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

Abdelrazig K. Tarboush; Jing Ge; Zhigui Lin

doi:10.3934/mbe.2018068

Article Contents

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

This issue Previous Article State feedback impulsive control of computer worm and virus with saturated incidence Next Article Review of stability and stabilization for impulsive delayed systems

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

* Corresponding author: zglin68@hotmail.com (Z. G. Lin)
* Corresponding author: zglin68@hotmail.com (Z. G. Lin)

Received: June 04, 2018

Revised: August 12, 2018

Published: November 2018

The second author is supported by the NNSF of China (Grant No. 11701206) and the third author is supported by the NNSF of China (Grant No. 11771381).

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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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Mathematics Subject Classification: Primary: 35K57, 92B05; Secondary: 35J60.

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Figure 1. Phase diagrams of $I_b(x)$ and $I_m(x)$ showing the existence of a positive solution of (4) for small cross-diffusion ($\beta_1 = 0.001$ and $ \beta_2 = 0.002$)

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Figure 2. Phase diagrams of $I_b(x,t)$ and $I_m(x,t)$ shows that the solution of (3) exists and stabilizes to a positive steady-state for small cross-diffusion ($\beta_1 = 0.132$ and $ \beta_2 = 0.11$)

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Figure 3. Phase diagrams of $I_b(x)$ and $I_m(x)$ shows that the global solution of (3) does not exist for big cross-diffusion ($\beta_1 = 0.133$ and $ \beta_2 = 0.11$)

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