Free boundary problem for a nonlocal time-periodic diffusive competition model

Qiaoling Chen; Fengquan Li; Sanyi Tang; Feng Wang; Qiaoling Chen; Fengquan Li; Sanyi Tang; Feng Wang

doi:10.3934/mbe.2023735

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 9: 16471-16505. doi: 10.3934/mbe.2023735

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Free boundary problem for a nonlocal time-periodic diffusive competition model

1.
School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710062, China
2.
School of Science, Xi'an Polytechnic University, Xi'an 710048, China
3.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
4.
School of Mathematics and Statistics, Xidian University, Xi'an 710071, China

Academic Editor: Zhi-Cheng Wang

Received: 19 June 2023 Revised: 26 July 2023 Accepted: 08 August 2023 Published: 16 August 2023

In this paper we consider a free boundary problem for a nonlocal time-periodic competition model. One species is assumed to adopt nonlocal dispersal, and the other one adopts mixed dispersal, which is a combination of both random dispersal and nonlocal dispersal. We first prove the global well-posedness of solutions to the free boundary problem with more general growth functions, and then discuss the spreading and vanishing phenomena. Moreover, under the weak competition condition, we study the long-time behaviors of solutions for the spreading case.
- free boundary problem,
- competition model,
- time-periodic,
- mixed dispersal,
- spreading and vanishing
Citation: Qiaoling Chen, Fengquan Li, Sanyi Tang, Feng Wang. Free boundary problem for a nonlocal time-periodic diffusive competition model[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 16471-16505. doi: 10.3934/mbe.2023735

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Abstract

In this paper we consider a free boundary problem for a nonlocal time-periodic competition model. One species is assumed to adopt nonlocal dispersal, and the other one adopts mixed dispersal, which is a combination of both random dispersal and nonlocal dispersal. We first prove the global well-posedness of solutions to the free boundary problem with more general growth functions, and then discuss the spreading and vanishing phenomena. Moreover, under the weak competition condition, we study the long-time behaviors of solutions for the spreading case.

