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Effects of environmental heterogeneity on species spreading via numerical analysis of some free boundary models
1. | School of Science and Technology, University of New England, Armidale, NSW 2351, Australia |
2. | Department of Mathematics, Pabna University of Science and Technology, Pabna-6600, Bangladesh |
This paper investigates the effect of environmental heterogeneity on species spreading via numerical simulation of suitable reaction-diffusion models with free boundaries. We focus on the changes of long-time dynamics (establishment or extinction) and spreading speeds of the species as the parameters describing the heterogeneity of the environment are varied. For the single species model in time-periodic environment and in space-periodic environment theoretically treated in [
References:
[1] |
S. Altizer, A. Dobson, P. Hosseini and et al.,
Seasonality and the dynamics of infectious diseases, Ecology Letters, 9 (2006), 467-484.
doi: 10.1111/j.1461-0248.2005.00879.x. |
[2] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Partial Differential Equations and Related Topics, Lecture Notes in Math., Springer, Berlin, 446 (1975), 5–49. |
[3] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[4] |
H. Berestycki and F. Hamel,
Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.
doi: 10.1002/cpa.3022. |
[5] |
H. Berestycki, F. Hamel and G. Nadin,
Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189.
doi: 10.1016/j.jfa.2008.06.030. |
[6] |
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. |
[7] |
H. Berestycki, F. Hamel and L. Roques,
Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51 (2005), 75-113.
doi: 10.1007/s00285-004-0313-3. |
[8] |
G. Bunting, Y. Du and K. Krakowski,
Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.
doi: 10.3934/nhm.2012.7.583. |
[9] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003.
doi: 10.1002/0470871296. |
[10] |
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. |
[11] |
J.-F. Cao, Y. Du, F. Li and W.-T. Li,
The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.
doi: 10.1016/j.jfa.2019.02.013. |
[12] |
E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier,
Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115.
doi: 10.1017/S0956792598003660. |
[13] |
W. Ding, Y. Du and X. Liang,
Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1539-1573.
doi: 10.1016/j.anihpc.2019.01.005. |
[14] |
Y. Du and Z. Guo,
The Stefan problem for the Fisher–KPP equation, J. Differential Equations, 253 (2012), 996-1035.
doi: 10.1016/j.jde.2012.04.014. |
[15] |
Y. Du, Z. 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. |
[16] |
Y. Du and X. Liang,
Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305.
doi: 10.1016/j.anihpc.2013.11.004. |
[17] |
Y. Du and Z. 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. |
[18] |
Y. Du and B. Lou,
Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.
doi: 10.4171/JEMS/568. |
[19] |
Y. Du, H. Matano and K. Wang,
Regularity and asymptotic behavior of nonlinear Stefan problems, Arch. Ration. Mech. Anal., 212 (2014), 957-1010.
doi: 10.1007/s00205-013-0710-0. |
[20] |
Y. Du, H. Matsuzawa and M. Zhou,
Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.
doi: 10.1137/130908063. |
[21] |
Y. Du, H. Matsuzawa and M. Zhou,
Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103 (2015), 741-787.
doi: 10.1016/j.matpur.2014.07.008. |
[22] |
Y. Du and W. Ni,
Analysis of a West Nile virus model with nonlocal diffusion and free boundaries, Nonlinearity, 33 (2020), 4407-4448.
doi: 10.1088/1361-6544/ab8bb2. |
[23] |
Y. Du, M. Wang and M. Zhou,
Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.
doi: 10.1016/j.matpur.2016.06.005. |
[24] |
Y. Du and C. H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differential Equations, 57 (2018), 52, 36 pp.
doi: 10.1007/s00526-018-1339-5. |
[25] |
M. E. Smaily, F. Hamel and L. Roques, Homogenization and influence of fragmentation in a biological invasion model, Discrete Contin. Dyn. Syst., 25 (2009), 321–342. doi: 10.3934/dcds.2009.25.321
|
[26] |
R. A. Fisher,
The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[27] |
J. Gärtner and M. I. Freidlin,
On the propagation of concentration waves in periodic and random media, Dokl. Akad. Nauk SSSR, 249 (1979), 521-525.
