American Institute of Mathematical Sciences

Advanced
October  2019, 24(10): 5633-5671. doi: 10.3934/dcdsb.2019076

Spatial dynamics of a Lotka-Volterra model with a shifting habitat

Yueding Yuan 1,2, , Yang Wang 3, and  Xingfu Zou 1,3,,

1. 

School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China
2. 

School of Mathematics and Computer Sciences, Yichun University, Yichun 336000, Jiangxi, China
3. 

Department of Applied Mathematics, Western University, London, Ontario, N6A 5B7, Canada

* Corresponding author

Received  October 2018 Published  April 2019

Fund Project: Research was partially supported by National Natural Science Foundation of China (No. 11561068) and China Postdoctoral Science Foundation (2016M592442) and NSERC of Canada (No. RGPIN-2016-04665).

In this paper, we study a Lotka-Volterra competition-diffusion model that describes the growth, spread and competition of two species in a shifting habitat. Our results show that (Ⅰ) if the competition between the two species are either mutually strong or mutually weak against each other, the spatial dynamics mainly depend on environment worsening speed c and the spreading speed of each species in the absence of the other in the best possible environment; (Ⅱ) if one species is a strong competitor and the other is a weak competitor, then the interplay of the species' competing strengths and the spreading speeds also has an effect on the spatial dynamics. Particularly, we find that a strong but slower competitor can co-persist with a weak but faster competitor, provided that the environment worsening speed is not too fast.

Keywords: Climate change, competition, reaction-diffusion, Lotka-Volterra model, coexistence, shifting habitat, spreading speed.
Mathematics Subject Classification: Primary: 92D40, 92D25; Secondary: 35K57, 93C10.
Citation: Yueding Yuan, Yang Wang, Xingfu Zou. Spatial dynamics of a Lotka-Volterra model with a shifting habitat. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5633-5671. doi: 10.3934/dcdsb.2019076
References:
[1]

D. G. Aronson and H. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve impulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Math., Springer-Verlag, Berlin, 446 (1975), 5–49. doi: 10.1007/BFb0070595.  Google Scholar

[2]

H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.  doi: 10.1007/s11538-008-9367-5.  Google Scholar

[3]

H. Berestycki and J. Fang, Forced waves of the Fisher-KPP equation in a shifting environment, J. Differential Equations, 264 (2018), 2157-2183.  doi: 10.1016/j.jde.2017.10.016.  Google Scholar

[4]

B. A. Bradley, D. S. Wilcove and M. Oppenheimer, Climate change increases risk of plant invasion in the eastern United States, Biological Invasions, 12 (2010), 1855-1872.  doi: 10.1007/s10530-009-9597-y.  Google Scholar

[5]

R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338.  doi: 10.1007/BF00167155.  Google Scholar

[6]

R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145.  doi: 10.1007/s002850050122.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, J. Wiley, Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[8]

C. Carrére, Spreading speeds for a two-species competition-diffusion system, J. Differential Equations, 264 (2018), 2133-2156.  doi: 10.1016/j.jde.2017.10.017.  Google Scholar

[9]

J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.  Google Scholar

[10]

J. Fang, Y. Lou and J. Wu, Can pathogen spread keep pace with its host invasion?, SIAM J. Appl. Math., 76 (2016), 1633-1657.  doi: 10.1137/15M1029564.  Google Scholar

[11]

A. Friedman, A strong maximum principle for weakly subparabolic functions, Pacific J. Math., 11 (1961), 175-184.  doi: 10.2140/pjm.1961.11.175.  Google Scholar

[12]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N. J., 1964.  Google Scholar

[13]

L. Girardin and K.-Y. Lam, Invasion of an empty habitat by two competitors: spreading properties of monostable two-species competition–diffusion systems, preprint, arXiv: 1803.00454. Google Scholar

[14]

P. Gonzalez, R. P. Neilson, J. M. Lenihan and R. J. Drapek, Global patterns in the vulnerability of ecosystems to vegetation shifts due to climate change, Glob. Ecol. Biogeogr., 19 (2010), 755-768.  doi: 10.1111/j.1466-8238.2010.00558.x.  Google Scholar

[15]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Popul. Biol., 24 (1983), 244-251.  doi: 10.1016/0040-5809(83)90027-8.  Google Scholar

