The spreading fronts in a mutualistic model with advection

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, preprint, arXiv:1301.5373, (2013).
[10]	Y. H. Du and L. Ma, Logistic type equations on $\mathbb{R}^N2$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	H. L. Smith, Monotone Dynamical Systems, American Math. Soc., Providence, 1995.
[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.
[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.
[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.
[28]	J. X. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, preprint, arXiv:1301.5373, (2013).
[10]	Y. H. Du and L. Ma, Logistic type equations on $\mathbb{R}^N2$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	H. L. Smith, Monotone Dynamical Systems, American Math. Soc., Providence, 1995.
[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.
[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.
[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.
[28]	J. X. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.
[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.
[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.
[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.

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