|
[28] |
J. S. Guo and C. H. Wu,
Dynamics for a two-species competition–diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.
doi: 10.1088/0951-7715/28/1/1. |
[29] |
F. Hamel, J. Fayard and L. Roques,
Spreading speeds in slowly oscillating environments, Bull. Math. Biol., 72 (2010), 1166-1191.
doi: 10.1007/s11538-009-9486-7. |
[30] |
F. Hamel, G. Nadin and L. Roques, A viscosity solution method for the spreading speed formula in slowly varying media, Indiana Univ. Math. J., 60 (2011), 1229–1247, https://www.jstor.org/stable/24903869.
doi: 10.1512/iumj.2011.60.4370. |
[31] |
A. Hastings, K. Cuddington, K. F. Davies, C. J. Dugaw, S. Elmendorf, A. Freestone, S. Harrison, M. Holland, J. Lambrinos, U. Malvadkar and B. A. Melbourne,
The spatial spread of invasions: New developments in theory and evidence, Ecology Letters, 8 (2005), 91-101.
doi: 10.1111/j.1461-0248.2004.00687.x. |
[32] |
D. Hilhorst, M. Mimura and R. Schätzle,
Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285.
doi: 10.1016/S1468-1218(02)00009-3. |
[33] |
H. Izuhara, H. Monobe and C.-H. Wu, The formation of spreading front: The singular limit of three-component reaction-diffusion models, J. Math. Biol., 82 (2021), Paper No. 38, 33 pp.
doi: 10.1007/s00285-021-01591-5. |
[34] |
K. Khan, S. Liu, T. Schaerf and Y. Du, Invasive behaviour under competition via a free boundary model: A numerical approach, J. Math. Biol., 83 (2021), Paper no. 23, 43 pp.
doi: 10.1007/s00285-021-01641-y. |
[35] |
N. Kinezaki, K. Kawasaki and N. Shigesada,
Spatial dynamics of invasion in sinusoidally varying environments, Population Ecology, 48 (2006), 263-270.
doi: 10.1007/s10144-006-0263-2. |
[36] |
J. Kingsland, How might climate change affect the spread of viruses?, Medical News Today, 2020. |
[37] |
A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov,
A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ. Math. Mech., 1 (1937), 1-25.
|
[38] |
S. Liu and X. Liu,
Numerical methods for a two-species competition-diffusion model with free boundaries, Mathematics, 6 (2018), 72.
doi: 10.3390/math6050072. |
[39] |
S. Liu, Y. Du and X. Liu,
Numerical studies of a class of reaction–diffusion equations with Stefan conditions, Int. J. Comput. Math., 97 (2020), 959-979.
doi: 10.1080/00207160.2019.1599868. |
[40] |
F. Lutscher,
Non-local dispersal and averaging in heterogeneous landscapes, Appl. Anal., 89 (2010), 1091-1108.
doi: 10.1080/00036811003735816. |
[41] |
F. Lutscher, M. A. Lewis and E. McCauley,
Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160.
doi: 10.1007/s11538-006-9100-1. |
[42] |
G. A. Maciel and F. Lutscher,
Allee effects and population spread in patchy landscapes, J. Biol. Dyn., 9 (2015), 109-123.
doi: 10.1080/17513758.2015.1027309. |
[43] |
M. Moriyama, M. W. J. Hugentobler and A. Iwasaki,
Seasonality of respiratory viral infections, Annu. Rev. Virol., 7 (2020), 83-101.
doi: 10.1146/annurev-virology-012420-022445. |
[44] |
G. Nadin,
The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator, SIAM J. Math. Anal., 41 (2009/10), 2388-2406.
doi: 10.1137/080743597. |
[45] |
L. Roques and F. Hamel,
Mathematical analysis of the optimal habitat configurations for species persistence, Math. Biosci., 210 (2007), 34-59.
doi: 10.1016/j.mbs.2007.05.007. |
[46] |
W. Shen,
Traveling waves in time dependent bistable equations, Diff. Integral Eqns., 19 (2006), 241-278.
|
[47] |
N. Shigesada, K. Kawasaki and E. Teramoto,
Traveling periodic waves in heterogeneous environments, Theoret. Population Biol., 30 (1986), 143-160.
doi: 10.1016/0040-5809(86)90029-8. |
[48] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, UK, 1997.
