Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment

Abdelrazig K. Tarboush; Jing Ge; Zhigui Lin; Abdelrazig K. Tarboush; Jing Ge; Zhigui Lin

doi:10.3934/mbe.2018068

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 6: 1479-1494. doi: 10.3934/mbe.2018068

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Coexistence of a cross-diffusive 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: 05 June 2018 Revised: 13 August 2018 Published: 01 December 2018
MSC : Primary: 35K57, 92B05; Secondary: 35J60

This paper is concerned with a strongly-coupled elliptic system, which describes a West Nile virus (WNv) model with cross-diffusion in a heterogeneous environment. The basic reproduction number is introduced through the next generation infection operator and some related eigenvalue problems. The existence of coexistence states is presented by using a method of upper and lower solutions. The true positive solutions are obtained by monotone iterative schemes. Our results show that a cross-diffusive WNv model possesses at least one coexistence solution if the basic reproduction number is greater than one and the cross-diffusion rates are small enough, while if the basic reproduction number is less than or equal to one, the model has no positive solution. To illustrate the impact of cross-diffusion and environmental heterogeneity on the transmission of WNv, some numerical simulations are given.
- West Nile virus,
- strongly-coupled elliptic systems,
- heterogeneous environment,
- basic reproduction number,
- coexistence
Citation: Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin. Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment[J]. Mathematical Biosciences and Engineering, 2018, 15(6): 1479-1494. doi: 10.3934/mbe.2018068

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Abstract

This paper is concerned with a strongly-coupled elliptic system, which describes a West Nile virus (WNv) model with cross-diffusion in a heterogeneous environment. The basic reproduction number is introduced through the next generation infection operator and some related eigenvalue problems. The existence of coexistence states is presented by using a method of upper and lower solutions. The true positive solutions are obtained by monotone iterative schemes. Our results show that a cross-diffusive WNv model possesses at least one coexistence solution if the basic reproduction number is greater than one and the cross-diffusion rates are small enough, while if the basic reproduction number is less than or equal to one, the model has no positive solution. To illustrate the impact of cross-diffusion and environmental heterogeneity on the transmission of WNv, some numerical simulations are given.

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