[16]

A. Hastings, Spatial heterogeneity and ecological models, Ecology, 71 (1990), 426-428.  doi: 10.2307/1940296.  Google Scholar

[17]

X. He and W. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032.  Google Scholar

[18]

X. He and W. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system II: The general case, J. Differential Equations, 254 (2013), 4088-4108.  doi: 10.1016/j.jde.2013.02.009.  Google Scholar

[19]

X. He and W. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity I, Comm. Pure Appl. Math., 69 (2016), 981-1014.  doi: 10.1002/cpa.21596.  Google Scholar

[20]

C. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations, 259 (2015), 1967-1989.  doi: 10.1016/j.jde.2015.03.025.  Google Scholar

[21]

H. Hu and X. Zou, Existence of an extinction wave in the Fisher equation with a shifting habitat, Proc. Amer. Math. Soc., 145 (2017), 4763-4771.  doi: 10.1090/proc/13687.  Google Scholar

[22]

J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271 (2002), 455-466.  doi: 10.1016/S0022-247X(02)00135-X.  Google Scholar

[23]

V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.  Google Scholar

[24]

V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, J. Differential Equations, 211 (2005), 135-161.  doi: 10.1016/j.jde.2004.06.003.  Google Scholar

[25]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[26]

S. A. Levin, Dispersion and population interactions, Amer. Natur., 108 (1974), 207-228.  doi: 10.1086/282900.  Google Scholar

[27]

B. Li, S. Bewick, J. Shang and W. F. Fagan, Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math., 74 (2014), 1397-1417.  doi: 10.1137/130938463.  Google Scholar

[28]

G. Lin and W.-T. Li, Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, European J. Appl. Math., 23 (2012), 669-689.  doi: 10.1017/S0956792512000198.  Google Scholar

[29]

S. R. Loarie, P. B. Duffy, H. Hamilton, G. P. Asner, C. B. Field and D. D. Ackerly, The velocity of climate change, Nature, 462 (2009), 1052-1055.  doi: 10.1038/nature08649.  Google Scholar

[30] T. E. Lovejoy and L. Hannah, Climate Change and Biodiversity, Yale University Press, New Haven, 2005.  doi: 10.2307/j.ctv8jnzw1.  Google Scholar
[31]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[32]

Y. Lou, Some reaction diffusion models in spatial ecology (in Chinese), Sci. Sin. Math., 45 (2015), 1619-1634. Google Scholar

[33]

J. P. McCarty, Ecological consequences of recent climate change, Conserv. Biol., 15 (2001), 320-331.  doi: 10.1046/j.1523-1739.2001.015002320.x.  Google Scholar

[34]

A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, second ed., Interdisciplinary Applied Mathematics, vol. 14, Springer, New York, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[35]

S. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theor. Popul. Biol., 21 (1982), 92-113.  doi: 10.1016/0040-5809(82)90008-9.  Google Scholar

[36]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.  doi: 10.1006/jmaa.1996.0111.  Google Scholar

[37]

C. Parmesan, N. Ryrholm, C. Stefanescu, J. K. Hill, C. D. Thomas, H. Descimon, B. Huntley, L. Kaila, J. Kullberg, T. Tammaru, W. J. Tennent, J. A Thomas and M. Warren, Poleward shifts in geographical ranges of butterfly species associated with regional warming, Nature, 399 (1999), 579-583.  doi: 10.1038/21181.  Google Scholar

[38]

C. Parmesan, Ecological and evolutionary responses to recent climate change, Ann. Rev. Ecology Evolution Systematics, 37 (2006), 637-669.  doi: 10.1146/annurev.ecolsys.37.091305.110100.  Google Scholar

[39]

C. L. Parr, E. F. Gray and W. J. Bond, Cascading biodiversity and functional consequences of a global change-induced biome switch, Divers. Distrib., 18 (2012), 493-503.  doi: 10.1111/j.1472-4642.2012.00882.x.  Google Scholar

[40]

A. B. Potapov and M. A. Lewis, Climate and competition: The effect of moving range boundaries on habitat invisibility, Bull. Math. Biol., 66 (2004), 975-1008.  doi: 10.1016/j.bulm.2003.10.010.  Google Scholar

[41]