![]() |
[49] |
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. |
[50] |
R. Wu and X.-Q. Zhao,
Spatial invasion of a birth pulse population with nonlocal dispersal, SIAM J. Appl. Math., 79 (2019), 1075-1097.
doi: 10.1137/18M1209805. |
[51] |
J. Xin,
Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
[52] |
M. Zhao, Y. Zhang, W.-T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347–3386.
doi: 10.1016/j.jde.2020.02.029. |
show all references
References:
[1] |
S. Altizer, A. Dobson, P. Hosseini and et al.,
Seasonality and the dynamics of infectious diseases, Ecology Letters, 9 (2006), 467-484.
doi: 10.1111/j.1461-0248.2005.00879.x. |
[2] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Partial Differential Equations and Related Topics, Lecture Notes in Math., Springer, Berlin, 446 (1975), 5–49. |
[3] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[4] |
H. Berestycki and F. Hamel,
Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.
doi: 10.1002/cpa.3022. |
[5] |
H. Berestycki, F. Hamel and G. Nadin,
Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189.
doi: 10.1016/j.jfa.2008.06.030. |
[6] |
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. |
[7] |
H. Berestycki, F. Hamel and L. Roques,
Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51 (2005), 75-113.
doi: 10.1007/s00285-004-0313-3. |
[8] |
G. Bunting, Y. Du and K. Krakowski,
Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.
doi: 10.3934/nhm.2012.7.583. |
[9] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003.
doi: 10.1002/0470871296. |
[10] |
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. |
[11] |
J.-F. Cao, Y. Du, F. Li and W.-T. Li,
The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.
doi: 10.1016/j.jfa.2019.02.013. |
[12] |
E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier,
Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115.
doi: 10.1017/S0956792598003660. |
[13] |
W. Ding, Y. Du and X. Liang,
Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1539-1573.
doi: 10.1016/j.anihpc.2019.01.005. |
[14] |
Y. Du and Z. Guo,
The Stefan problem for the Fisher–KPP equation, J. Differential Equations, 253 (2012), 996-1035.
doi: 10.1016/j.jde.2012.04.014. |
[15] |
Y. Du, Z. 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. |
[16] |
Y. Du and X. Liang,
Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305.
doi: 10.1016/j.anihpc.2013.11.004. |
[17] |
Y. Du and Z. 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. |
[18] |
Y. Du and B. Lou,
Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.
doi: 10.4171/JEMS/568. |
[19] |
Y. Du, H. Matano and K. Wang,
Regularity and asymptotic behavior of nonlinear Stefan problems, Arch. Ration. Mech. Anal., 212 (2014), 957-1010.
doi: 10.1007/s00205-013-0710-0. |
[20] |
Y. Du, H. Matsuzawa and M. Zhou,
Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.
doi: 10.1137/130908063. |
[21] |
Y. Du, H. Matsuzawa and M. Zhou,
Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103 (2015), 741-787.
doi: 10.1016/j.matpur.2014.07.008. |
[22] |
Y. Du and W. Ni,
Analysis of a West Nile virus model with nonlocal diffusion and free boundaries, Nonlinearity, 33 (2020), 4407-4448.
doi: 10.1088/1361-6544/ab8bb2. |
[23] |
Y. Du, M. Wang and M. Zhou,
Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.
