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doi: 10.3934/dcdsb.2022077
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Effects of environmental heterogeneity on species spreading via numerical analysis of some free boundary models

Kamruzzaman Khan 1,2, , Timothy M. Schaerf 1, and  Yihong Du 1,,

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

*Corresponding author: Yihong Du

Dedicated to the memory of Professor Masayasu Mimura

Received  September 2021 Revised  January 2022 Early access April 2022

Fund Project: This research was supported by the Australian Research Council and a University of New England PhD scholarship

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 [15,16], we obtain more detailed properties here. Among other results, our numerical simulation suggests that, in a time-periodic or space-periodic environment, moderate increase of the oscillation scale enhances the chances of establishment as well as the spreading speed of the species. We also numerically examine a related model with two competing species, which was treated in [34,28,24] recently and reduces to the single species free boundary model when one of the species is absent. Our numerical results, obtained by varying the parameters in the time-periodic and space-periodic terms of the model, suggest that heterogeneity of the environment enhances the invasion of the two species (as in the single species model), although there are subtle differences of the influences felt by the two. Some intriguing phenomena revealed in our simulations suggest that heterogeneity of the environment decreases the level of predictability of the competition outcome.

Keywords: Longtime dynamics, spreading speed, heterogeneous environment, reaction-diffusion models with free boundary.
Mathematics Subject Classification: Primary: 35K51, 92D25; Secondary: 65M22.
Citation: Kamruzzaman Khan, Timothy M. Schaerf, Yihong Du. Effects of environmental heterogeneity on species spreading via numerical analysis of some free boundary models. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022077
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 MR2525180" target="_blank">10.3934/dcds.2009.25.xx. 10.3934/dcds.2009.25.321 MR2525180

[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 MR2525180" target="_blank">10.3934/dcds.2009.25.xx. 10.3934/dcds.2009.25.321 MR2525180

[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.

Figure 2.1.  The graph of the population density function $ u(x,t) $ at time $ t = 120 $ (solid green curve) almost coincides with that of $ u(x,t) $ at time $ t = 121 $ (red dotted curve), indicating time-periodic variation of the population induced by the time-periodic environment during the spreading process
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Figure 2.2.  Evolution of the population density $ u(x,t) $ up to time $ t = 120 $ in the simulated time-periodic environment. Note that in the graph, $ u(x,t) $ was extended to 0 for $ x>s(t) $
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Figure 2.3.  Graph of spreading front function $ s(t) $
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Figure 2.4.  Spreading speed as a function of period $ T $ in a time-periodic environment
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Figure 2.5.  Profiles of $ u(x,t) $ in a space-periodic environment with period $ L = 1 $ and magnitude of oscillation $ \sigma = 1 $. Snapshots are taken at times $ t = 84.8 $, $ t = 102.4 $, and $ t = 120 $, indicating space-periodic variation of the population induced by the space-periodic environment during the spreading process
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Figure 2.6.  Evolution of the population density function $ u(x,t) $ up to time $ t = 120 $ in the simulated space-periodic environment. Note that $ u(x,t) $ was extended to 0 for $ x>s(t) $
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Figure 2.7.  Corresponding spreading front function $ s(t) $ up to time $ t = 120 $
