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.

[2]

O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.

[3]

J. Fang and X. -Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.

[4]

P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.

[5]

B. Gilding and R. Kersner, Traveling Waves in Nonlinear Diffusion Convection Reaction, Basel: Birkhäuser Verlag, 2004.

[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.

[7]

J. K. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977.

[8]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.

[9]

M. KotM. A. Lewis and P. Van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.

[10] J. Kuijt, The Biology of Parasitic Flowering Plants, University of California Press, Berkeley, 1969.
[11]

J. D. Murray, Mathematical Biology: Ⅰ and Ⅱ, Spriner-Verlag, New York, 2002.

[12]

S. X. PanW. T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.

[13]

J. Radciliffe and L. Rass, Spatial deterministic epidemics, Mathematical Surveys and Monographs. Amer. Math. Soc. Providence, RI, 102, 2003.

[14]

W. X. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060.

[15]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Uinversity Press, 1997.

[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.

[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.

[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).

[19]

C. C. WangR. S. LiuJ. P. Shi and D. C. Martinez, Spatiotemporal mutualistic model of mistletoes and birds, J. Math. Biol., 68 (2014), 1479-1520.

[20]

C. C. WangR. S. LiuJ. 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.

[21]

D. M. Watson, Mistletoe-a keystone resource in forests and woodlands worldwide, Annual Review of Ecology and Systematics, 32 (2001), 219-249.

[22]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.

[23]

P. X. WengH. 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.

[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.

[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.

[26]

S. L. WuW. 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.

[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.

[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.

[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.

[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.

[31]

G. B. ZhangW. 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.

[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.

[33]

X. -Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128.

show all references

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.

[2]

O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.

[3]

J. Fang and X. -Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.

[4]

P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.

[5]

B. Gilding and R. Kersner, Traveling Waves in Nonlinear Diffusion Convection Reaction, Basel: Birkhäuser Verlag, 2004.

[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.

[7]

J. K. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977.

[8]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.

[9]

M. KotM. A. Lewis and P. Van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.

[10] J. Kuijt, The Biology of Parasitic Flowering Plants, University of California Press, Berkeley, 1969.
[11]

J. D. Murray, Mathematical Biology: Ⅰ and Ⅱ, Spriner-Verlag, New York, 2002.

[12]

S. X. PanW. T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.

[13]

J. Radciliffe and L. Rass, Spatial deterministic epidemics, Mathematical Surveys and Monographs. Amer. Math. Soc. Providence, RI, 102, 2003.

[14]

W. X. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060.

[15]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Uinversity Press, 1997.

[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.

[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.

[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).

[19]

C. C. WangR. S. LiuJ. P. Shi and D. C. Martinez, Spatiotemporal mutualistic model of mistletoes and birds, J. Math. Biol., 68 (2014), 1479-1520.

[20]

C. C. WangR. S. LiuJ. 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.

[21]

D. M. Watson, Mistletoe-a keystone resource in forests and woodlands worldwide, Annual Review of Ecology and Systematics, 32 (2001), 219-249.

[22]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.

[23]

P. X. WengH. 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.

[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.

[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.

[26]

S. L. WuW. 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.

[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.

[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.

[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.

[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.

[31]

G. B. ZhangW. 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.

[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.

[33]

X. -Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128.

Figure 1.  The solution of system (5) with parameters in Table 1 and initial value condition: $u_{10}(t, x)=0.1$, $u_{20}(t, x)=0.1$, $t\in[-1,0]$
Figure 2.  (1) $\lambda_{1}(\nu)=-d_{m}+\frac{\alpha e^{-d_{i}\tau}}{\omega}\bar{k}(\nu)e^{-\lambda_{1}(\nu)\tau}$;
(2) $\lambda_{2}(\nu)=D\bar{J}(\nu)-1-D$
Figure 3.  (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)$\mathbf{\Phi}_{2}(\nu)=\frac{D\bar{J}(\nu)-1-D}{\nu}$
Figure 4.  (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) $\mathbf{ \Phi}_{1}(\nu)=-\frac{d_{m}}{\nu}+ \frac{\alpha e^{-d_{i}\tau}}{\omega }\frac{\bar{k}(\nu)}{\nu}e^{-\lambda_1(\nu)\tau}$
Figure 5.  (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) $\mathbf{\Phi}_{2}(\nu)=\frac{D\bar{J}(\nu)-1-D}{\nu}$
Figure 6.  The traveling wave solution found with parameters in Table 1 and initial value condition: $u_{10}(t, x)=0.001$, $u_{20}(t, x)=0.001$, $t\in[-1,0]$
Table 1.  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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