doi: 10.1016/j.matpur.2016.06.005. |
[24] |
Y. Du and C. H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differential Equations, 57 (2018), 52, 36 pp.
doi: 10.1007/s00526-018-1339-5. |
[25] |
M. E. Smaily, F. Hamel and L. Roques, Homogenization and influence of fragmentation in a biological invasion model, Discrete Contin. Dyn. Syst., 25 (2009), 321–342. doi: 10.3934/dcds.2009.25.321
|
[26] |
R. A. Fisher,
The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[27] |
J. Gärtner and M. I. Freidlin,
On the propagation of concentration waves in periodic and random media, Dokl. Akad. Nauk SSSR, 249 (1979), 521-525.
|
[28] |
J. S. Guo and C. H. Wu,
Dynamics for a two-species competition–diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.
doi: 10.1088/0951-7715/28/1/1. |
[29] |
F. Hamel, J. Fayard and L. Roques,
Spreading speeds in slowly oscillating environments, Bull. Math. Biol., 72 (2010), 1166-1191.
doi: 10.1007/s11538-009-9486-7. |
[30] |
F. Hamel, G. Nadin and L. Roques, A viscosity solution method for the spreading speed formula in slowly varying media, Indiana Univ. Math. J., 60 (2011), 1229–1247, https://www.jstor.org/stable/24903869.
doi: 10.1512/iumj.2011.60.4370. |
[31] |
A. Hastings, K. Cuddington, K. F. Davies, C. J. Dugaw, S. Elmendorf, A. Freestone, S. Harrison, M. Holland, J. Lambrinos, U. Malvadkar and B. A. Melbourne,
The spatial spread of invasions: New developments in theory and evidence, Ecology Letters, 8 (2005), 91-101.
doi: 10.1111/j.1461-0248.2004.00687.x. |
[32] |
D. Hilhorst, M. Mimura and R. Schätzle,
Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285.
doi: 10.1016/S1468-1218(02)00009-3. |
[33] |
H. Izuhara, H. Monobe and C.-H. Wu, The formation of spreading front: The singular limit of three-component reaction-diffusion models, J. Math. Biol., 82 (2021), Paper No. 38, 33 pp.
doi: 10.1007/s00285-021-01591-5. |
[34] |
K. Khan, S. Liu, T. Schaerf and Y. Du, Invasive behaviour under competition via a free boundary model: A numerical approach, J. Math. Biol., 83 (2021), Paper no. 23, 43 pp.
doi: 10.1007/s00285-021-01641-y. |
[35] |
N. Kinezaki, K. Kawasaki and N. Shigesada,
Spatial dynamics of invasion in sinusoidally varying environments, Population Ecology, 48 (2006), 263-270.
doi: 10.1007/s10144-006-0263-2. |
[36] |
J. Kingsland, How might climate change affect the spread of viruses?, Medical News Today, 2020. |
[37] |
A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov,
A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ. Math. Mech., 1 (1937), 1-25.
|
[38] |
S. Liu and X. Liu,
Numerical methods for a two-species competition-diffusion model with free boundaries, Mathematics, 6 (2018), 72.
doi: 10.3390/math6050072. |
[39] |
S. Liu, Y. Du and X. Liu,
Numerical studies of a class of reaction–diffusion equations with Stefan conditions, Int. J. Comput. Math., 97 (2020), 959-979.
doi: 10.1080/00207160.2019.1599868. |
[40] |
F. Lutscher,
Non-local dispersal and averaging in heterogeneous landscapes, Appl. Anal., 89 (2010), 1091-1108.
doi: 10.1080/00036811003735816. |
[41] |
F. Lutscher, M. A. Lewis and E. McCauley,
Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160.
doi: 10.1007/s11538-006-9100-1. |
[42] |
G. A. Maciel and F. Lutscher,
Allee effects and population spread in patchy landscapes, J. Biol. Dyn., 9 (2015), 109-123.
doi: 10.1080/17513758.2015.1027309. |
[43] |
M. Moriyama, M. W. J. Hugentobler and A. Iwasaki,
Seasonality of respiratory viral infections, Annu. Rev. Virol., 7 (2020), 83-101.
