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September  2015, 20(7): 2089-2105. doi: 10.3934/dcdsb.2015.20.2089

The spreading fronts in a mutualistic model with advection

Mei Li 1, and  Zhigui Lin 2,

1. 

School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China
2. 

School of Mathematical Science, Yangzhou University, Yangzhou 225002

Received  September 2014 Revised  March 2015 Published  July 2015

This paper is concerned with a system of semilinear parabolic equations with two free boundaries, which describe the spreading fronts of the invasive species in a mutualistic ecological model. The advection term is introduced to model the behavior of the invasive species in one dimension space. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that for small advection, two free boundaries tend monotonically to finite limits or infinities at the same time, and a spreading-vanishing dichotomy holds, namely, either the expanding environment is limited and the invasive species dies out, or the invasive species spreads to all new environment and establishes itself in a long run. Moreover, some rough estimates of the spreading speed are also given when spreading happens.
Keywords: Reaction-diffusion systems, free boundary, mutualistic model, spreading and vanishing..
Mathematics Subject Classification: Primary: 35R35; Secondary: 35K6.
Citation: Mei Li, Zhigui Lin. The spreading fronts in a mutualistic model with advection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2089-2105. doi: 10.3934/dcdsb.2015.20.2089
References:
[1]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26.  Google Scholar

[2]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146. doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

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[4]

Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011.  Google Scholar

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Y. H. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142. doi: 10.1016/j.jfa.2013.07.016.  Google Scholar

[6]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.  Google Scholar

[7]

Y. H. Du and Z. G. Lin, Erratum: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996. doi: 10.1137/110822608.  Google Scholar

[8]

Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132. doi: 10.3934/dcdsb.2014.19.3105.  Google Scholar

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Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, preprint, arXiv:1301.5373, (2013). Google Scholar

[10]

Y. H. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289.  Google Scholar

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R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[12]

H. Gu, Z. G. Lin and B. D. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Amer. Math. Soc., 143 (2015), 1109-1117. doi: 10.1090/S0002-9939-2014-12214-3.  Google Scholar

[13]

H. Gu, Z. G. Lin and B. D. Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett., 37 (2014), 49-53. doi: 10.1016/j.aml.2014.05.015.  Google Scholar

[14]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0.  Google Scholar

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F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb R^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238.  Google Scholar

[16]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.  Google Scholar

[17]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Ètude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique}, Bull. Univ. Moscou Sér. Internat., A1 (1937), 1-26; English transl. in Dynamics of Curved Fronts, (ed. P. Pelcé), Academic Press, 1988, 105-130. Google Scholar

[18]

C. X. Lei, Z. G. Lin and Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166. doi: 10.1016/j.jde.2014.03.015.  Google Scholar

[19]

C. X. Lei, Z. G. Lin and H. Y. Wang, The free boundary problem describing information diffusion in online social networks, J. Differential Equations, 254 (2013), 1326-1341. doi: 10.1016/j.jde.2012.10.021.  Google Scholar

[20]

Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.  Google Scholar

[21]

R. M. May, Simple mathematical models with very complicated dynamics, The Theory of Chaotic Attractors, (2004), 85-93. doi: 10.1007/978-0-387-21830-4_7.  Google Scholar

[22]

J. Memmott, P. G. Craze, H. M. Harman, P. Syrett and S. V. Fowler, The effect of propagule size on the invasion of an alien insect, J. Anim. Ecol., 74 (2005), 50-62. doi: 10.1111/j.1365-2656.2004.00896.x.  Google Scholar

[23]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst. A, 33 (2013), 2007-2031. doi: 10.3934/dcds.2013.33.2007.  Google Scholar

[24]

H. L. Smith, Monotone Dynamical Systems, American Math. Soc., Providence, 1995.  Google Scholar

[25]

M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[26]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.  Google Scholar

[27]

H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6.  Google Scholar

[28]

J. X. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.  Google Scholar

[29]

J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Analysis: Real World Appl., 16 (2014), 250-263. doi: 10.1016/j.nonrwa.2013.10.003.  Google Scholar

[30]

P. Zhou and Z. G. Lin, Global existence and blowup of a nonlocal problem in space with free boundary, J. Funct. Anal., 262 (2012), 3409-3429. doi: 10.1016/j.jfa.2012.01.018.  Google Scholar

[31]

P. Zhou and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954. doi: 10.1016/j.jde.2013.12.008.  Google Scholar

show all references

References:
[1]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26.  Google Scholar

[2]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146. doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley and Sons Ltd., Chichester, UK, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011.  Google Scholar

[5]

Y. H. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142. doi: 10.1016/j.jfa.2013.07.016.  Google Scholar

[6]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.  Google Scholar

[7]

Y. H. Du and Z. G. Lin, Erratum: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996. doi: 10.1137/110822608.  Google Scholar

[8]

Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132. doi: 10.3934/dcdsb.2014.19.3105.  Google Scholar

[9]

Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, preprint, arXiv:1301.5373, (2013). Google Scholar

[10]

Y. H. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289.  Google Scholar

[11]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[12]

H. Gu, Z. G. Lin and B. D. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Amer. Math. Soc., 143 (2015), 1109-1117. doi: 10.1090/S0002-9939-2014-12214-3.  Google Scholar

[13]

H. Gu, Z. G. Lin and B. D. Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett., 37 (2014), 49-53. doi: 10.1016/j.aml.2014.05.015.  Google Scholar

[14]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0.  Google Scholar

[15]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb R^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238.  Google Scholar

[16]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.  Google Scholar

[17]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Ètude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique}, Bull. Univ. Moscou Sér. Internat., A1 (1937), 1-26; English transl. in Dynamics of Curved Fronts, (ed. P. Pelcé), Academic Press, 1988, 105-130. Google Scholar

[18]

C. X. Lei, Z. G. Lin and Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166. doi: 10.1016/j.jde.2014.03.015.  Google Scholar

[19]

C. X. Lei, Z. G. Lin and H. Y. Wang, The free boundary problem describing information diffusion in online social networks, J. Differential Equations, 254 (2013), 1326-1341. doi: 10.1016/j.jde.2012.10.021.  Google Scholar

[20]

Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.  Google Scholar

[21]

R. M. May, Simple mathematical models with very complicated dynamics, The Theory of Chaotic Attractors, (2004), 85-93. doi: 10.1007/978-0-387-21830-4_7.  Google Scholar

[22]

J. Memmott, P. G. Craze, H. M. Harman, P. Syrett and S. V. Fowler, The effect of propagule size on the invasion of an alien insect, J. Anim. Ecol., 74 (2005), 50-62. doi: 10.1111/j.1365-2656.2004.00896.x.  Google Scholar

[23]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst. A, 33 (2013), 2007-2031. doi: 10.3934/dcds.2013.33.2007.  Google Scholar

[24]

H. L. Smith, Monotone Dynamical Systems, American Math. Soc., Providence, 1995.  Google Scholar

[25]

M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[26]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.  Google Scholar

[27]

H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6.  Google Scholar

[28]

J. X. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.  Google Scholar

[29]

J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Analysis: Real World Appl., 16 (2014), 250-263. doi: 10.1016/j.nonrwa.2013.10.003.  Google Scholar

[30]

P. Zhou and Z. G. Lin, Global existence and blowup of a nonlocal problem in space with free boundary, J. Funct. Anal., 262 (2012), 3409-3429. doi: 10.1016/j.jfa.2012.01.018.  Google Scholar

[31]

P. Zhou and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954. doi: 10.1016/j.jde.2013.12.008.  Google Scholar

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