# American Institute of Mathematical Sciences

August  2017, 22(6): 2207-2231. doi: 10.3934/dcdsb.2017093

## Bility and traveling wavefronts for a convolution model of mistletoes and birds with nonlocal diffusion

 School of Mathematics, South China Normal University, Guangzhou, Guangdong 510631, China

Received  December 2014 Revised  June 2017 Published  March 2017

Fund Project: Supported by the NSF of China (11171120) and the Natural Science Foundation of Guangdong Province (2016A030313426)

A convolution model of mistletoes and birds with nonlocal diffusion is considered in this paper. We first consider the stability of the constant steady states of the model by linearized method, and then the existence of traveling solutions. The main aim of this article is to challenge the hardness lying in the construction of upper-lowers for wave profile system. With the help of an additional condition, we at last obtain a pair of upper-lower solutions. A constant $c_{*}>0$ is obtained such that traveling wavefronts exist for $c\geq c_{*}$. Amongst the construction, we take advantage of the relation between two components of principle eigenvector for the linearized system to control the two components of upper solution. The method seems novel. Some simulations and discussions are given to illustrate the applications of our main results and the effect of parameters on $c_{*}$. A comparison for $c_{*}$ is also given with two different kernel functions.

Citation: Huimin Liang, Peixuan Weng, Yanling Tian. Bility and traveling wavefronts for a convolution model of mistletoes and birds with nonlocal diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2207-2231. doi: 10.3934/dcdsb.2017093
##### References:
 [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 165, 2010.Google Scholar [2] O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130. Google Scholar [3] J. Fang and X. -Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704. Google Scholar [4] P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.Google Scholar [5] B. Gilding and R. Kersner, Traveling Waves in Nonlinear Diffusion Convection Reaction, Basel: Birkhäuser Verlag, 2004.Google Scholar [6] S. A. Gourley and J. H. Wu, Delayed nonlocal diffusive systems in biological invasion and disease spread, Nonlinear Dynamic and Evolution Equations, Fields Inst. Commun. Amer. Math. Soc. Providence, RI, 48 (2006), 137-200. Google Scholar [7] J. K. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977.Google Scholar [8] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. Google Scholar [9] M. Kot, M. A. Lewis and P. Van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042. Google Scholar [10] J. Kuijt, The Biology of Parasitic Flowering Plants, University of California Press, Berkeley, 1969. Google Scholar [11] J. D. Murray, Mathematical Biology: Ⅰ and Ⅱ, Spriner-Verlag, New York, 2002.Google Scholar [12] S. X. Pan, W. T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392. Google Scholar [13] J. Radciliffe and L. Rass, Spatial deterministic epidemics, Mathematical Surveys and Monographs. Amer. Math. Soc. Providence, RI, 102, 2003.Google Scholar [14] W. X. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060. Google Scholar [15] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Uinversity Press, 1997.Google Scholar [16] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 41,1995.Google Scholar [17] H. R. Thieme and X. -Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed Reaction-Diffusion models, J. Differential Equations, 195 (2003), 430-470. Google Scholar [18] A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Amer. Math. Soc. Providence, RI, 140, (1994).Google Scholar [19] C. C. Wang, R. S. Liu, J. P. Shi and D. C. Martinez, Spatiotemporal mutualistic model of mistletoes and birds, J. Math. Biol., 68 (2014), 1479-1520. Google Scholar [20] C. C. Wang, R. S. Liu, J. P. Shi and D. C. Martinez, Traveling waves of a mutualistic model of mistletoes and birds, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 1743-1765. Google Scholar [21] D. M. Watson, Mistletoe-a keystone resource in forests and woodlands worldwide, Annual Review of Ecology and Systematics, 32 (2001), 219-249. Google Scholar [22] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. Google Scholar [23] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. Google Scholar [24] P. X. Weng and Z. T. Xu, Survey on progress for asymptotic speed of propagation and traveling wave solutions of some types of evolution equations (in Chinese), Advances in Mathematics (China), 39 (2010), 1-22. Google Scholar [25] J. H. Wu and X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delays, J. Dynam. Differential Equations, 13 (2001), 651-687. Google Scholar [26] S. L. Wu, W. T. Li and S. Y. Liu, Asymptotic stability of traveling wave fronts in nonlocal reaction-diffusion equations with delay, J. Math. Anal. Appl., 360 (2009), 439-458. Google Scholar [27] S. L. Wu and S. Y. Liu, Asymptotic speed of spread and traveling fronts for a nonlocal reaction-diffusion model with distributed delay, Appl. Math. Model., 33 (2009), 2757-2765. Google Scholar [28] Z. Q. Xu and P. X. Weng, Traveling waves in a convolution model with infinite distributed delay and non-monotonicity, Nonlinear Anal. Real World Appl., 12 (2011), 633-647. Google Scholar [29] Z. X. Yu and R. Yuan, Traveling waves of a nonlocal dispersal delayed age-structured population model, Japan J. Indust. Appl. Math., 30 (2013), 165-184. Google Scholar [30] X. J. Yu, P. X. Weng and Y. H. Huang, Traveling wavefronts of competing pioneer and climax model with nonlocal diffusion, Abstr. Appl. Anal. , (2013), Art. ID 725495, 12 pp.Google Scholar [31] G. B. Zhang, W. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling, 49 (2009), 1021-1029. Google Scholar [32] X. -Q. Zhao, Spatial dynamics of some evolution system in biology, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific Publishing Co. Pte. Ltd. Singapore, (2009), 332-363.Google Scholar [33] X. -Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128. Google Scholar