doi: 10.1146/annurev-virology-012420-022445. |
[44] |
G. Nadin,
The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator, SIAM J. Math. Anal., 41 (2009/10), 2388-2406.
doi: 10.1137/080743597. |
[45] |
L. Roques and F. Hamel,
Mathematical analysis of the optimal habitat configurations for species persistence, Math. Biosci., 210 (2007), 34-59.
doi: 10.1016/j.mbs.2007.05.007. |
[46] |
W. Shen,
Traveling waves in time dependent bistable equations, Diff. Integral Eqns., 19 (2006), 241-278.
|
[47] |
N. Shigesada, K. Kawasaki and E. Teramoto,
Traveling periodic waves in heterogeneous environments, Theoret. Population Biol., 30 (1986), 143-160.
doi: 10.1016/0040-5809(86)90029-8. |
[48] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, UK, 1997.
![]() |
[49] |
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. |
[50] |
R. Wu and X.-Q. Zhao,
Spatial invasion of a birth pulse population with nonlocal dispersal, SIAM J. Appl. Math., 79 (2019), 1075-1097.
doi: 10.1137/18M1209805. |
[51] |
J. Xin,
Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
[52] |
M. Zhao, Y. Zhang, W.-T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347–3386.
doi: 10.1016/j.jde.2020.02.029. |























Oscillation magnitude |
Spreading range of |
0.2 | |
0.4 | |
0.6 | |
0.8 | |
1.0 | |
1.5 | |
2.0 | |
5.0 |
Oscillation magnitude |
Spreading range of |
0.2 | |
0.4 | |
0.6 | |
0.8 | |
1.0 | |
1.5 | |
2.0 | |
5.0 |
Time period | Spreading range of |
0.2 | |
0.4 | |
0.6 | |
0.8 | |
1.0 | |
2.0 | |
5.0 | |
10.0 | |
20.0 |
Time period | Spreading range of |
0.2 | |
0.4 | |
0.6 | |
0.8 | |
1.0 | |
2.0 | |
5.0 | |
10.0 | |
20.0 |
Oscillation magnitude | Spreading range of |
0.2 | |
0.4 | |
0.6 | |
0.8 | |
1.0 | |
1.5 | |
2.0 | |
5.0 |
Oscillation magnitude | Spreading range of |
0.2 | |
0.4 | |
0.6 | |
0.8 | |
1.0 | |
1.5 | |
2.0 | |
5.0 |
Period | Spreading range of |
0.2 | |
0.4 | |
0.6 | |
0.8 | |
1.0 | |
2.0 | |
5.0 | |
10.0 | |
20.0 |
Period | Spreading range of |
0.2 | |
0.4 | |
0.6 | |
0.8 | |
1.0 | |
2.0 | |
5.0 | |
10.0 | |
20.0 |
| Environments | Both | Only | Chase-and-run coexistence | Only |
0.10 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
0.50 | Homogeneous | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
0.75 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
1.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
2.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
3.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
4.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
5.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
6.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
7.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
8.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
9.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA |
| Environments | Both | Only | Chase-and-run coexistence | Only |
0.10 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
0.50 | Homogeneous | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
0.75 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
1.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
2.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
3.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
4.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
5.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
6.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
7.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
8.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
9.0 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA |
| Environments | Both | Only | Chase-and-run coexistence | Only |
