doi: 10.3934/dcdsb.2019076

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

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.

Citation: Yueding Yuan, Yang Wang, Xingfu Zou. Spatial dynamics of a Lotka-Volterra model with a shifting habitat. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019076
References:
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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. BerestyckiO. DiekmannC. 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

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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

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B. A. BradleyD. 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. DockeryV. HutsonK. 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. FangY. 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. GonzalezR. P. NeilsonJ. 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

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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

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

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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

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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. HutsonY. 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. HutsonY. 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. HutsonS. MartinezK. 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. LiS. BewickJ. 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. LoarieP. B. DuffyH. HamiltonG. P. AsnerC. 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.
[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. ParmesanN. RyrholmC. StefanescuJ. K. HillC. D. ThomasH. DescimonB. HuntleyL. KailaJ. KullbergT. TammaruW. J. TennentJ. 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. ParrE. 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. SandelL. ArgeB. DalsgaardR. G. DaviesK. J. GastonW. 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. WaltherE. PostP. ConveyA. MenzelC. ParmesanT. J. C. BeebeeJ. M. FromentinO. 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. XiaZ. WuS. Yan and W. Shu, Theory of Real Variable Function and Functional Analysis, second ed, Higher Education Press, Beijing, 2010.
[49]

Z. ZhangW. 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. BerestyckiO. DiekmannC. 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. BradleyD. 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. DockeryV. HutsonK. 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. FangY. 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. GonzalezR. P. NeilsonJ. 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. HutsonY. 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. HutsonY. 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. HutsonS. MartinezK. 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. LiS. BewickJ. 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. LoarieP. B. DuffyH. HamiltonG. P. AsnerC. 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.
[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. ParmesanN. RyrholmC. StefanescuJ. K. HillC. D. ThomasH. DescimonB. HuntleyL. KailaJ. KullbergT. TammaruW. J. TennentJ. 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. ParrE. 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. SandelL. ArgeB. DalsgaardR. G. DaviesK. J. GastonW. 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. WaltherE. PostP. ConveyA. MenzelC. ParmesanT. J. C. BeebeeJ. M. FromentinO. 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. XiaZ. WuS. Yan and W. Shu, Theory of Real Variable Function and Functional Analysis, second ed, Higher Education Press, Beijing, 2010.
[49]

Z. ZhangW. 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 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 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 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 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 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 $
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