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Figure 2.8.  The blue solid line represents the speed of the expanding front at $ t = 120 $ in a homogeneous environment, the green solid line depicts the average speed of the expanding front at $ t = 120 $ in the simulated time-periodic environment and the red solid line stands for the average speed of the expanding front at $ t = 120 $ in the simulated space-periodic environment
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Figure 2.9.  Spreading speed as a function of period $ L $ in a space-periodic environment
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Figure 3.10.  Evolution of the population density functions $ u(x,t) $ and $ v(x,t) $ up to time $ t = 120 $ in the simulated time-periodic environment. Note that the extensions $ u(x,t) = 0 $ for $ x>s_1(t) $ and $ v(x, t) = 0 $ for $ x>s_2(t) $ were used in the graph
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Figure 3.11.  Spreading fronts of the species up to time $ t = 120 $
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Figure 3.12.  Average speeds of $ u $ and $ v $ with $ \sigma_1 = \sigma_2 \in [0.1,2] $: The average speed of $ u $ at $ t = 120 $ (i.e., $ \overline{s_1'(120)} $) (the green curve in left figure) increases while the average speed of $ v $ at $ t = 120 $ (i.e., $ \overline{s_2'(120)} $ (magenta line in right figure) decreases as $ \sigma_1 = \sigma_2 $ increases in $ [0.1,2] $. Compared with the corresponding speed of $ u $ and $ v $ in a homogeneous environment (the blue and red lines in the left and right figures, respectively), the speed of $ u $ is slightly bigger in the time-periodic environment, while the speed of $ v $ is slightly smaller
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Figure 3.13.  Average speeds of $ u $ and $ v $ with $ \sigma_1 = 1 $ and $ \sigma_2 \in [0.1,2] $: The average speed of $ u $ at $ t = 120 $ (i.e., $ \overline{s_1'(120)} $) decreases very slowly (green curve in left figure) as $ \sigma_2 $ is increased in $ [0.1,2] $, but it is bigger than the corresponding speed in the homogeneous environment (blue line); the average speed of $ v $ at $ t = 120 $ (i.e., $ \overline{s_2'(120)} $) also decreases (magenta curve in right figure) as $ \sigma_2 $ increases in [0.1, 2], but in contrast, it is smaller than the speed of $ v $ in homogeneous environment (red line)
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Figure 3.14.  Average speeds of $ u $ and $ v $ with $ \sigma_2 = 1 $ and $ \sigma_1 \in [0.1,2] $: The average speed of $ u $ at $ t = 120 $ (i.e., $ \overline{s_1'(120)} $) increases (green curve in left figure) when $ \sigma_1 $ increases in $ [0.1,2] $, and it is bigger than the corresponding speed of $ u $ in a homogeneous environment (i.e., $ \sigma_1 = \sigma_2 = 0 $); while the average speed of $ v $ at $ t = 120 $ (i.e., $ \overline{s_2'(120)} $) stays constant (magenta line in right figure) as $ \sigma_1 $ varies in $ [0.1, 2] $, and in contrast, it is smaller than the corresponding speed of $ v $ (i.e., $ s_2'(120) $) in a homogeneous environment
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Figure 3.15.  Moving fronts of $ u $ and $ v $ for a selection of values of $ T $ in Table 3.8
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Figure 3.16.  Spreading speeds of $ u $ and $ v $ for a selection of values of $ T $ in Table 3.8
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Figure 3.17.  Population distribution of $ u $ and $ v $ at time $ t = 10 $ with $ T = 0.6, 1.0, 1.1, 1.5, 1.8,1.9, 10 $
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Figure 3.18.  Population distribution of $ u $ and $ v $ at time $ t = 0, 10, 30, 50,120 $ with $ T = 0.6, 1.5, 2, 10 $
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Figure 3.19.  Profiles of $ u(x,t) $ and $ v(x,t) $ in a space-periodic environment of period $ L = 1 $ and magnitude of the oscillations $ \sigma_1 = \sigma_2 = 1 $