0.10 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
0.50 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
0.75 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
1.0 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
2.0 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
3.0 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
4.0 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
5.0 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA |
| Environments | Both | Only | Chase-and-run coexistence | Only |
0.10 | Homogeneous: | NA | NA | ||
Time-Periodic: | NA | NA | |||
Space-Periodic: | NA | NA | |||
0.50 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
0.75 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
1.0 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
2.0 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
3.0 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
4.0 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA | ||||
5.0 | Homogeneous: | NA | |||
Time-Periodic: | NA | ||||
Space-Periodic: | NA |
Time-Periodic Case | Homogeneous Case | ||
Vanishing of |
Chase-and-run coexistence | ||
Time-Periodic Case | Homogeneous Case | ||
Vanishing of |
Chase-and-run coexistence | ||
| Time-Periodic Case | Homogeneous Case | |||
1 | 0.602 | 1 | 1 | chase-and-run coexistence | vanishing of |
0.1 | 1 | vanishing of | |||
0.869 | 1 | 1 | vanishing of | chase-and-run coexistence | |
0.01 | 0.01 | chase-and-run coexistence | |||
2 | 0.611 | 1 | 1 | chase-and-run coexistence | vanishing of |
0.1 | 1 | vanishing of | |||
1.819 | 1 | 1 | vanishing of | chase-and-run coexistence | |
0.01 | 0.01 | chase-and-run coexistence | |||
3 | 0.609 | 1 | 1 | chase-and-run coexistence | vanishing of |
0.1 | 1 | vanishing of | |||
2.799 | 1 | 1 | vanishing of | chase-and-run coexistence | |
0.01 | 0.01 | chase-and-run coexistence | |||
4 | 0.606 | 1 | 1 | chase-and-run coexistence | vanishing of |
0.1 | 1 | vanishing of | |||
3.782 | 1 | 1 | vanishing of | chase-and-run coexistence | |
0.01 | 0.01 | chase-and-run coexistence | |||
5 | 0.604 | 1 | 1 | chase-and-run coexistence | vanishing of |
0.1 | 1 | vanishing of | |||
4.768 | 1 | 1 | vanishing of | chase-and-run coexistence | |
0.01 | 0.01 | chase-and-run coexistence |
| Time-Periodic Case | Homogeneous Case | |||
1 | 0.602 | 1 | 1 | chase-and-run coexistence | vanishing of |
0.1 | 1 | vanishing of | |||
0.869 | 1 | 1 | vanishing of | chase-and-run coexistence | |
0.01 | 0.01 | chase-and-run coexistence | |||
2 | 0.611 | 1 | 1 | chase-and-run coexistence | vanishing of |
0.1 | 1 | vanishing of | |||
1.819 | 1 | 1 | vanishing of | chase-and-run coexistence | |
0.01 | 0.01 | chase-and-run coexistence | |||
3 | 0.609 | 1 | 1 | chase-and-run coexistence | vanishing of |
0.1 | 1 | vanishing of | |||
2.799 | 1 | 1 | vanishing of | chase-and-run coexistence | |
0.01 | 0.01 | chase-and-run coexistence | |||
4 | 0.606 | 1 | 1 | chase-and-run coexistence | vanishing of |
0.1 | 1 | vanishing of | |||
3.782 | 1 | 1 | vanishing of | chase-and-run coexistence | |
0.01 | 0.01 | chase-and-run coexistence | |||
5 | 0.604 | 1 | 1 | chase-and-run coexistence | vanishing of |
0.1 | 1 | vanishing of | |||
4.768 | 1 | 1 | vanishing of | chase-and-run coexistence | |
0.01 | 0.01 | chase-and-run coexistence |
Space-Periodic Case | Homogeneous Case | ||||
1 | 1.168 | 1 | 1 | vanishing of |
chase-and-run coexistence |
2 | chase-and-run coexistence | ||||
2 | 2.182 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
3 | 3.202 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
