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Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion
The spreading fronts in a mutualistic model with advection
1. | School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China |
2. | School of Mathematical Science, Yangzhou University, Yangzhou 225002 |
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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