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Figure 3.20.  Evolution of the population density functions $ u(x,t) $ and $ v(x,t) $ up to time $ t = 120 $ in the simulated space-periodic environment. The extensions $ u(x,t) = 0 $ for $ x>s_1(t) $ and $ v(x,t) = 0 $ for $ x>s_2(t) $ were used in the graph
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Figure 3.21.  Corresponding spreading front functions of the species in Figure 3.19
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Figure 3.22.  Spreading speeds of $ u $ (left) and $ v $ (right) in homogeneous and space-periodic environments with period $ L = 1 $ and magnitude of the oscillations $ \sigma_1 = 1.0 $, $ \sigma_2 = 1.0 $
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Figure 3.23.  Average speeds of $ u $ and $ v $ with $ \sigma_1 = \sigma_2 \in [0.1,2] $: The average speeds of $ u $ (green curve in left figure) and $ v $ (magenta curve in right figure) both increase as $ \sigma_1 = \sigma_2 $ are increased. They are both bigger than the corresponding speeds in a homogeneous environment
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Figure 3.24.  Average speeds of $ u $ and $ v $ with $ \sigma_1 = 1 $ and $ \sigma_2 \in [0.1,2] $: The average speed of $ u $ decreases (green curve in left figure) when $ \sigma_2 $ is increased in $ [0.1,2] $, while the average speed of $ v $ increases with $ \sigma_2 $ (magenta curve in right figure). Both speeds are greater than the corresponding speeds in a homogeneous environment
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Figure 3.25.  Average speeds of $ u $ and $ v $ with $ \sigma_2 = 1 $ and $ \sigma_1 \in [0.1,2] $: The average speed of $ u $ increases (green curve in left figure) when $ \sigma_1 $ is increased in $ [0.1,2] $, while the average speed of $ v $ does not change with $ \sigma_1 $ (magenta line in right figure). Both speeds are greater than the corresponding speeds in a homogeneous environment except that of $ u $ for $ \sigma_1\leq 0.4 $
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Figure 3.26.  Changes of spreading speeds as $ L $ varies given in Table 3.9
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Figure 3.27.  Moving fronts of $ u $ and $ v $ for a selection of values of $ L $ in Table 3.9
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Figure 3.28.  Spreading speeds of $ u $ and $ v $ for a selection of values of $ L $ in Table 3.9
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Figure 3.29.  Moving fronts of $ u $ and $ v $ with $ L = 5, 10 $
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Figure 3.30.  Spreading speeds of $ u $ and $ v $ with $ L = 5, 10 $
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Figure 3.31.  Population distribution of $ u $ and $ v $ at time $ t = 0, 10, 40, 80,120 $ with $ L = 1, 2, 3, 5 $
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Table 2.1.  Spreading range of $\lambda$ with time period $T = 1$ and changing magnitude $\sigma$
Oscillation magnitude $\sigma$ Spreading range of $\lambda$
0.2 $(0.445, \infty)$
0.4 $(0.445, \infty)$
0.6 $(0.445, \infty)$
0.8 $(0.435, \infty)$
1.0 $(0.435, \infty)$
1.5 $(0.435, \infty)$
2.0 $(0.435, \infty)$
5.0 $(0.405, \infty)$
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Oscillation magnitude $\sigma$ Spreading range of $\lambda$
0.2 $(0.445, \infty)$
0.4 $(0.445, \infty)$
0.6 $(0.445, \infty)$
0.8 $(0.435, \infty)$
1.0 $(0.435, \infty)$
1.5 $(0.435, \infty)$
2.0 $(0.435, \infty)$
5.0 $(0.405, \infty)$
Table 2.2.  Spreading range of $\lambda$ with $\sigma = 1$ and varying time period $T$
Time period $T$ Spreading range of $\lambda$
0.2$(0.445, \infty)$
0.4$(0.445, \infty)$
0.6$(0.445, \infty)$
0.8$(0.435, \infty)$
1.0$(0.435, \infty)$
2.0$(0.435, \infty)$
5.0$(0.415, \infty)$
10.0$(0.385, \infty)$
20.0$(0.365, \infty)$
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Time period $T$ Spreading range of $\lambda$