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##### References:
 [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 165, 2010.Google Scholar [2] O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130. Google Scholar [3] J. Fang and X. -Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704. Google Scholar [4] P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.Google Scholar [5] B. Gilding and R. Kersner, Traveling Waves in Nonlinear Diffusion Convection Reaction, Basel: Birkhäuser Verlag, 2004.Google Scholar [6] S. A. Gourley and J. H. Wu, Delayed nonlocal diffusive systems in biological invasion and disease spread, Nonlinear Dynamic and Evolution Equations, Fields Inst. Commun. Amer. Math. Soc. Providence, RI, 48 (2006), 137-200. Google Scholar [7] J. K. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977.Google Scholar [8] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. Google Scholar [9] M. Kot, M. A. Lewis and P. Van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042. Google Scholar [10] J. Kuijt, The Biology of Parasitic Flowering Plants, University of California Press, Berkeley, 1969. Google Scholar [11] J. D. Murray, Mathematical Biology: Ⅰ and Ⅱ, Spriner-Verlag, New York, 2002.Google Scholar [12] S. X. Pan, W. T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392. Google Scholar [13] J. Radciliffe and L. Rass, Spatial deterministic epidemics, Mathematical Surveys and Monographs. Amer. Math. Soc. Providence, RI, 102, 2003.Google Scholar [14] W. X. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060. Google Scholar [15] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Uinversity Press, 1997.Google Scholar [16] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 41,1995.Google Scholar [17] H. R. Thieme and X. -Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed Reaction-Diffusion models, J. Differential Equations, 195 (2003), 430-470. Google Scholar [18] A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Amer. Math. Soc. Providence, RI, 140, (1994).Google Scholar [19] C. C. Wang, R. S. Liu, J. P. Shi and D. C. Martinez, Spatiotemporal mutualistic model of mistletoes and birds, J. Math. Biol., 68 (2014), 1479-1520. Google Scholar [20] C. C. Wang, R. S. Liu, J. P. Shi and D. C. Martinez, Traveling waves of a mutualistic model of mistletoes and birds, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 1743-1765. Google Scholar [21] D. M. Watson, Mistletoe-a keystone resource in forests and woodlands worldwide, Annual Review of Ecology and Systematics, 32 (2001), 219-249. Google Scholar [22] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. Google Scholar [23] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. Google Scholar [24] P. X. Weng and Z. T. Xu, Survey on progress for asymptotic speed of propagation and traveling wave solutions of some types of evolution equations (in Chinese), Advances in Mathematics (China), 39 (2010), 1-22. Google Scholar [25] J. H. Wu and X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delays, J. Dynam. Differential