4 | 4.2202 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
5 | 5.233 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
6 | 6.244 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
7 | 7.253 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
10 | 10.276 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence |
Space-Periodic Case | Homogeneous Case | ||||
1 | 1.168 | 1 | 1 | vanishing of |
chase-and-run coexistence |
2 | chase-and-run coexistence | ||||
2 | 2.182 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
3 | 3.202 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
4 | 4.2202 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
5 | 5.233 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
6 | 6.244 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
7 | 7.253 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence | ||||
10 | 10.276 | 1 | 1 | vanishing of |
vanishing of |
2 | chase-and-run coexistence |
Time-Periodic Case | Homogeneous Case | |
Chase-and-run coexistence | Chase-and-run coexistence | |
Vanishing of |
||
Chase-and-run coexistence | ||
Time-Periodic Case | Homogeneous Case | |
Chase-and-run coexistence | Chase-and-run coexistence | |
Vanishing of |
||
Chase-and-run coexistence | ||
Space-Periodic Case | Homogeneous Case | |
Chase-and-run coexistence | Chase-and-run coexistence | |
Vanishing of |
||
Chase-and-run coexistence | ||
Vanishing of |
||
Space-Periodic Case | Homogeneous Case | |
Chase-and-run coexistence | Chase-and-run coexistence | |
Vanishing of |
||
Chase-and-run coexistence | ||
Vanishing of |
||
Time-Periodic |
Time-Periodic |
|
0.045093 | 0.364358 | |
0.045099 | 0.364336 | |
0.045118 | 0.364294 | |
0.045158 | 0.364233 | |
0.045224 | 0.364154 | |
0.056211 | 0 | |
0.056282 | 0 | |
0.056449 | 0 | |
0.056545 | 0 | |
0.056648 | 0 | |
0.056874 | 0 | |
0.045852 | 0.363676 | |
0.045949 | 0.363617 | |
0.048875 | 0.362510 | |
0.050970 | 0.362933 | |
0.051696 | 0.363868 |
Time-Periodic |
Time-Periodic |
|
0.045093 | 0.364358 | |
0.045099 | 0.364336 | |
0.045118 | 0.364294 | |
0.045158 | 0.364233 | |
0.045224 | 0.364154 | |
0.056211 | 0 | |
0.056282 | 0 | |
0.056449 | 0 | |
0.056545 | 0 | |
0.056648 | 0 | |
0.056874 | 0 | |
0.045852 | 0.363676 | |
0.045949 | 0.363617 | |
0.048875 | 0.362510 | |
0.050970 | 0.362933 | |
0.051696 | 0.363868 |
Space period |
Average speed of |
Average speed of |
0.045131 | 0.364630 | |
0.045402 | 0.366188 | |
0.045908 | 0.369344 | |
0.046770 | 0.374485 | |
0.058183 | 0.013643 ( |
|
0.057517 | 0.012500 ( |
|
0.058908 |
0.006666 |
|
0.059048 |
0.007000 |
|
0.059164 |
0.007333 |
|
0.047889 |
0.381331 |
|
0.047982 |
0.382087 |
|
0.047892 |
0.382278 |
|
0.047668 |
0.381776 |
|
0.047319 |
0.380748 |
|
0.058552 |
0.011000 |
|
0.058332 |
0.011333 |
|
0.056601 |
0.013333 |
|
0.052802 |
0.016666 |
|
0.020768 |
0.020000 |
|
0.030000 |
0.030000 |
|
0.038608 |
0.034420 |
|
* These average speeds are at t = 300. ** These average speeds are at t = 500. |
Space period |
Average speed of |
Average speed of |
0.045131 | 0.364630 | |
0.045402 | 0.366188 | |
0.045908 | 0.369344 | |
0.046770 | 0.374485 | |
0.058183 | 0.013643 ( |
|
0.057517 | 0.012500 ( |
|
0.058908 |
0.006666 |
|
0.059048 |
0.007000 |
|
0.059164 |
0.007333 |
|
0.047889 |
0.381331 |
|
0.047982 |
0.382087 |
|
0.047892 |
0.382278 |
|
0.047668 |
0.381776 |
|
0.047319 |
0.380748 |
|
0.058552 |
0.011000 |
|
0.058332 |
0.011333 |
|
0.056601 |
0.013333 |
|
0.052802 |
0.016666 |
|
0.020768 |
0.020000 |
|
0.030000 |
0.030000 |
|
0.038608 |
0.034420 |
|
* These average speeds are at t = 300. ** These average speeds are at t = 500. |
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