0.2$(0.445, \infty)$
0.4$(0.445, \infty)$
0.6$(0.445, \infty)$
0.8$(0.435, \infty)$
1.0$(0.435, \infty)$
2.0$(0.435, \infty)$
5.0$(0.415, \infty)$
10.0$(0.385, \infty)$
20.0$(0.365, \infty)$
Table 2.3.  Spreading range of $\lambda$ with space-period $L = 1$ and varying $\sigma$
Oscillation magnitude $\sigma$ Spreading range of $\lambda$
0.2$(0.435, \infty)$
0.4$(0.425, \infty)$
0.6$(0.415, \infty)$
0.8$(0.405, \infty)$
1.0$(0.395, \infty)$
1.5$(0.365, \infty)$
2.0$(0.335, \infty)$
5.0$(0.215, \infty)$
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Oscillation magnitude $\sigma$ Spreading range of $\lambda$
0.2$(0.435, \infty)$
0.4$(0.425, \infty)$
0.6$(0.415, \infty)$
0.8$(0.405, \infty)$
1.0$(0.395, \infty)$
1.5$(0.365, \infty)$
2.0$(0.335, \infty)$
5.0$(0.215, \infty)$
Table 2.4.  Spreading range of $\lambda$ with $\sigma = 1$ and varying $L$
Period $L$ Spreading range of $\lambda$
0.2$(0.435, \infty)$
0.4$(0.425, \infty)$
0.6$(0.415, \infty)$
0.8$(0.405, \infty)$
1.0$(0.395, \infty)$
2.0$(0.335, \infty)$
5.0$(0.365, \infty)$
10.0$(0.395, \infty)$
20.0$(0.415, \infty)$
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Period $L$ Spreading range of $\lambda$
0.2$(0.435, \infty)$
0.4$(0.425, \infty)$
0.6$(0.415, \infty)$
0.8$(0.405, \infty)$
1.0$(0.395, \infty)$
2.0$(0.335, \infty)$
5.0$(0.365, \infty)$
10.0$(0.395, \infty)$
20.0$(0.415, \infty)$
Table 3.1.  Change of long-time behaviour as the initial functions are varied in homogeneous, time-periodic and space-periodic environments - A
$\lambda_1$ Environments Both $u$ and $v$ vanishing Only $v$ vanishing Chase-and-run coexistence Only $u$ vanishing
0.10Homogeneous:$\lambda_2\le 0.223$NANA$\lambda_2\ge 0.224$
Time-Periodic:$\lambda_2\le 0.212$NANA$\lambda_2\ge 0.213$
Space-Periodic:$\lambda_2\le 0.190$NANA$\lambda_2\ge 0.191$
0.50HomogeneousNA$\lambda_2 \le 0.367$$0.368 \le \lambda_2 \le 0.375$$\lambda_2\ge 0.376$
Time-Periodic:NA$\lambda_2 \le 0.380$$0.381 \le \lambda_2 \le 0.390$$\lambda_2\ge 0.391$
Space-Periodic:NA$\lambda_2 \le 0.444$$0.445 \le \lambda_2 \le 0.583$$\lambda_2 \ge 0.584$
0.75Homogeneous:NA$\lambda_2\le 0.840$$\lambda_2\ge 0.841$NA
Time-Periodic:NA$\lambda_2\le 0.847$$\lambda_2\ge 0.848$NA
Space-Periodic:NA$\lambda_2\le 0.887$$\lambda_2\ge 0.888$NA
1.0Homogeneous:NA$\lambda_2\le 1.151$$\lambda_2\ge 1.152$NA
Time-Periodic:NA$\lambda_2\le 1.152$$\lambda_2\ge 1.153$NA
Space-Periodic:NA$\lambda_2\le 1.168$$\lambda_2\ge 1.169$NA
2.0Homogeneous:NA$\lambda_2\le 2.184$$\lambda_2\ge 2.185$NA
Time-Periodic:NA$\lambda_2\le 2.186$$\lambda_2\ge 2.187$NA
Space-Periodic:NA$\lambda_2\le 2.182$$\lambda_2\ge 2.183$NA
3.0Homogeneous:NA$\lambda_2\le 3.204$$\lambda_2\ge 3.205$NA
Time-Periodic:NA$\lambda_2\le 3.206$$\lambda_2\ge 3.207$NA
Space-Periodic:NA$\lambda_2\le 3.202$$\lambda_2\ge 3.203$NA
4.0Homogeneous:NA$\lambda_2\le 4.220$$\lambda_2\ge 4.221$NA
Time-Periodic:NA$\lambda_2\le 4.223$$\lambda_2\ge 4.224$NA
Space-Periodic:NA$\lambda_2\le 4.2202$$\lambda_2\ge 4.2203$NA
5.0Homogeneous:NA$\lambda_2\le 5.234$$\lambda_2\ge 5.235$NA
Time-Periodic:NA$\lambda_2\le 5.237$$\lambda_2\ge 5.238$NA
Space-Periodic:NA$\lambda_2\le 5.233$$\lambda_2\ge 5.234$NA
6.0Homogeneous:NA$\lambda_2\le 6.246$$\lambda_2\ge 6.247$NA
Time-Periodic:NA$\lambda_2\le 6.248$$\lambda_2\ge 6.249$NA
Space-Periodic:NA$\lambda_2\le 6.244$$\lambda_2\ge 6.245$NA
7.0Homogeneous:NA$\lambda_2\le 7.256$$\lambda_2\ge 7.257$NA
Time-Periodic:NA$\lambda_2\le 7.258$$\lambda_2\ge 7.259$NA
Space-Periodic:NA$\lambda_2\le 7.253$$\lambda_2\ge 7.254$NA
8.0Homogeneous:NA$\lambda_2\le 8.264$$\lambda_2\ge 8.265$NA
Time-Periodic:NA$\lambda_2\le 8.267$$\lambda_2\ge 8.268$NA
Space-Periodic:NA$\lambda_2\le 8.262$$\lambda_2\ge 8.263$NA
9.0Homogeneous:NA$\lambda_2\le 9.272$$\lambda_2\ge 9.273$NA
Time-Periodic:NA$\lambda_2\le 9.274$$\lambda_2\ge 9.275$NA
Space-Periodic:NA$\lambda_2\le 9.269$$\lambda_2\ge 9.270$NA