References

[1]	C.Y. Kao, Y. Lou, W. X. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. B, 17 (2012), 2047–2072. https://doi.org/10.3934/dcdsb.2012.17.2047 doi: 10.3934/dcdsb.2012.17.2047
[2]	X. L. Bai, F. Li, Classification of global dynamics of competition models with nonlocal dispersals Ⅰ: Symmetric kernels, Calc. Var. Partial Differ. Equations, 57 (2018), 144. https://doi.org/10.1007/s00526-018-1419-6 doi: 10.1007/s00526-018-1419-6
[3]	Y. H. Du, M. X. Wang, M. Zhao, Two species nonlocal diffusion systems with free boundaries, Discrete Contin. Dyn. Syst., 42 (2022), 1127–1162. https://doi.org/10.3934/dcds.2021149 doi: 10.3934/dcds.2021149
[4]	J. P. Wang, M. X. Wang, Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior, Discrete Contin. Dyn. Syst. B, 25 (2020), 4721–4736. https://doi.org/10.3934/dcdsb.2020121 doi: 10.3934/dcdsb.2020121
[5]	J. P. Wang, M. X. Wang, Free boundary problems with nonlocal and local diffusions Ⅰ: Global solution, J. Math. Anal. Appl., 490 (2020), 123974. https://doi.org/10.1016/j.jmaa.2020.123974 doi: 10.1016/j.jmaa.2020.123974
[6]	J. F. Cao, W. T. Li, J. Wang, M. Zhao, The Dynamics of a Lotka-Volterra competition model with nonlocal diffusion and free boundaries, Adv. Differ. Equations, 26 (2021), 163–200. https://doi.org/10.57262/ade026-0304-163 doi: 10.57262/ade026-0304-163
[7]	Y. H. Du, W. J. Ni, Analysis of a West Nile virus model with nonlocal diffusion and free boundaries, Nonlinearity, 33 (2020), 4407–4448. https://doi.org/10.1088/1361-6544/ab8bb2 doi: 10.1088/1361-6544/ab8bb2
[8]	M. Zhao, Y. Zhang, W. T. Li, Y. H. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differ. Equations, 269 (2020), 3347–3386. https://doi.org/10.1016/j.jde.2020.02.029 doi: 10.1016/j.jde.2020.02.029
[9]	J. F. Cao, Y. Du, F. Li, W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772–2814. https://doi.org/10.1016/j.jfa.2019.02.013 doi: 10.1016/j.jfa.2019.02.013
[10]	Y. H. Du, Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377–405. https://doi.org/10.1137/090771089 doi: 10.1137/090771089
[11]	Y. H. Du, F. Li, M. L. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, J. Math. Pures Appl., 154 (2021), 30–66. https://doi.org/10.1016/j.matpur.2021.08.008 doi: 10.1016/j.matpur.2021.08.008
[12]	W. Y. Zhang, Z. H. Liu, L. Zhou, Dynamics of a nonlocal diffusive Logistic model with free boundaries in time periodic environment, Discrete Contin. Dyn. Syst. B, 26 (2021), 3767–3784. https://doi.org/10.3934/dcdsb.2020256 doi: 10.3934/dcdsb.2020256
[13]	W. W. Ding, R. Peng, L. Wei, The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment, J. Differ. Equations, 263 (2017), 2736–2779. https://doi.org/10.1016/j.jde.2017.04.013 doi: 10.1016/j.jde.2017.04.013
[14]	Y. H. Du, Z. M. Guo, R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089–2142. https://doi.org/10.1016/j.jfa.2013.07.016 doi: 10.1016/j.jfa.2013.07.016
[15]	C. X. Lei, Z. G. Lin, Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differ. Equations, 257 (2014), 145–166. https://doi.org/10.1016/j.jde.2014.03.015 doi: 10.1016/j.jde.2014.03.015
[16]	M. X. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483–508. https://doi.org/10.1016/j.jfa.2015.10.014 doi: 10.1016/j.jfa.2015.10.014
[17]	P. Zhou, D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differ. Equations, 256 (2014), 1927–1954. https://doi.org/10.1016/j.jde.2013.12.008 doi: 10.1016/j.jde.2013.12.008
[18]	H. Gu, Z. G. Lin, B. D. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Am. Math. Soc., 143 (2015), 1109–1117. https://doi.org/10.1090/S0002-9939-2014-12214-3 doi: 10.1090/S0002-9939-2014-12214-3
[19]	C. H. Wu, Biased movement and the ideal free distribution in some free boundary problems, J. Differ. Equations, 265 (2018), 4251–4282. https://doi.org/10.1016/j.jde.2018.06.002 doi: 10.1016/j.jde.2018.06.002
[20]	N. K. Sun, J. Fang, Propagation dynamics of Fisher-KPP equation with time delay and free boundaries, Calc. Var. Partial Differ. Equations, 58 (2019), 148. https://doi.org/10.1007/s00526-019-1599-8 doi: 10.1007/s00526-019-1599-8
[21]	Y. H. Du, B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673–2724. https://doi.org/10.4171/JEMS/568 doi: 10.4171/JEMS/568
[22]	J. L. Ren, D. D. Zhu, On a reaction-advection-diffusion equation with double free boundaries and mth-order Fisher non-linearity, IMA J. Appl. Math., 84 (2019), 197–227. https://doi.org/10.1093/imamat/hxy057 doi: 10.1093/imamat/hxy057
[23]	G. Bunting, Y. H. Du, K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks Heterogen. Media, 7 (2012), 583–603. https://doi.org/10.3934/nhm.2012.7.583 doi: 10.3934/nhm.2012.7.583
[24]	X. Liu, B. D. Lou, On a reaction-diffusion equation with Robin and free boundary conditions, J. Differ. Equations, 259 (2015), 423–453. https://doi.org/10.1016/j.jde.2015.02.012 doi: 10.1016/j.jde.2015.02.012
[25]	Y. Kaneko, Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction-diffusion equations, Nonlinear Anal. Real World Appl., 18 (2014), 121–140. https://doi.org/10.1016/j.nonrwa.2014.01.008 doi: 10.1016/j.nonrwa.2014.01.008
