation with delay, Funkcialaj Ekvacioj, 2004, 47, 395-422. doi: 10.1619/fesi.47.395 CrossRef Google Scholar
[2]	L. Berezansky, E. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: main results and open problems, Appl. Math. Model, 2010, 34, 1405-1417. doi: 10.1016/j.apm.2009.08.027 CrossRef Google Scholar
[3]	S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differerential Equations, 1996, 124(1), 80-107. doi: 10.1006/jdeq.1996.0003 CrossRef Google Scholar
[4]	S. Chen, Y. Lou and J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differerential Equations, 2018, 264, 5333-5359. doi: 10.1016/j.jde.2018.01.008 CrossRef Google Scholar
[5]	S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 2012, 253(12), 3440-3470. doi: 10.1016/j.jde.2012.08.031 CrossRef Google Scholar
[6]	S. Chen and J. Yu, Stability and bifurcations in a nonlocal delayed reaction-diffusion population model, J. Differential Equations, 2016, 260(1), 218-240. doi: 10.1016/j.jde.2015.08.038 CrossRef Google Scholar
[7]	S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 2015, 259(4), 1409-1448. doi: 10.1016/j.jde.2015.03.006 CrossRef Google Scholar
[8]	S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, J. Differential Equations, 2016, 260(1), 781-817. doi: 10.1016/j.jde.2015.09.031 CrossRef Google Scholar
[9]	M. S. Gurney, S. P Blythe and R. M. Nisbet, Nicholson's bowflies revisited, Nature, 1980, 287, 17-21. doi: 10.1038/287017a0 CrossRef Google Scholar
[10]	B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. Google Scholar
[11]	Z. Jin and R. Yuan, Hopf bifurcation in a reaction-diffusion-advection equation with nonlocal delay effect, J. Differerential Equations, 2021, 271(1), 533-562. Google Scholar
[12]	J. Li, Global attractivity in Nicholson's blowflies, Applied Mathematics, 1996, 11(4), 425-434. doi: 10.1007/BF02662882 CrossRef Google Scholar
[13]	K. Liao and Y. Lou, The effect of time delay in a two-patch model with random dispersal, Bull. Math. Biol., 2014, 76, 335-376. doi: 10.1007/s11538-013-9921-7 CrossRef Google Scholar
[14]	Z. Ma, H. Huo and H. Xiang, Hopf bifurcation for a delyed predator-prey diffusion system with Dirichlet boundary condition, Applied Mathematics and Computation, 2017, 311, 1-18. Google Scholar
[15]	A. J. Nicholson, Compensatory reactions of populations to stresses, and their evolutionary significance, Australian Journal of Zoology, 1954, 2, 1-8. doi: 10.1071/ZO9540001 CrossRef Google Scholar
[16]	S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynam. Contin. Discrete Impulsive Syst. Ser. A Math. Anal., 2003, 10, 863-874. Google Scholar
[17]	H. L. Smith, Monotone Dynamical Systems: An introduction to the theory of competitive and cooperative systems, American Mathematical Society, Providence, Rhode Island, 2008. Google Scholar
[18]	H. L. Smith and H. Thieme, Strongly order preserving semiflows generated by functional differential equations, J. Differential Equations, 1991, 93, 332-363. doi: 10.1016/0022-0396(91)90016-3 CrossRef Google Scholar
[19]	J. W. H. So, J. Wu and Y. Yang, Numerical steady-state and Hopf bifurcation analysis on the diffusive Nicholson's blowflies equation, Appl. Math. Comput., 2000, 111, 33-51. Google Scholar
[20]	J. W. H. So and Y. Yang, Dirichlet problem for the diffusive Nicholson's blowflies equation, J. Differential Equations, 1998, 150(2), 317-348. doi: 10.1006/jdeq.1998.3489 CrossRef Google Scholar
[21]	Y. Su, J. Wei and J. Shi, Hopf bifurcations in a reaction?Cdiffusion population model with delay effect, J. Differential Equations, 2009, 247(4), 1156-1184. doi: 10.1016/j.jde.2009.04.017 CrossRef Google Scholar
[22]	Y. Su, J. Wei and J. Shi, Hopf bifurcation in a diffusive logistic equation with mixed delayed and instantaneous density dependence, J. Dynam. Differential Equations, 2012, 24(4), 897-925. doi: 10.1007/s10884-012-9268-z CrossRef Google Scholar
[23]	J. Wei and M. Li, Hopf bifurcation analysis in a delayde Nicholson's blowflies equation, Nonlinear Anal. TMA, 2005, 60, 1351-1367. doi: 10.1016/j.na.2003.04.002 CrossRef Google Scholar
[24]	J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 1998, 350, 4799-4838. doi: 10.1090/S0002-9947-98-02083-2 CrossRef Google Scholar
[25]	X. Yan and W. Li, Stability of bifurcating periodic solutions in a delayed reaction?Cdiffusion population model, Nonlinearity, 2010, 23(6), 1413-1431. doi: 10.1088/0951-7715/23/6/008 CrossRef Google Scholar
[26]	X. Yan and W. Li, Stability and Hopf bifurcations for a delayed diffusion system in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 2012, 17(1), 367-399. Google Scholar
[27]	Y. Yang and J. W. H. So, Dynamics of the diffusive Nicholson's blowflies equation. in: Wenxiong Chen, Shouchuan Hu. (Eds.), in: Proceedings of the International Conference on Dynamical Systems and Differential Equations, Vol. Ⅱ, Springfield, Missouri, U.S. A, 1996, An added volume to Discrete Contin. Dyn. Syst., 1998, 333-352. Google Scholar
[28]	T. Yi, Y Chen and J. Wu, Threshold dynamics of a delayed reaction diffusion equation subject to the Dirichlet condition, J. Biol. Dyn., 2009, 3, 331-341. doi: 10.1080/17513750802425656 CrossRef Google Scholar
[29]	X. Zhang, S. Song and J. Wu, Onset and termination of oscillation of disease spread through contaminated environment, Mathematical Biosciences and Engineering, 2017, 14(5/6), 1515-1533. doi: 10.3934/mbe.2017079 CrossRef Google Scholar
[30]	X. Zhao, Global attractivity in a class of non-monotone reaction-diffusion equations with time delay, Cand. Appl. Math. Quart., 2009, 17(1), 271-281. Google Scholar
[31]	L. Zhou, Y. Tang and S. Hussein, Stability and Hopf bifurcation for a delay competition diffusion system, Chaos, Solution and Fractals, 2002, 14, 1201-1225. doi: 10.1016/S0960-0779(02)00068-1 CrossRef Google Scholar