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$\lambda_1$ Environments Both $u$ and $v$ vanishing Only $v$ vanishing Chase-and-run coexistence Only $u$ vanishing
0.10Homogeneous:$\lambda_2\le 0.223$NANA$\lambda_2\ge 0.224$
Time-Periodic:$\lambda_2\le 0.212$NANA$\lambda_2\ge 0.213$
Space-Periodic:$\lambda_2\le 0.190$NANA$\lambda_2\ge 0.191$
0.50HomogeneousNA$\lambda_2 \le 0.367$$0.368 \le \lambda_2 \le 0.375$$\lambda_2\ge 0.376$
Time-Periodic:NA$\lambda_2 \le 0.380$$0.381 \le \lambda_2 \le 0.390$$\lambda_2\ge 0.391$
Space-Periodic:NA$\lambda_2 \le 0.444$$0.445 \le \lambda_2 \le 0.583$$\lambda_2 \ge 0.584$
0.75Homogeneous:NA$\lambda_2\le 0.840$$\lambda_2\ge 0.841$NA
Time-Periodic:NA$\lambda_2\le 0.847$$\lambda_2\ge 0.848$NA
Space-Periodic:NA$\lambda_2\le 0.887$$\lambda_2\ge 0.888$NA
1.0Homogeneous:NA$\lambda_2\le 1.151$$\lambda_2\ge 1.152$NA
Time-Periodic:NA$\lambda_2\le 1.152$$\lambda_2\ge 1.153$NA
Space-Periodic:NA$\lambda_2\le 1.168$$\lambda_2\ge 1.169$NA
2.0Homogeneous:NA$\lambda_2\le 2.184$$\lambda_2\ge 2.185$NA
Time-Periodic:NA$\lambda_2\le 2.186$$\lambda_2\ge 2.187$NA
Space-Periodic:NA$\lambda_2\le 2.182$$\lambda_2\ge 2.183$NA
3.0Homogeneous:NA$\lambda_2\le 3.204$$\lambda_2\ge 3.205$NA
Time-Periodic:NA$\lambda_2\le 3.206$$\lambda_2\ge 3.207$NA
Space-Periodic:NA$\lambda_2\le 3.202$$\lambda_2\ge 3.203$NA
4.0Homogeneous:NA$\lambda_2\le 4.220$$\lambda_2\ge 4.221$NA
Time-Periodic:NA$\lambda_2\le 4.223$$\lambda_2\ge 4.224$NA
Space-Periodic:NA$\lambda_2\le 4.2202$$\lambda_2\ge 4.2203$NA
5.0Homogeneous:NA$\lambda_2\le 5.234$$\lambda_2\ge 5.235$NA
Time-Periodic:NA$\lambda_2\le 5.237$$\lambda_2\ge 5.238$NA
Space-Periodic:NA$\lambda_2\le 5.233$$\lambda_2\ge 5.234$NA
6.0Homogeneous:NA$\lambda_2\le 6.246$$\lambda_2\ge 6.247$NA
Time-Periodic:NA$\lambda_2\le 6.248$$\lambda_2\ge 6.249$NA
Space-Periodic:NA$\lambda_2\le 6.244$$\lambda_2\ge 6.245$NA
7.0Homogeneous:NA$\lambda_2\le 7.256$$\lambda_2\ge 7.257$NA
Time-Periodic:NA$\lambda_2\le 7.258$$\lambda_2\ge 7.259$NA
Space-Periodic:NA$\lambda_2\le 7.253$$\lambda_2\ge 7.254$NA
8.0Homogeneous:NA$\lambda_2\le 8.264$$\lambda_2\ge 8.265$NA
Time-Periodic:NA$\lambda_2\le 8.267$$\lambda_2\ge 8.268$NA
Space-Periodic:NA$\lambda_2\le 8.262$$\lambda_2\ge 8.263$NA
9.0Homogeneous:NA$\lambda_2\le 9.272$$\lambda_2\ge 9.273$NA
Time-Periodic:NA$\lambda_2\le 9.274$$\lambda_2\ge 9.275$NA
Space-Periodic:NA$\lambda_2\le 9.269$$\lambda_2\ge 9.270$NA
Table 3.2.  Change of long-time behaviour as the initial functions are varied in homogeneous, time-periodic and space-periodic environments - B
$\lambda_2$ Environments Both $u$ and $v$ vanishing Only $u$ vanishing Chase-and-run coexistence Only $v$ vanishing
0.10Homogeneous:$\lambda_1\le 0.444$NANA$\lambda_1\ge 0.445$
Time-Periodic:$\lambda_1\le 0.438$NANA$\lambda_1\ge 0.439$
Space-Periodic:$\lambda_1\le 0.393$NANA$\lambda_1\ge 0.394$
0.50Homogeneous:NA$\lambda_1\le 0.541$$0.542 \le \lambda_1 \le 0.564$$\lambda_1 \ge 0.565$
Time-Periodic:NA$\lambda_1\le 0.535$$0.536 \le \lambda_1 \le 0.557$$\lambda_1\ge 0.558$
Space-Periodic:NA$\lambda_1 \le 0.487$$0.488 \le \lambda_1 \le 0.523$$\lambda_1 \ge 0.524$
0.75Homogeneous:NA$\lambda_1\le 0.585$$0.586 \le \lambda_1 \le 0.692$$\lambda_1 \ge 0.693$
Time-Periodic:NA$\lambda_1\le 0.579$$0.580 \le \lambda_1 \le 0.686$$\lambda_1\ge 0.687$
Space-Periodic:NA$\lambda_1 \le 0.516$$0.517 \le \lambda_1 \le 0.652$$\lambda_1 \ge 0.653$
1.0Homogeneous:NA$\lambda_1\le 0.602$$0.603 \le \lambda_1 \le 0.869$$\lambda_1 \ge 0.870$
Time-Periodic:NA$\lambda_1\le 0.597$$0.598 \le \lambda_1 \le 0.866$$\lambda_1\ge 0.867$
Space-Periodic:NA$\lambda_1 \le 0.528$$0.529 \le \lambda_1 \le 0.839$$\lambda_1 \ge 0.840$
2.0Homogeneous:NA$\lambda_1\le 0.611$$0.612 \le \lambda_1 \le 1.819$$\lambda_1 \ge 1.820$
Time-Periodic:NA$\lambda_1\le 0.606$$0.607 \le \lambda_1 \le 1.817$$\lambda_1\ge 1.818$
Space-Periodic:NA$\lambda_1 \le 0.533$$0.534 \le \lambda_1 \le 1.817$$\lambda_1 \ge 1.818$
3.0Homogeneous:NA$\lambda_1\le 0.609$$0.610 \le \lambda_1 \le 2.799$$\lambda_1 \ge 2.800$