[26]	Y. H. Du, Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. B, 19 (2014), 3105–3132. https://doi.org/10.3934/dcdsb.2014.19.3105 doi: 10.3934/dcdsb.2014.19.3105
[27]	Y. H. Du, M. X. Wang, M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253–287. https://doi.org/10.1016/j.matpur.2016.06.005 doi: 10.1016/j.matpur.2016.06.005
[28]	J. S. Guo, C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differ. Equations, 24 (2012), 873–895. https://doi.org/10.1007/s10884-012-9267-0 doi: 10.1007/s10884-012-9267-0
[29]	J. S. Guo, C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1–27. https://doi.org/10.1088/0951-7715/28/1/1 doi: 10.1088/0951-7715/28/1/1
[30]	J. Wang, L. Zhang, Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377–398. https://doi.org/10.1016/j.jmaa.2014.09.055 doi: 10.1016/j.jmaa.2014.09.055
[31]	Q. L. Chen, F. Q. Li, F. Wang, A reaction-diffusion-advection competition model with two free boundaries in heterogeneous time-periodic environment, IMA J. Appl. Math., 82 (2017), 445–470. https://doi.org/10.1093/imamat/hxw059 doi: 10.1093/imamat/hxw059
[32]	M. X. Wang, Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67 (2016), 132. https://doi.org/10.1007/s00033-016-0729-9 doi: 10.1007/s00033-016-0729-9
[33]	C. R. Tian, S. G. Ruan, On an advection-reaction-diffusion competition system with double free boundaries modeling invasion and competition of Aedes albopictus and Aedes aegypti mosquitoes, J. Differ. Equations, 265 (2018), 4016–4051. https://doi.org/10.1016/j.jde.2018.05.027 doi: 10.1016/j.jde.2018.05.027
[34]	L. Zhou, S. Zhang, Z. H. Liu, An evolutional free-boundary problem of a reaction-diffusion-advection system, Proc. R. Soc. Edinburgh Sect. A: Math., 147 (2017), 615–648. https://doi.org/10.1017/S0308210516000226 doi: 10.1017/S0308210516000226
[35]	M. X. Wang, On some free boundary problems of the prey-predator model, J. Differ. Equations, 256 (2014), 3365–3394. https://doi.org/10.1016/j.jde.2014.02.013 doi: 10.1016/j.jde.2014.02.013
[36]	M. X. Wang, Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differ. Equations, 264 (2018), 3527–3558. https://doi.org/10.1016/j.jde.2017.11.027 doi: 10.1016/j.jde.2017.11.027
[37]	J. F. Cao, W. T. Li, J. Wang, F. Y. Yang, A free boundary problem of a diffusive SIRS model with nonlinear incidence, Z. Angew. Math. Phys., 68 (2017), 39. https://doi.org/10.1007/s00033-017-0786-8 doi: 10.1007/s00033-017-0786-8
[38]	J. Ge, K. Kim, Z. G. Lin, H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differ. Equations, 259 (2015), 5486–5509. https://doi.org/10.1016/j.jde.2015.06.035 doi: 10.1016/j.jde.2015.06.035
[39]	Z. G. Lin, H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381–1409. https://doi.org/10.1007/s00285-017-1124-7 doi: 10.1007/s00285-017-1124-7
[40]	Q. L. Chen, F. Q. Li, Z. D. Teng, F. Wang, Global dynamics and asymptotic spreading speeds for a partially degenerate epidemic model with time-delay and free boundaries, J. Dyn. Differ. Equations, 34 (2022), 1209–1236. https://doi.org/10.1007/s10884-020-09934-4 doi: 10.1007/s10884-020-09934-4
[41]	Q. X. Ye, Z. Y. Li, M. X. Wang, Y. P. Wu, Introduction to Reaction Diffusion Equations (in Chinese), 2nd edition, Science Press, Beijing, 2011.
[42]	R. Wang, Y. H. Du, Long-time dynamics of a diffusive epidemic model with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2201–2238. https://doi.org/10.3934/dcdsb.2020360 doi: 10.3934/dcdsb.2020360
[43]	J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differ. Equations, 249 (2010), 2921–2953. https://doi.org/10.1016/j.jde.2010.07.003 doi: 10.1016/j.jde.2010.07.003
[44]	Z. W. Shen, H. Vo, Nonlocal dispersal equations in time-periodic media: Principal spectral theory, limiting properties and long-time dynamics, J. Differ. Equations, 267 (2019), 1423–1466. https://doi.org/10.1016/j.jde.2019.02.013 doi: 10.1016/j.jde.2019.02.013
[45]	P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Harlow, Essex: Longman Scientific and Technical, 1991.
[46]	P. Bates, G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428–440. https://doi.org/10.1016/j.jmaa.2006.09.007 doi: 10.1016/j.jmaa.2006.09.007
[47]	M. Marcus, L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: The subcritical case, Arch. Ration. Mech. Anal., 144 (1998), 201–231. https://doi.org/10.1007/s002050050116 doi: 10.1007/s002050050116
[48]	J. P. Gao, S. G. Guo, W. X. Shen, Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media, Discrete Contin. Dyn. Syst. B, 26 (2021), 2645–2676. https://doi.org/10.3934/dcdsb.2020199 doi: 10.3934/dcdsb.2020199
[49]	F. X. Chen, Almost periodic traveling waves of nonlocal evolution equations, Nonlinear Anal., 50 (2002), 807–838. https://doi.org/10.1016/S0362-546X(01)00787-8 doi: 10.1016/S0362-546X(01)00787-8
[50]	H. M. Huang, M. X. Wang, A nonlocal SIS epidemic problem with double free boundaries, Z. Angew. Math. Phys., 70 (2019), 109. https://doi.org/10.1007/s00033-019-1156-5 doi: 10.1007/s00033-019-1156-5
[51]	M. X. Wang, J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differ. Equations, 26 (2014), 655–672. https://doi.org/10.1007/s10884-014-9363-4 doi: 10.1007/s10884-014-9363-4

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