Equations, 13 (2001), 651-687. Google Scholar [26] S. L. Wu, W. T. Li and S. Y. Liu, Asymptotic stability of traveling wave fronts in nonlocal reaction-diffusion equations with delay, J. Math. Anal. Appl., 360 (2009), 439-458. Google Scholar [27] S. L. Wu and S. Y. Liu, Asymptotic speed of spread and traveling fronts for a nonlocal reaction-diffusion model with distributed delay, Appl. Math. Model., 33 (2009), 2757-2765. Google Scholar [28] Z. Q. Xu and P. X. Weng, Traveling waves in a convolution model with infinite distributed delay and non-monotonicity, Nonlinear Anal. Real World Appl., 12 (2011), 633-647. Google Scholar [29] Z. X. Yu and R. Yuan, Traveling waves of a nonlocal dispersal delayed age-structured population model, Japan J. Indust. Appl. Math., 30 (2013), 165-184. Google Scholar [30] X. J. Yu, P. X. Weng and Y. H. Huang, Traveling wavefronts of competing pioneer and climax model with nonlocal diffusion, Abstr. Appl. Anal. , (2013), Art. ID 725495, 12 pp.Google Scholar [31] G. B. Zhang, W. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling, 49 (2009), 1021-1029. Google Scholar [32] X. -Q. Zhao, Spatial dynamics of some evolution system in biology, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific Publishing Co. Pte. Ltd. Singapore, (2009), 332-363.Google Scholar [33] X. -Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128. Google Scholar
The solution of system (5) with parameters in Table 1 and initial value condition: $u_{10}(t, x)=0.1$, $u_{20}(t, x)=0.1$, $t\in[-1,0]$
(1) $\lambda_{1}(\nu)=-d_{m}+\frac{\alpha e^{-d_{i}\tau}}{\omega}\bar{k}(\nu)e^{-\lambda_{1}(\nu)\tau}$;
(2) $\lambda_{2}(\nu)=D\bar{J}(\nu)-1-D$
(1) $\mathbf{\Phi}_{1}(\nu)=-\frac{d_{m}}{\nu}+\frac{\alpha e^{-d_{i}\tau}}{\omega}\frac{\bar{k}(\nu)}{\nu}e^{-\nu\mathbf{\Phi}_{1}(\nu)\tau}$;
(2)$\mathbf{\Phi}_{2}(\nu)=\frac{D\bar{J}(\nu)-1-D}{\nu}$
(1) $\lambda_1(\nu)=-d_{m}+ \frac{\alpha e^{-d_{i}\tau}}{\omega }\bar{k}(\nu)e^{-\lambda_1(\nu)\tau}$;
(2) $\hat{\lambda}(\nu)=d\bar{k}(\nu)+D\bar{J}(\nu)-1-D$;
(3) $\mathbf{ \Phi}_{1}(\nu)=-\frac{d_{m}}{\nu}+ \frac{\alpha e^{-d_{i}\tau}}{\omega }\frac{\bar{k}(\nu)}{\nu}e^{-\lambda_1(\nu)\tau}$
(1) $\mathbf{\Phi}_{1}(\nu)=-\frac{d_{m}}{\nu}+\frac{\alpha e^{-d_{i}\tau}}{\omega\nu}e^{-\nu\mathbf{\Phi}_{1}(\nu)\tau}$;
(2) $\mathbf{\Phi}_{2}(\nu)=\frac{D\bar{J}(\nu)-1-D}{\nu}$
The traveling wave solution found with parameters in Table 1 and initial value condition: $u_{10}(t, x)=0.001$, $u_{20}(t, x)=0.001$, $t\in[-1,0]$
Parameter values for simulations
 $k(y)$ $J(y)$ $\bar{k}(\nu)$ $\bar{J}(\nu)$ $d_{m}$ $\omega$ $\alpha$ $d$ $d_{i}$ $\tau$ $D$ $\frac{1}{\sqrt{4\pi}}e^{-\frac{y^{2}}{4}}$ $\frac{1}{\sqrt{4\pi}}e^{-\frac{y^{2}}{4}}$ $e^{\nu^{2}}$ $e^{\nu^{2}}$ 0.1 1 0.7 0.3 0.3 1 0.5
 $k(y)$ $J(y)$ $\bar{k}(\nu)$ $\bar{J}(\nu)$ $d_{m}$ $\omega$ $\alpha$ $d$ $d_{i}$ $\tau$ $D$ $\frac{1}{\sqrt{4\pi}}e^{-\frac{y^{2}}{4}}$ $\frac{1}{\sqrt{4\pi}}e^{-\frac{y^{2}}{4}}$ $e^{\nu^{2}}$ $e^{\nu^{2}}$ 0.1 1 0.7 0.3 0.3 1 0.5
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