Time-Periodic:NA$\lambda_1\le 0.603$$0.604 \le \lambda_1 \le 2.796$$\lambda_1\ge 2.797$
Space-Periodic:NA$\lambda_1 \le 0.530$$0.531 \le \lambda_1 \le 2.800$$\lambda_1 \ge 2.801$
4.0Homogeneous:NA$\lambda_1\le 0.606$$0.607 \le \lambda_1 \le 3.782$$\lambda_1 \ge 3.783$
Time-Periodic:NA$\lambda_1\le 0.601$$0.602 \le \lambda_1 \le 3.779$$\lambda_1\ge 3.780$
Space-Periodic:NA$\lambda_1 \le 0.527$$0.528 \le \lambda_1 \le 3.784$$\lambda_1 \ge 3.785$
5.0Homogeneous:NA$\lambda_1\le 0.604$$0.605 \le \lambda_1 \le 4.768$$\lambda_1 \ge 4.769$
Time-Periodic:NA$\lambda_1\le 0.599$$0.600 \le \lambda_1 \le 4.765$$\lambda_1\ge 4.766$
Space-Periodic:NA$\lambda_1 \le 0.525$$0.526 \le \lambda_1 \le 4.770$$\lambda_1 \ge 4.771$
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$\lambda_2$ Environments Both $u$ and $v$ vanishing Only $u$ vanishing Chase-and-run coexistence Only $v$ vanishing
0.10Homogeneous:$\lambda_1\le 0.444$NANA$\lambda_1\ge 0.445$
Time-Periodic:$\lambda_1\le 0.438$NANA$\lambda_1\ge 0.439$
Space-Periodic:$\lambda_1\le 0.393$NANA$\lambda_1\ge 0.394$
0.50Homogeneous:NA$\lambda_1\le 0.541$$0.542 \le \lambda_1 \le 0.564$$\lambda_1 \ge 0.565$
Time-Periodic:NA$\lambda_1\le 0.535$$0.536 \le \lambda_1 \le 0.557$$\lambda_1\ge 0.558$
Space-Periodic:NA$\lambda_1 \le 0.487$$0.488 \le \lambda_1 \le 0.523$$\lambda_1 \ge 0.524$
0.75Homogeneous:NA$\lambda_1\le 0.585$$0.586 \le \lambda_1 \le 0.692$$\lambda_1 \ge 0.693$
Time-Periodic:NA$\lambda_1\le 0.579$$0.580 \le \lambda_1 \le 0.686$$\lambda_1\ge 0.687$
Space-Periodic:NA$\lambda_1 \le 0.516$$0.517 \le \lambda_1 \le 0.652$$\lambda_1 \ge 0.653$
1.0Homogeneous:NA$\lambda_1\le 0.602$$0.603 \le \lambda_1 \le 0.869$$\lambda_1 \ge 0.870$
Time-Periodic:NA$\lambda_1\le 0.597$$0.598 \le \lambda_1 \le 0.866$$\lambda_1\ge 0.867$
Space-Periodic:NA$\lambda_1 \le 0.528$$0.529 \le \lambda_1 \le 0.839$$\lambda_1 \ge 0.840$
2.0Homogeneous:NA$\lambda_1\le 0.611$$0.612 \le \lambda_1 \le 1.819$$\lambda_1 \ge 1.820$
Time-Periodic:NA$\lambda_1\le 0.606$$0.607 \le \lambda_1 \le 1.817$$\lambda_1\ge 1.818$
Space-Periodic:NA$\lambda_1 \le 0.533$$0.534 \le \lambda_1 \le 1.817$$\lambda_1 \ge 1.818$
3.0Homogeneous:NA$\lambda_1\le 0.609$$0.610 \le \lambda_1 \le 2.799$$\lambda_1 \ge 2.800$
Time-Periodic:NA$\lambda_1\le 0.603$$0.604 \le \lambda_1 \le 2.796$$\lambda_1\ge 2.797$
Space-Periodic:NA$\lambda_1 \le 0.530$$0.531 \le \lambda_1 \le 2.800$$\lambda_1 \ge 2.801$
4.0Homogeneous:NA$\lambda_1\le 0.606$$0.607 \le \lambda_1 \le 3.782$$\lambda_1 \ge 3.783$
Time-Periodic:NA$\lambda_1\le 0.601$$0.602 \le \lambda_1 \le 3.779$$\lambda_1\ge 3.780$
Space-Periodic:NA$\lambda_1 \le 0.527$$0.528 \le \lambda_1 \le 3.784$$\lambda_1 \ge 3.785$
5.0Homogeneous:NA$\lambda_1\le 0.604$$0.605 \le \lambda_1 \le 4.768$$\lambda_1 \ge 4.769$
Time-Periodic:NA$\lambda_1\le 0.599$$0.600 \le \lambda_1 \le 4.765$$\lambda_1\ge 4.766$
Space-Periodic:NA$\lambda_1 \le 0.525$$0.526 \le \lambda_1 \le 4.770$$\lambda_1 \ge 4.771$
Table 3.3.  From coexistence to vanishing of $v$ due to time-periodic variation ($T = 1, \; \sigma_1 = \sigma_2 = 1$)
$\lambda_1$ $\lambda_2$ Time-Periodic Case Homogeneous Case
$1$ $1.152$ Vanishing of $v$ Chase-and-run coexistence
$2$ $2.185$
$3$ $3.205$
$4$ $4.223$
$5$ $5.235$
$6$ $6.247$
$7$ $7.257$
$10$ $10.280$
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$\lambda_1$ $\lambda_2$ Time-Periodic Case Homogeneous Case
$1$ $1.152$ Vanishing of $v$ Chase-and-run coexistence
$2$ $2.185$
$3$ $3.205$
$4$ $4.223$
$5$ $5.235$
$6$ $6.247$
$7$ $7.257$
$10$ $10.280$
Table 3.4.  Change of long-time behaviour in time-periodic environment as $\sigma_1$ and $\sigma_2$ vary, with a selection of initial data
$\lambda_2$$\lambda_1$$\sigma_1$$\sigma_2$ Time-Periodic Case Homogeneous Case
10.60211chase-and-run coexistencevanishing of $u$
0.11vanishing of $u$
0.86911vanishing of $v$chase-and-run coexistence
0.010.01chase-and-run coexistence
20.61111chase-and-run coexistencevanishing of $u$
0.11vanishing of $u$
1.81911vanishing of $v$chase-and-run coexistence
