Hostname: page-component-7f64f4797f-l842n Total loading time: 0 Render date: 2025-11-05T07:51:45.062Z Has data issue: false hasContentIssue false
  • English
  • Français

Impact of birth pulse and environment shift on population survival and propagation

Part of: Parabolic equations and systems Partial differential equations Representations of solutions

Published online by Cambridge University Press:  17 October 2025

Yaobin Tang ,
Binxiang Dai Open the ORCID record for Binxiang Dai [Opens in a new window]  and
Jianhong Wu Open the ORCID record for Jianhong Wu [Opens in a new window]
Yaobin Tang
Affiliation:
Central South University , China e-mail: tangyaobin@csu.edu.cn
Binxiang Dai*
Affiliation:
Central South University , China e-mail: tangyaobin@csu.edu.cn
Jianhong Wu
Affiliation:
York University , Canada e-mail: wujh@yorku.ca
*
e-mail: bxdai@csu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the propagation dynamics of a single species with a birth pulse and living in a shifting environment driven by climate change. We describe how birth pulse and environment shift jointly impact the propagation properties. We show that a moderate environment shifting speed promotes the spatial–temporal propagation represented by a stable forced KPP wave, and that the birth pulse shrinks the survival region.

Keywords

Shifting environmentimpulsive equationforced KPP wavesspreading properties

MSC classification

Primary: 35K57: Reaction-diffusion equations 35C07: Traveling wave solutions 35B40: Asymptotic behavior of solutions 92D25: Population dynamics (general)

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 27
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This research was supported by the National Natural Science Foundation of China (Grant No. 12271525); the NSERC-Sanofi Industrial Research Research Chair Program “Vaccine Mathematics, Modelling and Manufacturing” (Grant No. 517504); the Discovery Grant of the Natural Science and Engineering Research Council of Canada (Grant No. 105588); and the Fundamental Research Funds for the Central Universities of Central South University (Grant No. CX20230218).