B. Sandel, L. Arge, B. Dalsgaard, R. G. Davies, K. J. Gaston, W. J. Sutherland and J.-C. Svenning, The influence of late quaternary climate-change velocity on species endemism, Science, 334 (2011), 660-664.   Google Scholar

[42]

S. Scheiter and S. I. Higgins, Impacts of climate change on the vegetation of Africa: An adaptive dynamic vegetation modelling approach, Glob. Change Biol., 15 (2009), 2224-2246.  doi: 10.1111/j.1365-2486.2008.01838.x.  Google Scholar

[43]

M. M. Tang and P. C. Fife, Propagation fronts in competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77.  doi: 10.1007/BF00283257.  Google Scholar

[44]

J. H. van Vuuren, The existence of traveling plane waves in a general class of competition- diffusion systems, IMA J. Appl. Math., 55 (1995), 135-148.  doi: 10.1093/imamat/55.2.135.  Google Scholar

[45]

G. R. Walther, E. Post, P. Convey, A. Menzel, C. Parmesan, T. J. C. Beebee, J. M. Fromentin, O. Hoegh-Guldberg and F. Bairlein, Ecological responses to recent climate change, Nature, 416 (2002), 389-395.  doi: 10.1038/416389a.  Google Scholar

[46]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.  doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

[47]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[48] D. Xia, Z. Wu, S. Yan and W. Shu, Theory of Real Variable Function and Functional Analysis, second ed, Higher Education Press, Beijing, 2010.   Google Scholar
[49]

Z. Zhang, W. Wang and J. Yang, Persistence versus extinction for two competing species under climate change, Nonlinear Analysis: Modelling and Control, 22 (2017), 285-302.  doi: 10.15388/NA.2017.3.1.  Google Scholar

[50]

Y. Zhou and M. Kot, Discrete-time growth-dispersal models with shifting species ranges, Theor. Ecol., 4 (2011), 13-25.  doi: 10.1007/s12080-010-0071-3.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve impulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Math., Springer-Verlag, Berlin, 446 (1975), 5–49. doi: 10.1007/BFb0070595.  Google Scholar

[2]

H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.  doi: 10.1007/s11538-008-9367-5.  Google Scholar

[3]

H. Berestycki and J. Fang, Forced waves of the Fisher-KPP equation in a shifting environment, J. Differential Equations, 264 (2018), 2157-2183.  doi: 10.1016/j.jde.2017.10.016.  Google Scholar

[4]

B. A. Bradley, D. S. Wilcove and M. Oppenheimer, Climate change increases risk of plant invasion in the eastern United States, Biological Invasions, 12 (2010), 1855-1872.  doi: 10.1007/s10530-009-9597-y.  Google Scholar

[5]

R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338.  doi: 10.1007/BF00167155.  Google Scholar

[6]

R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145.  doi: 10.1007/s002850050122.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, J. Wiley, Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[8]

C. Carrére, Spreading speeds for a two-species competition-diffusion system, J. Differential Equations, 264 (2018), 2133-2156.  doi: 10.1016/j.jde.2017.10.017.  Google Scholar

[9]

J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.  Google Scholar

[10]

J. Fang, Y. Lou and J. Wu, Can pathogen spread keep pace with its host invasion?, SIAM J. Appl. Math., 76 (2016), 1633-1657.  doi: 10.1137/15M1029564.  Google Scholar

[11]

A. Friedman, A strong maximum principle for weakly subparabolic functions, Pacific J. Math., 11 (1961), 175-184.  doi: 10.2140/pjm.1961.11.175.  Google Scholar

[12]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N. J., 1964.  Google Scholar

[13]

L. Girardin and K.-Y. Lam, Invasion of an empty habitat by two competitors: spreading properties of monostable two-species competition–diffusion systems, preprint, arXiv: 1803.00454. Google Scholar

[14]

P. Gonzalez, R. P. Neilson, J. M. Lenihan and R. J. Drapek, Global patterns in the vulnerability of ecosystems to vegetation shifts due to climate change, Glob. Ecol. Biogeogr., 19 (2010), 755-768.  doi: 10.1111/j.1466-8238.2010.00558.x.  Google Scholar

[15]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Popul. Biol., 24 (1983), 244-251.  doi: 10.1016/0040-5809(83)90027-8.  Google Scholar

[16]