0.010.01chase-and-run coexistence
30.60911chase-and-run coexistencevanishing of $u$
0.11vanishing of $u$
2.79911vanishing of $v$chase-and-run coexistence
0.010.01chase-and-run coexistence
40.60611chase-and-run coexistencevanishing of $u$
0.11vanishing of $u$
3.78211vanishing of $v$chase-and-run coexistence
0.010.01chase-and-run coexistence
50.60411chase-and-run coexistencevanishing of $u$
0.11vanishing of $u$
4.76811vanishing of $v$chase-and-run coexistence
0.010.01chase-and-run coexistence
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$\lambda_2$$\lambda_1$$\sigma_1$$\sigma_2$ Time-Periodic Case Homogeneous Case
10.60211chase-and-run coexistencevanishing of $u$
0.11vanishing of $u$
0.86911vanishing of $v$chase-and-run coexistence
0.010.01chase-and-run coexistence
20.61111chase-and-run coexistencevanishing of $u$
0.11vanishing of $u$
1.81911vanishing of $v$chase-and-run coexistence
0.010.01chase-and-run coexistence
30.60911chase-and-run coexistencevanishing of $u$
0.11vanishing of $u$
2.79911vanishing of $v$chase-and-run coexistence
0.010.01chase-and-run coexistence
40.60611chase-and-run coexistencevanishing of $u$
0.11vanishing of $u$
3.78211vanishing of $v$chase-and-run coexistence
0.010.01chase-and-run coexistence
50.60411chase-and-run coexistencevanishing of $u$
0.11vanishing of $u$
4.76811vanishing of $v$chase-and-run coexistence
0.010.01chase-and-run coexistence
Table 3.5.  Change of long-time behaviour in space-periodic environment as $\sigma_1$ and $\sigma_2$ vary, with a selection of initial data
$ \lambda_1 $ $ \lambda_2 $ $ \sigma_1 $ $ \sigma_2 $ Space-Periodic Case Homogeneous Case
1 1.168 1 1 vanishing of $ v $ chase-and-run coexistence
2 chase-and-run coexistence
2 2.182 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
3 3.202 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
4 4.2202 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
5 5.233 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
6 6.244 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
7 7.253 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
10 10.276 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
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$ \lambda_1 $ $ \lambda_2 $ $ \sigma_1 $ $ \sigma_2 $ Space-Periodic Case Homogeneous Case
1 1.168 1 1 vanishing of $ v $ chase-and-run coexistence
2 chase-and-run coexistence
2 2.182 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
3 3.202 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
4 4.2202 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
5 5.233 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
6 6.244 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
7 7.253 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
10 10.276 1 1 vanishing of $ v $ vanishing of $ v $
2 chase-and-run coexistence
Table 3.6.  Change of long-time behaviour in a time-periodic environment when the time period $T$ is varied while $\sigma_1 = \sigma_2 = 1$ and $\lambda_1 = 1$, $\lambda_2 = 1.153$
$ T $ Time-Periodic Case Homogeneous Case
$ 0.2 $ Chase-and-run coexistence Chase-and-run coexistence
$ 0.4 $
$ 0.6 $
$ 0.8 $
$ 1.0 $
$ 1.1 $ Vanishing of $ v $
$ 1.5 $
$ 1.8 $
$ 1.9 $ Chase-and-run coexistence
$ 2 $
$ 5 $
$ 10 $
$ 15 $
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$ T $ Time-Periodic Case Homogeneous Case
$ 0.2 $ Chase-and-run coexistence Chase-and-run coexistence
$ 0.4 $
$ 0.6 $
$ 0.8 $
$ 1.0 $
$ 1.1 $ Vanishing of $ v $
$ 1.5 $
$ 1.8 $
$ 1.9 $ Chase-and-run coexistence
$ 2 $
$ 5 $
$ 10 $
$ 15 $
Table 3.7.  Change of long-time behaviour in space-periodic environment when the space period $L$ is varied while $\sigma_1 = \sigma_2 = 1$ and $\lambda_1 = 1$, $\lambda_2 = 1.169$
$ L $ Space-Periodic Case Homogeneous Case
$ 0.2 $ Chase-and-run coexistence Chase-and-run coexistence
$ 0.4 $
$ 0.6 $
$ 0.8 $
$ 1.0 $
$ 1.1 $
$ 1.5 $
$ 1.6 $ Vanishing of $ v $
$ 1.8 $
$ 2 $
$ 2.2 $
$ 2.3 $ Chase-and-run coexistence
$ 2.5 $
$ 2.8 $