References

Alan Pounds, J., Bustamante, M. R., Coloma, L. A., Consuegra, J. A., Fogden, M. P. L., Foster, P. N., La Marca, E., Masters, K. L., Merino-Viteri, A., Puschendorf, R., Ron, S. R., Sánchez-Azofeifa, G. A., Still, C. J., and Young, B. E., Widespread amphibian extinctions from epidemic disease driven by global warming . Nature 439(2006), no. 7073, 161–167.Google Scholar
Aronson, D. G. and Weinberger, H. F., Nonlinear diffusion in population genetics, combustion and nerve propagation . In: J. A. Goldstein (ed.), Partial differential equations and related topics: Ford Foundation Sponsored Program at Tulane University, Springer, Berlin, Heidelberg, 1974, pp. 5–49.Google Scholar
Aronson, D. G. and Weinberger, H. F., Multidimensional nonlinear diffusion arising in population genetics . Adv. Math. 30(1978), no. 1, 33–76.Google Scholar
Bai, Z., Lou, Y., and Zhao, X. Q., Spatial dynamics of species with annually synchronized emergence of adults . J. Nonlinear Sci. 32(2022), no. 6, 78.Google Scholar
Berestycki, H., Diekmann, O., Nagelkerke, C. J., and Zegeling, P. A., Can a species keep pace with a shifting climate? Bull. Math. Biol. 71(2009), 399–429.Google Scholar
Berestycki, H. and Fang, J., Forced waves of the Fisher-KPP equation in a shifting environment . J. Differ. Equ. 264(2018), no. 3, 2157–2183.Google Scholar
Berestycki, H. and Rossi, L., Reaction-diffusion equations for population dynamics with forced speed I: The case of the whole space . Discrete Contin. Dyn. Syst. 21(2008), no. 1, 41–67.Google Scholar
Buedo-Fernández, S. and Faria, T., Positive periodic solutions for impulsive differential equations with infinite delay and applications to integro-differential equations . Math. Meth. Appl. Sci. 43(2020), no. 6, 3052–3075.Google Scholar
Cantrell, R. S. and Cosner, C., Spatial ecology via reaction-diffusion equations. John Wiley & Sons, Chichester, 2004.Google Scholar
Fang, J., Peng, R., and Zhao, X. Q., Propagation dynamics of a reaction-diffusion equation in a time-periodic shifting environment . J. Math. Pure Appl. 147(2021), 1–28.Google Scholar
Fang, J. and Zhao, X. Q., Traveling waves for monotone semiflows with weak compactness . SIAM J. Math. Anal. 46(2014), no. 6, 3678–3704.Google Scholar
Faria, T. and Figueroa, R., Positive periodic solutions for systems of impulsive delay differential equations . Discrete Contin. Dyn. Syst. Ser. B 28(2023), no. 1, 170–196.Google Scholar
Fazly, M., Lewis, M., and Wang, H., On impulsive reaction-diffusion models in higher dimensions . SIAM J. Appl. Math. 77(2017), no. 1, 224–246.Google Scholar
Fazly, M., Lewis, M., and Wang, H., Analysis of propagation for impulsive reaction-diffusion models . SIAM J. Appl. Math. 80(2020), no. 1, 521–542.Google Scholar
Lewis, M. A. and Li, B., Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models . Bull. Math. Biol. 74(2012), 2383–2402.Google Scholar
Li, B., Bewick, S., Shang, J., and Fagan, W. F., Persistence and spread of a species with a shifting habitat edge . SIAM J. Appl. Math. 74(2014), no. 5, 1397–1417.Google Scholar
Liang, X., Yi, Y., and Zhao, X. Q., Spreading speeds and traveling waves for periodic evolution systems . J. Differ. Equ. 231(2006), no. 1, 57–77.Google Scholar
Lin, Y. and Wang, Q. R., Spreading speed and traveling wave solutions in impulsive reaction-diffusion models . Commun. Nonlinear Sci. Numer. Simul. 23(2015), nos. 1–3, 185–191.Google Scholar
Meng, Y., Ge, J., and Lin, Z., Dynamics of a free boundary problem modelling species invasion with impulsive harvesting . Discrete Contin. Dyn. Syst. B 27(2022), no. 12, 7689–7720.Google Scholar
Meng, Y., Lin, Z., and Pedersen, M., Effects of impulsive harvesting and an evolving domain in a diffusive logistic model . Nonlinearity 34(2021), no. 10, 7005–7029.Google Scholar
Murray, J. D., Mathematical biology I: An introduction. Springer, Berlin, 2002.Google Scholar
Murray, J. D., Mathematical biology II: Spatial models and biomedical applications. Springer, New York, NY, 2003.Google Scholar
Okubo, A. and Levin, S. A., Diffusion and ecological problems: Modern perspectives. Springer, New York, NY, 2001.Google Scholar
Peng, R. and Wei, D., The periodic-parabolic logistic equation on ${\mathbb{R}}^N$ . Discrete Contin. Dyn. Syst. 32(2012), no. 2, 619–641.Google Scholar
Potapov, A. B. and Lewis, M. A., Climate and competition: The effect of moving range boundaries on habitat invasibility . Bull. Math. Biol. 66(2004), no. 5, 975–1008.Google Scholar
Shen, W., Shen, Z., Xue, S., and Zhou, D., Population dynamics under climate change: Persistence criterion and effects of fluctuations . J. Math. Biol. 84(2022), no. 4, 30.Google Scholar
Shen, W. and Xue, S., Forced waves of parabolic-elliptic Keller-Segel models in shifting environments . J. Dyn. Differ. Equ. 34(2022), no. 4, 3057–3088.Google Scholar
Shigesada, N. and Kawasaki, K., Biological invasions: Theory and practice. Oxford University Press, Oxford, 1997.Google Scholar
Skellam, J. G., Random dispersal in theoretical populations . Biometrika 38(1951), nos. 1–2, 196–218.Google Scholar
Stanton, J. C., Shoemaker, K. T., Pearson, R. G., and Akçakaya, H. R., Warning times for species extinctions due to climate change . Glob. Change Biol. 21(2015), no. 3, 1066–1077.Google Scholar
Thomas, C. D., Hill, J. K., Anderson, B. J., Bailey, S., Beale, C. M., Bradbury, R. B., Bulman, C. R., Crick, H. Q. P., Eigenbrod, F., Griffiths, H. M., Kunin, W. E., Oliver, T. H., Walmsley, C. A., Watts, K., Worsfold, N. T., and Yardley, T., A framework for assessing threats and benefits to species responding to climate change . Methods Ecol. Evol. 2(2011), no. 2, 125–142.Google Scholar
Vasilyeva, O., Lutscher, F., and Lewis, M., Analysis of spread and persistence for stream insects with winged adult stages . J. Math. Biol. 72(2016), 851–875.Google Scholar
Vo, H. H., Persistence versus extinction under a climate change in mixed environments . J. Differ. Equ. 259(2015), no. 10, 4947–4988.Google Scholar
Wang, M., Zhang, Y., and Huang, Q., A stage-structured continuous-/discrete-time population model: persistence and spatial spread . Bull. Math. Biol. 84(2022), no. 11, 135.Google Scholar
Wang, Z., An, Q., and Wang, H., Properties of traveling waves in an impulsive reaction-diffusion model with overcompensation . Z. Angew. Math. Phys. 74(2023), no. 3, 114.Google Scholar
Wang, Z., Salmaniw, Y., and Wang, H., Persistence and propagation of a discrete-time map and PDE hybrid model with strong Allee effect . Nonlinear Anal. Real World Appl. 61(2021), 103336.Google Scholar
Wang, Z. and Wang, H., Persistence and propagation of a PDE and discrete-time map hybrid animal movement model with habitat shift driven by climate change . SIAM J. Appl. Math. 80(2020), no. 6, 2608–2630.Google Scholar
Wang, Z. and Wang, H., Bistable traveling waves in impulsive reaction-advection-diffusion models . J. Differ. Equ. 285(2021), 17–39.Google Scholar
Wu, R. and Zhao, X. Q., Spatial invasion of a birth pulse population with nonlocal dispersal . SIAM J. Appl. Math. 79(2019), no. 3, 1075–1097.Google Scholar
Wu, R. and Zhao, X. Q., The evolution dynamics of an impulsive hybrid population model with spatial heterogeneity . Commun. Nonlinear Sci. Numer. Simul. 107(2022), 106181.Google Scholar
Zhang, G. B. and Zhao, X. Q., Propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat . J. Differ. Equ. 268(2020), no. 6, 2852–2885.Google Scholar
Zhang, Y., Yi, T., and Chen, Y., Spreading dynamics of an impulsive reaction-diffusion model with shifting environments . J. Differ. Equ. 381(2024), 1–19.Google Scholar
Zhang, Y., Yi, T., and Wu, J., Global population propagation dynamics of reaction-diffusion models with shifting environment for non-monotone kinetics and birth pulse . J. Differ. Equ. 402(2024), 290–314.Google Scholar
Zhao, X. Q., Dynamical systems in population biology. 2nd ed., Springer, New York, NY, 2017.Google Scholar