
Mathematical Biosciences and Engineering, 2018, 15(6): 14791494. doi: 10.3934/mbe.2018068.
Article
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
Coexistence of a crossdiffusive West Nile virus model in a heterogenous environment
1. School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2. Department of Mathematics, Faculty of Education, University of Khartoum, Khartoum 321, Sudan
3. School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China
Received: , Accepted: , Published:
Keywords: West Nile virus; stronglycoupled elliptic systems; heterogeneous environment; basic reproduction number; coexistence
Citation: Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin. Coexistence of a crossdiffusive West Nile virus model in a heterogenous environment. Mathematical Biosciences and Engineering, 2018, 15(6): 14791494. doi: 10.3934/mbe.2018068
References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reactiondiffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 120.
 [2] P. ÁlvarezCaudevilla and J. LópezGómez, Asymptotic behaviour of principal eigenvalues for a class of cooperative systems, J. Differential Equations, 244 (2008), 10931113.
 [3] D. S. Asnis, R. Conetta, A. A. Teixeira, G. Waldman and B. A. Sampson, The West Nile virus outbreak of 1999 in New York: The flushing hospital experience, Clinical Infect Dis., 30 (2000), 413418.
 [4] K. W. Blaynech, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile virus, Bull. Math. Biol., 72 (2010), 10061028.
 [5] E. Braverman and Md. Kamrujjaman, Competitivecooperative models with various diffusion strategies, Comput. Math. Appl., 72 (2016), 653662.
 [6] B. Chen and R. Peng, Coexistence states of a strongly coupled preypredator model, J. Partial Diff. Eqs., 18 (2005), 154166.
 [7] V. Chevalier, A. Tran and B. Durand, Predictive modeling of west nile virus transmission risk in the mediterranean basin, Int. J. Environ. Res. Public Health, 11 (2014), 6790.
 [8] G. CruzPacheco, L. Esteva and C. Vargas, Seasonality and outbreaks in west nile virus infection, Bull. Math. Biol., 71 (2009), 13781393.
 [9] D. G. de Figueiredo and E. Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal., 17 (1986), 836849.
 [10] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365382.
 [11] S. Fu, L. Zhang and P. Hu, Global behavior of solutions in a LotkaVolterra predatorprey model with preystage structure, Nonlinear Anal. Real World Appl., 14 (2013), 20272045.
 [12] W. Gan and Z. Lin, Coexistence and asymptotic periodicity in a competitorcompetitormutualist model, J. Math. Anal. Appl., 337 (2008), 10891099.
 [13] D. Horstmann, Remarks on some LotkaVolterra type crossdiffusion models, Nonlinear Anal. Real World Appl., 8 (2007), 90117.
 [14] M. Iida, M. Mimura and H. Ninomiya, Diffusion, crossdiffusion and competitive iteraction, J. Math. Biol., 53 (2006), 617641.
 [15] D. J. Jamieson, J. E. Ellis, D. B. Jernigan and T. A. Treadwell, Emerging infectious disease outbreaks: Old lessons and new challenges for obstetriciangynecologists, Am. J. Obstet. Gynecol., 194 (2006), 15461555.
 [16] Y. Jia, J. Wu and H. Xu, Positive solutions of LotkaVolterra competition model with crossdiffusion, Comput. Math. Appl., 68 (2014), 12201228.
 [17] A. Jüngel and I. V. Stelzer, Entropy structure of a crossdiffusion tumorgrowth model, Math. Models Methods Appl. Sci., 22 (2012), 1250009, 26pp.
 [18] K. I. Kim and Z. G. Lin, Coexistence of three species in a strongly coupled elliptic system, Nonlinear Anal., 55 (2003), 313333.
 [19] W. Ko and K. Ryu, On a predatorprey system with crossdiffusion representing the tendency of prey to keep away from its predators, Appl. Math. Lett., 21 (2008), 11771183.
 [20] K. Kuto and Y. Yamada, Multiple coexistence states for a preypredator system with crossdiffusion, J. Differential Equations, 197 (2004), 315348.
 [21] M. Lewis, J. Renclawowicz and P. Driessche, Travalling waves and spread rates for a west nile virus model, Bull. Math. Biol., 68 (2006), 323.
 [22] S. Li, J. Wu and S. Liu, Effect of crossdiffusion on the stationary problem of a Leslie preypredator model with a protection zone, Calc. Var. Partial Differential Equations, 56 (2017), Art. 82, 35 pp.
 [23] Z. G. Lin and H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 13811409.
 [24] Y. Lou, W. M. Ni and Y. Wu, On the global existence of a crossdiffusion system, Discrete Contin. Dynam. Sys A, 4 (1998), 193203.
 [25] Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the LotkaVolterra competition with crossdiffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435458.
 [26] D. Nash, F. Mostashari and A. Fine, etc., The Outbreak of West Nile Virus Infection in New York city area in 1999, N. Engl. Med., 344 (2001), 18071814.
 [27] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
 [28] C. V. Pao, Strongly coupled elliptic systems and applications to LotkaVolterra models with crossdiffusion, Nonlinear Anal., 60 (2005), 11971217.
 [29] K. A. Rahman, R. Sudarsan and H. J. Eberl, A mixedculture biofilm model with crossdiffusion, Bull. Math. Biol., 77 (2015), 20862124.
 [30] K. Ryu and I. Ahn, Positive steadystates for two interacting species models with linear selfcross diffusions, Discrete Contin. Dyn. Syst., 9 (2003), 10491061.
 [31] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 8399.
 [32] S. Shim, Long time properties of preypredator system with cross diffusion, Comm. Korean Math. Soc., 21 (2006), 293320.
 [33] G. Sweers, Strong positivity in $C(\overline \Omega)$ for elliptic systems, Math. Z., 209 (1992), 251271.
 [34] A. K. Tarboush, Z. G. Lin and M. Y. Zhang, Spreading and vanishing in a West Nile virus model with expanding fronts, Sci. China Math., 60 (2017), 841860.
 [35] P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 2948.
 [36] H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus, Math. Biosci., 272 (2010), 2028.
 [37] W. D. Wang and X. Q. Zhao, Basic reproduction numbers for reactiondiffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 16521673.
 [38] Z. Wen and S. Fu, Turing instability for a competitorcompetitormutualist model with nonlinear crossdiffusion effects, Chaos Solitons Fractals, 91 (2016), 379385.
 [39] M. J. Wonham, T. C. Beck and M. A. Lewis, An epidemiology model for West Nile virus: Invansion analysis and control applications, Proc. R. Soc. Lond B, 271 (2004), 501507.
 [40] Y. P. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289340.
 [41] X. Q. Zhao, Dynamical Systems in Population Biology, Second edition, CMS Books in Mathematics/Ouvrages de Math´ee´matiques de la SMC. Springer, Cham, 2017.
 [42] H. Zhou and Z. G. Lin, Coexistence in a stroungly coupled system describing a twospecies cooperative model, Appl. Math. Lett., 20 (2007), 11261130.
Reader Comments
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)
Associated material
Metrics
Other articles by authors
Related pages
Tools
your name: * your email: *