$ 3.0 $
$ 3.2 $
$ 3.3 $ Vanishing of $ v $
$ 5 $
$ 10 $
$ 15 $
$ 20 $
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$ L $ Space-Periodic Case Homogeneous Case
$ 0.2 $ Chase-and-run coexistence Chase-and-run coexistence
$ 0.4 $
$ 0.6 $
$ 0.8 $
$ 1.0 $
$ 1.1 $
$ 1.5 $
$ 1.6 $ Vanishing of $ v $
$ 1.8 $
$ 2 $
$ 2.2 $
$ 2.3 $ Chase-and-run coexistence
$ 2.5 $
$ 2.8 $
$ 3.0 $
$ 3.2 $
$ 3.3 $ Vanishing of $ v $
$ 5 $
$ 10 $
$ 15 $
$ 20 $
Table 3.8.  Average spreading speeds of $u$ and $v$ at time $t = 120$ with $\sigma_1 = \sigma_2 = 1$ and different values of $T$
$T$ Time-Periodic $ \overline{s_1'(120)}$ Time-Periodic $\overline{s'_2(120)}$
$0.2$ 0.045093 0.364358
$0.4$ 0.045099 0.364336
$0.6$ 0.045118 0.364294
$0.8$ 0.045158 0.364233
$1.0$ 0.045224 0.364154
$1.1$ 0.056211 0
$1.2$ 0.056282 0
$1.4$ 0.056449 0
$1.5$ 0.056545 0
$1.6$ 0.056648 0
$1.8$ 0.056874 0
$1.9$ 0.045852 0.363676
$2$ 0.045949 0.363617
$5$ 0.048875 0.362510
$10$ 0.050970 0.362933
$15$ 0.051696 0.363868
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$T$ Time-Periodic $ \overline{s_1'(120)}$ Time-Periodic $\overline{s'_2(120)}$
$0.2$ 0.045093 0.364358
$0.4$ 0.045099 0.364336
$0.6$ 0.045118 0.364294
$0.8$ 0.045158 0.364233
$1.0$ 0.045224 0.364154
$1.1$ 0.056211 0
$1.2$ 0.056282 0
$1.4$ 0.056449 0
$1.5$ 0.056545 0
$1.6$ 0.056648 0
$1.8$ 0.056874 0
$1.9$ 0.045852 0.363676
$2$ 0.045949 0.363617
$5$ 0.048875 0.362510
$10$ 0.050970 0.362933
$15$ 0.051696 0.363868
Table 3.9.  Average spreading speeds of $u$ and $v$ with $\sigma_1 = \sigma_2 = 1$ and different values of the length of space-period $L$ at time $t = 120$
Space period $L$ Average speed of $u$ $\overline{s_1'(120)}/\overline{s_1'(300)}/\overline{s_1'(500)} $ Average speed of $v$ $\overline{s'_2(120)}/\overline{s'_2(300)}/\overline{s'_2(500)}$
$0.2$ 0.045131 0.364630
$0.6$ 0.045402 0.366188
$1.0$ 0.045908 0.369344
$1.5$ 0.046770 0.374485
$1.6$ 0.058183 0.013643 ($v$ is vanishing)
$1.8$ 0.057517 0.012500 ($v$ is vanishing)
$2.0$ 0.058908$^*$ 0.006666$^*$ ($v$ is vanishing)
$2.1$ 0.059048$^*$ 0.007000$^*$ ($v$ is vanishing)
$2.2$ 0.059164$^*$ 0.007333$^*$ ($v$ is vanishing)
$2.3$ 0.047889$^*$ 0.381331$^*$
$2.5$ 0.047982$^*$ 0.382087$^*$
$2.8$ 0.047892$^*$ 0.382278$^*$
$3.0$ 0.047668$^*$ 0.381776$^*$
$3.2$ 0.047319$^*$ 0.380748$^*$
$3.3$ 0.058552$^*$ 0.011000$^*$ ($v$ is vanishing)
$3.4$ 0.058332$^*$ 0.011333$^*$ ($v$ is vanishing)
$4$ 0.056601$^*$ 0.013333$^*$ ($v$ is vanishing)
$5$ 0.052802$^*$ 0.016666$^*$ ($v$ is vanishing)
$10$ 0.020768$^{**}$ 0.020000$^{**}$ ($v$ is vanishing)
$15$ 0.030000$^{**}$ 0.030000$^{**}$ ($v$ is vanishing)
$20$ 0.038608$^{**}$ 0.034420$^{**}$ ($v$ is vanishing)
* These average speeds are at t = 300.
** These average speeds are at t = 500.
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Space period $L$ Average speed of $u$ $\overline{s_1'(120)}/\overline{s_1'(300)}/\overline{s_1'(500)} $ Average speed of $v$ $\overline{s'_2(120)}/\overline{s'_2(300)}/\overline{s'_2(500)}$
$0.2$ 0.045131 0.364630
$0.6$ 0.045402 0.366188
$1.0$ 0.045908 0.369344
$1.5$ 0.046770 0.374485
$1.6$ 0.058183 0.013643 ($v$ is vanishing)
$1.8$ 0.057517 0.012500 ($v$ is vanishing)
$2.0$ 0.058908$^*$ 0.006666$^*$ ($v$ is vanishing)
$2.1$ 0.059048$^*$ 0.007000$^*$ ($v$ is vanishing)
$2.2$ 0.059164$^*$ 0.007333$^*$ ($v$ is vanishing)
$2.3$ 0.047889$^*$ 0.381331$^*$
$2.5$ 0.047982$^*$ 0.382087$^*$
$2.8$ 0.047892$^*$ 0.382278$^*$
$3.0$ 0.047668$^*$ 0.381776$^*$
$3.2$ 0.047319$^*$ 0.380748$^*$
$3.3$ 0.058552$^*$ 0.011000$^*$ ($v$ is vanishing)
$3.4$ 0.058332$^*$ 0.011333$^*$ ($v$ is vanishing)
$4$ 0.056601$^*$ 0.013333$^*$ ($v$ is vanishing)
$5$ 0.052802$^*$ 0.016666$^*$ ($v$ is vanishing)
$10$ 0.020768$^{**}$ 0.020000$^{**}$ ($v$ is vanishing)
$15$ 0.030000$^{**}$ 0.030000$^{**}$ ($v$ is vanishing)
$20$ 0.038608$^{**}$ 0.034420$^{**}$ ($v$ is vanishing)
* These average speeds are at t = 300.
** These average speeds are at t = 500.
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