A. Hastings, Spatial heterogeneity and ecological models, Ecology, 71 (1990), 426-428.  doi: 10.2307/1940296.  Google Scholar

[17]

X. He and W. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032.  Google Scholar

[18]

X. He and W. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system II: The general case, J. Differential Equations, 254 (2013), 4088-4108.  doi: 10.1016/j.jde.2013.02.009.  Google Scholar

[19]

X. He and W. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity I, Comm. Pure Appl. Math., 69 (2016), 981-1014.  doi: 10.1002/cpa.21596.  Google Scholar

[20]

C. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations, 259 (2015), 1967-1989.  doi: 10.1016/j.jde.2015.03.025.  Google Scholar

[21]

H. Hu and X. Zou, Existence of an extinction wave in the Fisher equation with a shifting habitat, Proc. Amer. Math. Soc., 145 (2017), 4763-4771.  doi: 10.1090/proc/13687.  Google Scholar

[22]

J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271 (2002), 455-466.  doi: 10.1016/S0022-247X(02)00135-X.  Google Scholar

[23]

V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.  Google Scholar

[24]

V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, J. Differential Equations, 211 (2005), 135-161.  doi: 10.1016/j.jde.2004.06.003.  Google Scholar

[25]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[26]

S. A. Levin, Dispersion and population interactions, Amer. Natur., 108 (1974), 207-228.  doi: 10.1086/282900.  Google Scholar

[27]

B. Li, S. Bewick, J. Shang and W. F. Fagan, Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math., 74 (2014), 1397-1417.  doi: 10.1137/130938463.  Google Scholar

[28]

G. Lin and W.-T. Li, Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, European J. Appl. Math., 23 (2012), 669-689.  doi: 10.1017/S0956792512000198.  Google Scholar

[29]

S. R. Loarie, P. B. Duffy, H. Hamilton, G. P. Asner, C. B. Field and D. D. Ackerly, The velocity of climate change, Nature, 462 (2009), 1052-1055.  doi: 10.1038/nature08649.  Google Scholar

[30] T. E. Lovejoy and L. Hannah, Climate Change and Biodiversity, Yale University Press, New Haven, 2005.  doi: 10.2307/j.ctv8jnzw1.  Google Scholar
[31]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[32]

Y. Lou, Some reaction diffusion models in spatial ecology (in Chinese), Sci. Sin. Math., 45 (2015), 1619-1634. Google Scholar

[33]

J. P. McCarty, Ecological consequences of recent climate change, Conserv. Biol., 15 (2001), 320-331.  doi: 10.1046/j.1523-1739.2001.015002320.x.  Google Scholar

[34]

A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, second ed., Interdisciplinary Applied Mathematics, vol. 14, Springer, New York, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[35]

S. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theor. Popul. Biol., 21 (1982), 92-113.  doi: 10.1016/0040-5809(82)90008-9.  Google Scholar

[36]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.  doi: 10.1006/jmaa.1996.0111.  Google Scholar

[37]

C. Parmesan, N. Ryrholm, C. Stefanescu, J. K. Hill, C. D. Thomas, H. Descimon, B. Huntley, L. Kaila, J. Kullberg, T. Tammaru, W. J. Tennent, J. A Thomas and M. Warren, Poleward shifts in geographical ranges of butterfly species associated with regional warming, Nature, 399 (1999), 579-583.  doi: 10.1038/21181.  Google Scholar

[38]

C. Parmesan, Ecological and evolutionary responses to recent climate change, Ann. Rev. Ecology Evolution Systematics, 37 (2006), 637-669.  doi: 10.1146/annurev.ecolsys.37.091305.110100.  Google Scholar

[39]

C. L. Parr, E. F. Gray and W. J. Bond, Cascading biodiversity and functional consequences of a global change-induced biome switch, Divers. Distrib., 18 (2012), 493-503.  doi: 10.1111/j.1472-4642.2012.00882.x.  Google Scholar

[40]

A. B. Potapov and M. A. Lewis, Climate and competition: The effect of moving range boundaries on habitat invisibility, Bull. Math. Biol., 66 (2004), 975-1008.  doi: 10.1016/j.bulm.2003.10.010.  Google Scholar

[41]

B. Sandel, L. Arge, B. Dalsgaard, R. G. Davies, K. J. Gaston, W. J. Sutherland and J.-C. Svenning, The influence of late quaternary climate-change velocity on species endemism, Science, 334 (2011), 660-664.   Google Scholar

[42]

S. Scheiter and S. I. Higgins, Impacts of climate change on the vegetation of Africa: An adaptive dynamic vegetation modelling approach, Glob. Change Biol., 15 (2009), 2224-2246.  doi: 10.1111/j.1365-2486.2008.01838.x.  Google Scholar

[43]

M. M. Tang and P. C. Fife, Propagation fronts in competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77.  doi: 10.1007/BF00283257.  Google Scholar

[44]

J. H. van Vuuren, The existence of traveling plane waves in a general class of competition- diffusion systems, IMA J. Appl. Math., 55 (1995), 135-148.  doi: 10.1093/imamat/55.2.135.  Google Scholar

[45]

G. R. Walther, E. Post, P. Convey, A. Menzel, C. Parmesan, T. J. C. Beebee, J. M. Fromentin, O. Hoegh-Guldberg and F. Bairlein, Ecological responses to recent climate change, Nature, 416 (2002), 389-395.  doi: 10.1038/416389a.  Google Scholar

[46]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.  doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

[47]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[48] D. Xia, Z. Wu, S. Yan and W. Shu, Theory of Real Variable Function and Functional Analysis, second ed, Higher Education Press, Beijing, 2010.   Google Scholar
[49]

Z. Zhang, W. Wang and J. Yang, Persistence versus extinction for two competing species under climate change, Nonlinear Analysis: Modelling and Control, 22 (2017), 285-302.  doi: 10.15388/NA.2017.3.1.  Google Scholar

[50]

Y. Zhou and M. Kot, Discrete-time growth-dispersal models with shifting species ranges, Theor. Ecol., 4 (2011), 13-25.  doi: 10.1007/s12080-010-0071-3.  Google Scholar

Figure 4.  Numerical simulations on (6) with (113)- (116): for $ c = 1.8 $ we have $ c> \hat{c}^*(\infty) $ but $ c <c^*(\infty) $, the two species can still co-persist by spreading to the right with the respective asymptotic speeds $ c^*_1(\infty) = 2 $ and $ c^*_2(\infty) = 2.2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Numerical simulations on (6) with (113)- (116) with $ a_2 = 0.36 $ replaced by $ a_2 = 3 $ and the environment worsening rate is very small $ (c = 1.8) $ in the sense that $ c<c_1^*(\infty) $, the two species can still be co-persist by spreading to the right with the respective asymptotic speeds $ c^*_1(\infty) = 2 $ and $ c^*_2(\infty) = 2.2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 1.  Numerical simulations on (6) with (113)- (116): when the environment worsening rate is too large ($ c = 2.21 >c^*_i(\infty) > \hat{c}^*_i(\infty) $ for $ i = 1,2 $), both species go to extinct in the habitat
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Numerical simulations on (6) with (113)- (116): when the environment worsening rate is neutral in the sense that $ c = 2.05 \in (c^*_1(\infty), c^*_2(\infty)) $, species 1 becomes extinct in the habitat and species 2 persist by spreading to the right with the asymptotic speed $ c^*_2(\infty) = 2.2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Numerical simulations on (6) with (113)- (116): when the environment worsening rate is very small ($ c = 1.65 $) in the sense that $ c < \hat{c}^*(\infty) $, both species co-persist by spreading to the right with the respective asymptotic speeds $ c^*_1(\infty) = 2 $ and $ c^*_2(\infty) = 2.2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Numerical simulations on (6) with (113)- (116) with $ a_1 = 0.19 $ and $ a_2 = 0.36 $ replaced by $ a_1 = 3 $ and $ a_2 = 2 $ respectively. The environment worsening rate is very small (c = 1.8) in the sense that $ c<c_1^*(\infty) $, species 1 becomes extinct in the habitat and species 2 persist by spreading to the right with the asymptotic speed $ c^*_2(\infty) = 2.2 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197
[2]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[3]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[4]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005
[5]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[6]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004
[7]

Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021067
[8]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973
[9]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[10]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[11]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[12]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[13]

Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161
[14]

Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021039
[15]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190
[16]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101
[17]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427
[18]

Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014
[19]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53
[20]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (244)
  • HTML views (470)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences