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doi: 10.3934/dcdsb.2021266
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Recent developments on spatial propagation for diffusion equations in shifting environments

1. 

School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

2. 

Center for Mathematical Sciences, China University of Geosciences, Wuhan 430074, China

3. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

4. 

School of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

* Corresponding author: Wan-Tong Li

Received  March 2021 Revised  September 2021 Early access November 2021

Fund Project: Wang was partially supported by NSF of China (11901543) and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (CUGSX01). Li was partially supported by NSF of China (11731005). Dong was partially supported by NSF of China (12101171) and the Zhejiang Province Natural Science Foundation (Q22A015337)

In this short review, we describe some recent developments on the spatial propagation for diffusion problems in shifting environments, including single species models, competition/cooperative models and chemotaxis models submitted to classical reaction-diffusion equations (with or without free boundaries), integro-difference equations, lattice differential equations and nonlocal dispersal equations. The considered topics may typically come from modeling the threats associated with global climate change and the worsening of the environment resulting from industrialization which lead to the shifting or translating of the habitat ranges, and also arise indirectly in studying the pathophoresis as well as some multi-stage invasion processes. Some open problems and potential research directions are also presented.

Citation: Jia-Bing Wang, Wan-Tong Li, Fang-Di Dong, Shao-Xia Qiao. Recent developments on spatial propagation for diffusion equations in shifting environments. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021266
References:
[1]

M. AlfaroH. Berestycki and G. Raoul, The effect of climate shift on a species submitted to dispersion, evolution, growth and nonlocal competition, SIAM J. Math. Anal., 49 (2017), 562-596.  doi: 10.1137/16M1075934.  Google Scholar

[2]

M. Alfaro and J. Coville, Propagation phenomena in monostable integro-differential equations: Acceleration or not?, J. Differential Equations, 263 (2017), 5727-5758.  doi: 10.1016/j.jde.2017.06.035.  Google Scholar

[3]

D. Aronson and H. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[4]

H. BerestyckiL. Desvillettes and O. Diekmann, Can climate change lead to gap formation?, Ecol. Complex., 20 (2014), 264-270.  doi: 10.1016/j.ecocom.2014.10.006.  Google Scholar

[5]

H. BerestyckiO. DiekmannC. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.  doi: 10.1007/s11538-008-9367-5.  Google Scholar

[6]

H. Berestycki and J. Fang, Forced waves of the Fisher-KPP equation in a shifting environment, J. Differential Equations, 264 (2018), 2157-2183.  doi: 10.1016/j.jde.2017.10.016.  Google Scholar

[7]

H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space, Discrete Contin. Dyn. Syst., 21 (2008), 41-67.  doi: 10.3934/dcds.2008.21.41.  Google Scholar

[8]

H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed. II. Cylindrical-type domains, Discrete Contin. Dyn. Syst., 25 (2009), 19-61.  doi: 10.3934/dcds.2009.25.19.  Google Scholar

[9]

J. Bouhours and T. Giletti, Spreading and vanishing for a monostable reaction-diffusion equation with forced speed, J. Dynam. Differential Equations, 31 (2019), 247-286.  doi: 10.1007/s10884-018-9643-5.  Google Scholar

[10]

J. Bouhours and T. Giletti, Extinction and spreading of a species under the joint influence of climate change and a weak Allee effect: A two-patch model, arXiv: 1601.06589v1. Google Scholar

[11]

J. Bouhours and G. Nadin, A variational approach to reaction-diffusion equations with forced speed in dimension $1$, Discrete Contin. Dyn. Syst., 35 (2015), 1843-1872.  doi: 10.3934/dcds.2015.35.1843.  Google Scholar

[12]

J. F. CaoY. DuF. 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.  Google Scholar

[13]

X. ChenJ. C. Tsai and Y. Wu, Longtime behavior of solutions of a SIS epidemiological model, SIAM J. Math. Anal., 49 (2017), 3925-3950.  doi: 10.1137/16M1108741.  Google Scholar

[14]

W. ChoiT. Giletti and J. S. Guo, Persistence of species in a predator-prey system with climate change and either nonlocal or local dispersal, J. Differential Equations, 302 (2021), 807-853.  doi: 10.1016/j.jde.2021.09.017.  Google Scholar

[15]

C. Cosner, Challenges in modeling biological invasions and population distributions in a changing climate, Ecol. Complex., 20 (2014), 258-263.  doi: 10.1016/j.ecocom.2014.05.007.  Google Scholar

[16]

J. CovilleJ. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[17]

J. Coville, Can a population survive in a shifting environment using non-local dispersion?, Nonlinear Anal., 212 (2021), 112416.  doi: 10.1016/j.na.2021.112416.  Google Scholar

[18]

J. Coville and F. Hamel, On generalized principal eigenvalues of nonlocal operators with a drift, Nonlinear Anal., 193 (2020), 111569.  doi: 10.1016/j.na.2019.07.002.  Google Scholar

[19]

P. De LeenheerW. Shen and A. Zhang, Persistence and extinction of nonlocal dispersal evolution equations in moving habitats, Nonlinear Anal. Real World Appl., 54 (2020), 103110.  doi: 10.1016/j.nonrwa.2020.103110.  Google Scholar

[20]

F. D. DongB. Li and W. T. Li, Forced waves in a Lotka-Volterra diffusion-competition model with a shifting habitat, J. Differential Equations, 276 (2021), 433-459.  doi: 10.1016/j.jde.2020.12.022.  Google Scholar

[21]

F. D. DongW. T. Li and J. B. Wang, Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application, Discrete Contin. Dyn. Syst., 37 (2017), 6291-6318.  doi: 10.3934/dcds.2017272.  Google Scholar

[22]

F. D. Dong, W. T. Li and J. B. Wang, Propagation phenomena for a nonlocal dispersal Lotka-Volterra competition model in shifting habitats, Preprint, 2021. Google Scholar

[23]

F. D. DongJ. ShangW. F. Fagan and B. Li, Persistence and spread of solutions in a two-species Lotka-Volterra competition-diffusion model with a shifting habitat, SIAM J. Appl. Math., 81 (2021), 1600-1622.  doi: 10.1137/20M1341064.  Google Scholar

[24]

Y. Du, Y. Hu and X. Liang, A climate shift model with free boundary: Enhanced invasion, J. Dynam. Differential Equations, 2021. doi: 10.1007/s10884-021-10031-3.  Google Scholar

[25]

Y. DuF. Li and M. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, J. Math. Pures Appl., 154 (2021), 30-66.  doi: 10.1016/j.matpur.2021.08.008.  Google Scholar

[26]

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.  Google Scholar

[27]

Y. Du and W. Ni, Semi-wave, traveling wave and spreading speed for monostable cooperative systems with nonlocal diffusion and free boundaries, arXiv: 2010.01244v1. Google Scholar

[28]

Y. Du and W. Ni, The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, arXiv: 2102.05286v1. Google Scholar

[29]

Y. DuL. Wei and L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, J. Dynam. Differential Equations, 30 (2018), 1389-1426.  doi: 10.1007/s10884-017-9614-2.  Google Scholar

[30]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar

[31]

A. DucrotT. GilettiJ. S. Guo and M. Shimojo, Asymptotic spreading speeds for a predator-prey system with two predators and one prey, Nonlinearity, 34 (2021), 669-704.  doi: 10.1088/1361-6544/abd289.  Google Scholar

[32]

A. DucrotT. Giletti and H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differential Equations, 58 (2019), 137.  doi: 10.1007/s00526-019-1576-2.  Google Scholar

[33]

A. DucrotJ. S. GuoG. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), 146.  doi: 10.1007/s00033-019-1188-x.  Google Scholar

[34]

S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.  Google Scholar

[35]

J. FangY. Lou and J. Wu, Can pathogen spread keep pace with its host invasion?, SIAM J. Appl. Math., 76 (2016), 1633-1657.  doi: 10.1137/15M1029564.  Google Scholar

[36]

J. FangR. Peng and X. Q. Zhao, Propagation dynamics of a reaction-diffusion equation in a time-periodic shifting environment, J. Math. Pures Appl., 147 (2021), 1-28.  doi: 10.1016/j.matpur.2021.01.001.  Google Scholar

[37]

G. Faye, T. Giletti and M. Holzer, Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity, arXiv: 2103.15466v1. Google Scholar

[38]

S. B. Fey and C. M. Herren, Temperature-mediated biotic interactions influence enemy release of nonnative species in warming environments, Ecology, 95 (2014), 2246-2256.  doi: 10.1890/13-1799.1.  Google Scholar

[39]

D. Finkelshtein and P. Tkachov, Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line, Appl. Anal., 98 (2019), 756-780.  doi: 10.1080/00036811.2017.1400537.  Google Scholar

[40]

R. A. Fisher, The wave of advance of advantageous genes, Annu. Eugen., 7 (1937), 353-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[41]

B. G. Freeman and A. M. C. Freeman, Rapid upslope shifts in new guinean birds illustrate strong distributional responses of tropical montane species to global warming, Proc. Natl. Acad. Sci. USA, 111 (2014), 4490-4494.  doi: 10.1073/pnas.1318190111.  Google Scholar

[42]

R. A. Gardner, Existence of travelling wave solutions of predator-prey systems via the connection index, SIAM J. Appl. Math., 44 (1984), 56-79.  doi: 10.1137/0144006.  Google Scholar

[43]

J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.  Google Scholar

[44]

P. GonzalezR. P. NeilsonJ. M. Lenihan and R. J. Drapek, Global patterns in the vulnerability of ecosystems to vegetation shifts due to climate change, Glob. Ecol. Biogeogr., 19 (2010), 755-768.  doi: 10.1111/j.1466-8238.2010.00558.x.  Google Scholar

[45]

J. S. GuoY. WangC. H. Wu and C. C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373.  Google Scholar

[46]

J. S. Guo and C. C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.  doi: 10.1016/j.jde.2015.09.036.  Google Scholar

[47]

F. Hamel, Reaction-diffusion problems in cylinders with no invariance by translation. II. Monotone perturbations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 555-596.  doi: 10.1016/S0294-1449(97)80126-6.  Google Scholar

[48] R. J. Hobbs, Invasive species in a changing world, Island Press, London, 2000.   Google Scholar
[49]

M. Holzer and A. Scheel, Accelerated fronts in a two-stage invasion process, SIAM J. Math. Anal., 46 (2014), 397-427.  doi: 10.1137/120887746.  Google Scholar

[50]

C. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations, 259 (2015), 1967-1989.  doi: 10.1016/j.jde.2015.03.025.  Google Scholar

[51]

C. HuJ. Shang and B. Li, Spreading speeds for reaction-diffusion equations with a shifting habitat, J. Dynam. Differential Equations, 32 (2020), 1941-1964.  doi: 10.1007/s10884-019-09796-5.  Google Scholar

[52]

H. HuL. Deng and J. Huang, Traveling wave of a nonlocal dispersal Lotka-Volterra cooperation model under shifting habitat, J. Math. Anal. Appl., 500 (2021), 125100.  doi: 10.1016/j.jmaa.2021.125100.  Google Scholar

[53]

H. HuT. Yi and X. Zou, On spatial-temporal dynamics of a Fisher-KPP equation with a shifting environment, Proc. Amer. Math. Soc., 148 (2020), 213-221.  doi: 10.1090/proc/14659.  Google Scholar

[54]

H. Hu and X. Zou, Existence of an extinction wave in the Fisher equation with a shifting habitat, Proc. Amer. Math. Soc., 145 (2017), 4763-4771.  doi: 10.1090/proc/13687.  Google Scholar

[55]

Y. Hu, X. Hao and Y. Du, Spreading via a free boundary model under shifting climate: Invasion of deteriorated environment, Comm. Contemp. Math., 2021. doi: 10.1142/S0219199720500777.  Google Scholar

[56]

Y. HuX. HaoX. Song and Y. Du, A free boundary problem for spreading under shifting climate, J. Differential Equations, 269 (2020), 5931-5958.  doi: 10.1016/j.jde.2020.04.024.  Google Scholar

[57]

S. F. Iglesias and S. Mirrahimi, Selection and mutation in a shifting and fluctuating environment, Commun. Math. Sci., 19 (2021), 1761-1798.  doi: 10.4310/CMS.2021.v19.n7.a1.  Google Scholar

[58]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[59]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[60]

A. N. KolmogorovI. G. Petrovski 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.   Google Scholar

[61]

K. Y. Lam and X. Yu, Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments, arXiv: 2101.06698v2. Google Scholar

[62]

C. Lei and Y. Du, Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 895-911.  doi: 10.3934/dcdsb.2017045.  Google Scholar

[63]

C. LeiH. NieW. Dong and Y. Du, Spreading of two competing species governed by a free boundary model in a shifting environment, J. Math. Anal. Appl., 462 (2018), 1254-1282.  doi: 10.1016/j.jmaa.2018.02.042.  Google Scholar

[64]

M. A. LewisN. G. Marculis and Z. Shen, Integrodifference equations in the presence of climate change: Persistence criterion, travelling waves and inside dynamics, J. Math. Biol., 77 (2018), 1649-1687.  doi: 10.1007/s00285-018-1206-1.  Google Scholar

[65]

B. LiS. BewickM. R. Barnard and W. F. Fagan, Persistence and spreading speeds of integro-difference equations with an expanding or contracting habitat, Bull. Math. Biol., 78 (2016), 1337-1379.  doi: 10.1007/s11538-016-0180-2.  Google Scholar

[66]

B. LiS. BewickJ. Shang and W. F. Fagan, Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math., 74 (2014), 1397-1417.  doi: 10.1137/130938463.  Google Scholar

[67]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[68]

B. Li and J. Wu, Traveling waves in integro-difference equations with a shifting habitat, J. Differentail Equations, 268 (2020), 4059-4078.  doi: 10.1016/j.jde.2019.10.018.  Google Scholar

[69]

F. LiJ. Coville and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., 37 (2017), 879-903.  doi: 10.3934/dcds.2017036.  Google Scholar

[70]

W. T. LiJ. B. Wang and X. Q. Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, J. Nonlinear Sci., 28 (2018), 1189-1219.  doi: 10.1007/s00332-018-9445-2.  Google Scholar

[71]

G. LiuT. Xu and J. Yin, Forced waves of reaction-diffusion model with density-dependent dispersal in shifting environments, J. Differential Equations, 282 (2021), 127-147.  doi: 10.1016/j.jde.2021.02.027.  Google Scholar

[72]

Y. MengZ. Yu and S. Zhang, Spatial dynamics of the lattice Lotka-Volterra competition system in a shifting habitat, Nonlinear Anal. Real World Appl., 60 (2021), 103287.  doi: 10.1016/j.nonrwa.2020.103287.  Google Scholar

[73]

G. F. Midgley and W. J. Bond, Future of african terrestrial biodiversity and ecosystems under anthropogenic climate change, Nat. Clim. Change, 5 (2015), 823-829.  doi: 10.1038/nclimate2753.  Google Scholar

[74]

D. L. Murray, M. J. L. Peers, Y. N. Majchrzak, M. Wehtje, C. Ferreira, R. S. A. Pickles, J. R. Row and D. H. Thornton, Continental divide: Predicting climate-mediated fragmentation and biodiversity loss in the boreal forest, PLoS ONE, 12 (2017), e0176706. doi: 10.1371/journal.pone.0176706.  Google Scholar

[75]

K. J. Painter, Mathematical models for chemotaxisand their applications in self-organisation phenomena, J. Theor. Biol., 481 (2019), 162-182.  doi: 10.1016/j.jtbi.2018.06.019.  Google Scholar

[76]

Y. PanY. Su and J. Wei, Accelerating propagation in a recursive system arising from seasonal population models with nonlocal dispersal, J. Differential Equations, 267 (2019), 150-179.  doi: 10.1016/j.jde.2019.01.009.  Google Scholar

[77]

L. Y. Pang and S. L. Wu, Propagation dynamics for lattice differential equations in a time-periodic shifting habitat, Z. Angew. Math. Phys., 72 (2021), 93.  doi: 10.1007/s00033-021-01522-w.  Google Scholar

[78]

C. L. ParrE. F. Gray and W. J. Bond, Cascading biodiversity and functional consequences of a global change-induced biome switch, Divers. Distrib., 18 (2012), 493-503.  doi: 10.1111/j.1472-4642.2012.00882.x.  Google Scholar

[79]

J. J. PolovinaJ. P. DunneP. A. Woodworth and E. A. Howell, Projected expansion of the subtropical biome and contraction of the temperate and equatorial upwelling biomes in the North Pacific under global warming, ICES J. Mar. Sci., 68 (2011), 986-995.  doi: 10.1093/icesjms/fsq198.  Google Scholar

[80]

A. B. Potapov and M. A. Lewis, Climate and competition: The effect of moving range boundaries on habitat invasibility, Bull. Math. Biol., 66 (2004), 975-1008.  doi: 10.1016/j.bulm.2003.10.010.  Google Scholar

[81]

S. X. Qiao, W. T. Li and J. B. Wang, Asymptotic propagations of a nonlocal dispersal population model with shifting habitats, European J. Appl. Math., (2021), 1–28. doi: 10.1017/S095679252100019X.  Google Scholar

[82]

S. X. QiaoW. T. Li and J. B. Wang, Multi-type forced waves in nonlocal dispersal KPP equations with shifting habitats, J. Math. Anal. Appl., 505 (2022), 125504.  doi: 10.1016/j.jmaa.2021.125504.  Google Scholar

[83]

S. X. QiaoJ. L. Zhu and J. B. Wang, Asymptotic behaviors of forced waves for the lattice Lotka-Volterra competition system with shifting habitats, Appl. Math. Lett., 118 (2021), 107168.  doi: 10.1016/j.aml.2021.107168.  Google Scholar

[84]

F. J. Rahel and J. D. Olden, Assessing the effects of climate change on aquatic invasive species, Conserv. Biol., 22 (2008), 521-533.  doi: 10.1111/j.1523-1739.2008.00950.x.  Google Scholar

[85]

N. RawalW. Shen and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst., 35 (2015), 1609-1640.  doi: 10.3934/dcds.2015.35.1609.  Google Scholar

[86]

L. RoquesA. RoquesH. Berestycki and A. Kretzschmar, A population facing climate change: Joint influences of Allee effects and environmental boundary geometry, Popul. Ecol., 50 (2008), 215-225.  doi: 10.1007/s10144-007-0073-1.  Google Scholar

[87]

M. SchefferM. HirotaM. HolmgrenE. H. Van Nes and F. S. Chapin, Thresholds for boreal biome transitions, Proc. Natl. Acad. Sci. USA, 109 (2012), 21384-21389.  doi: 10.1073/pnas.1219844110.  Google Scholar

[88]

W. Shen and S. Xue, Persistence and spreading speeds of parabolic-elliptic Keller-Segel models in shifting environments, J. Differential Equations, 269 (2020), 6236-6268.  doi: 10.1016/j.jde.2020.04.040.  Google Scholar

[89]

W. Shen and S. Xue, Forced waves of parabolic-elliptic Keller-Segel models in shifting environments, J. Dynam. Differential Equations, 2021. doi: 10.1007/s10884-020-09924-6.  Google Scholar

[90]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[91]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101.   Google Scholar

[92]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar

[93]

H. H. Vo, Persistence versus extinction under a climate change in mixed environments, J. Differential Equations, 259 (2015), 4947-4988.  doi: 10.1016/j.jde.2015.06.014.  Google Scholar

[94]

G. R. WaltherE. PostP. ConveyA. MenzelC. ParmesanT. J. C. BeebeeJ. M. FromentinO. Hoegh-Guldberg and F. Bairlein, Ecological responses to recent climate change, Nature, 416 (2002), 389-395.  doi: 10.1038/416389a.  Google Scholar

[95]

H. WangC. Pan and C. Ou, Existence of forced waves and gap formations for the lattice Lotka-Volterra competition system in a shifting environment, Appl. Math. Lett., 106 (2020), 106349.  doi: 10.1016/j.aml.2020.106349.  Google Scholar

[96]

H. Wang, C. Pan and C. Ou, Existence, uniqueness and stability of forced waves to the Lotka-Volterra competition system in a shifting environment, Stud. Appl. Math., 2021. doi: 10.1111/sapm.12438.  Google Scholar

[97]

J. B. Wang and W. T. Li, Wave propagation for a cooperative model with nonlocal dispersal under worsening habitats, Z. Angew. Math. Phys., 71 (2020), 147.  doi: 10.1007/s00033-020-01374-w.  Google Scholar

[98]

J. B. Wang and C. Wu, Forced waves and gap formations for a Lotka-Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Anal. Real World Appl., 58 (2021), 103208.  doi: 10.1016/j.nonrwa.2020.103208.  Google Scholar

[99]

J. B. Wang and X. Q. Zhao, Uniqueness and global stability of forced waves in a shifting environment, Proc. Amer. Math. Soc., 147 (2019), 1467-1481.  doi: 10.1090/proc/14235.  Google Scholar

[100]

L. Wei, G. Zhang and M. Zhou, Long time behavior for solutions of the diffusive logistic equation with advection and free boundary, Calc. Var. Partial Differ. Equ., 55 (2016), 34 pp. doi: 10.1007/s00526-016-1039-y.  Google Scholar

[101]

P. Weng and X. Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[102]

C. WuY. Wang and X. Zou, Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment, J. Differential Equations, 267 (2019), 4890-4921.  doi: 10.1016/j.jde.2019.05.019.  Google Scholar

[103]

C. WuY. Yang and Z. Wu, Existence and uniqueness of forced waves in a delayed reaction–diffusion equation in a shifting environment, Nonlinear Anal. Real World Appl., 57 (2021), 103198.  doi: 10.1016/j.nonrwa.2020.103198.  Google Scholar

[104]

W. B. XuW. T. Li and G. Lin, Nonlocal dispersal cooperative systems: Acceleration propagation among species, J. Differential Equations, 268 (2020), 1081-1105.  doi: 10.1016/j.jde.2019.08.039.  Google Scholar

[105]

W. B. XuW. T. Li and S. Ruan, Fast propagation for reaction-diffusion cooperative systems, J. Differential Equations, 265 (2018), 645-670.  doi: 10.1016/j.jde.2018.03.004.  Google Scholar

[106]

W. B. XuW. T. Li and S. Ruan, Spatial propagation in nonlocal dispersal Fisher-KPP equations, J. Funct. Anal., 280 (2021), 108957.  doi: 10.1016/j.jfa.2021.108957.  Google Scholar

[107]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953.  doi: 10.2977/prims/1260476648.  Google Scholar

[108]

Y. YangC. Wu and Z. Li, Forced waves and their asymptotics in a Lotka-Volterra cooperative model under climate change, Appl. Math. Comput., 353 (2019), 254-264.  doi: 10.1016/j.amc.2019.01.058.  Google Scholar

[109]

T. YiY. Chen and J. Wu, Asymptotic propagations of asymptotical monostable type equations with shifting habitats, J. Differential Equations, 269 (2020), 5900-5930.  doi: 10.1016/j.jde.2020.04.025.  Google Scholar

[110]

T. Yi and X. Q. Zhao, Propagation dynamics for monotone evolution systems without spatial translation invariance, J. Funct. Anal., 279 (2020), 108722.  doi: 10.1016/j.jfa.2020.108722.  Google Scholar

[111]

Y. YuanY. Wang and X. Zou, Spatial dynamics of a Lotka-Volterra model with a shifting habitat, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5633-5671.  doi: 10.3934/dcdsb.2019076.  Google Scholar

[112]

Y. Yuan and X. Zou, Spatial-temporal dynamics of a diffusive Lotka–Volterra competition model with a shifting habitat II: Case of faster diffuser being a weaker copetitor, J. Dynam. Differential Equations, 33 (2021), 2091-2132.  doi: 10.1007/s10884-020-09885-w.  Google Scholar

[113]

G. B. Zhang and X. Q. Zhao, Propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat, J. Differential Equations, 268 (2020), 2852-2885.  doi: 10.1016/j.jde.2019.09.044.  Google Scholar

[114]

Z. ZhangW. Wang and J. Yang, Persistence versus extinction for two competing species under a climate change, Nonlinear Anal. Model. Control, 22 (2017), 285-302.  doi: 10.15388/NA.2017.3.1.  Google Scholar

[115]

M. ZhaoY. ZhangW. 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.  Google Scholar

[116]

Y. Zhou and M. Kot, Discrete-time growth-dispersal models with shifting species ranges, Theor. Ecol., 4 (2011), 13-25.  doi: 10.1007/s12080-010-0071-3.  Google Scholar

[117]

Y. Zhou and M. Kot, Life on the move: Modeling the effects of climate-driven range shifts with integrodifference equations, Dispersal, Individual Movement and Spatial Ecology, 2071 (2013), 263-292.  doi: 10.1007/978-3-642-35497-7_9.  Google Scholar

[118]

J. L. Zhu, J. B. Wang and F. D. Dong, Existence of multi-type forced waves in the lattice competition system under shifting habitats, Preprint, 2021. Google Scholar

show all references

References:
[1]

M. AlfaroH. Berestycki and G. Raoul, The effect of climate shift on a species submitted to dispersion, evolution, growth and nonlocal competition, SIAM J. Math. Anal., 49 (2017), 562-596.  doi: 10.1137/16M1075934.  Google Scholar

[2]

M. Alfaro and J. Coville, Propagation phenomena in monostable integro-differential equations: Acceleration or not?, J. Differential Equations, 263 (2017), 5727-5758.  doi: 10.1016/j.jde.2017.06.035.  Google Scholar

[3]

D. Aronson and H. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[4]

H. BerestyckiL. Desvillettes and O. Diekmann, Can climate change lead to gap formation?, Ecol. Complex., 20 (2014), 264-270.  doi: 10.1016/j.ecocom.2014.10.006.  Google Scholar

[5]

H. BerestyckiO. DiekmannC. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.  doi: 10.1007/s11538-008-9367-5.  Google Scholar

[6]

H. Berestycki and J. Fang, Forced waves of the Fisher-KPP equation in a shifting environment, J. Differential Equations, 264 (2018), 2157-2183.  doi: 10.1016/j.jde.2017.10.016.  Google Scholar

[7]

H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space, Discrete Contin. Dyn. Syst., 21 (2008), 41-67.  doi: 10.3934/dcds.2008.21.41.  Google Scholar

[8]

H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed. II. Cylindrical-type domains, Discrete Contin. Dyn. Syst., 25 (2009), 19-61.  doi: 10.3934/dcds.2009.25.19.  Google Scholar

[9]

J. Bouhours and T. Giletti, Spreading and vanishing for a monostable reaction-diffusion equation with forced speed, J. Dynam. Differential Equations, 31 (2019), 247-286.  doi: 10.1007/s10884-018-9643-5.  Google Scholar

[10]

J. Bouhours and T. Giletti, Extinction and spreading of a species under the joint influence of climate change and a weak Allee effect: A two-patch model, arXiv: 1601.06589v1. Google Scholar

[11]

J. Bouhours and G. Nadin, A variational approach to reaction-diffusion equations with forced speed in dimension $1$, Discrete Contin. Dyn. Syst., 35 (2015), 1843-1872.  doi: 10.3934/dcds.2015.35.1843.  Google Scholar

[12]

J. F. CaoY. DuF. 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.  Google Scholar

[13]

X. ChenJ. C. Tsai and Y. Wu, Longtime behavior of solutions of a SIS epidemiological model, SIAM J. Math. Anal., 49 (2017), 3925-3950.  doi: 10.1137/16M1108741.  Google Scholar

[14]

W. ChoiT. Giletti and J. S. Guo, Persistence of species in a predator-prey system with climate change and either nonlocal or local dispersal, J. Differential Equations, 302 (2021), 807-853.  doi: 10.1016/j.jde.2021.09.017.  Google Scholar

[15]

C. Cosner, Challenges in modeling biological invasions and population distributions in a changing climate, Ecol. Complex., 20 (2014), 258-263.  doi: 10.1016/j.ecocom.2014.05.007.  Google Scholar

[16]

J. CovilleJ. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[17]

J. Coville, Can a population survive in a shifting environment using non-local dispersion?, Nonlinear Anal., 212 (2021), 112416.  doi: 10.1016/j.na.2021.112416.  Google Scholar

[18]

J. Coville and F. Hamel, On generalized principal eigenvalues of nonlocal operators with a drift, Nonlinear Anal., 193 (2020), 111569.  doi: 10.1016/j.na.2019.07.002.  Google Scholar

[19]

P. De LeenheerW. Shen and A. Zhang, Persistence and extinction of nonlocal dispersal evolution equations in moving habitats, Nonlinear Anal. Real World Appl., 54 (2020), 103110.  doi: 10.1016/j.nonrwa.2020.103110.  Google Scholar

[20]

F. D. DongB. Li and W. T. Li, Forced waves in a Lotka-Volterra diffusion-competition model with a shifting habitat, J. Differential Equations, 276 (2021), 433-459.  doi: 10.1016/j.jde.2020.12.022.  Google Scholar

[21]

F. D. DongW. T. Li and J. B. Wang, Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application, Discrete Contin. Dyn. Syst., 37 (2017), 6291-6318.  doi: 10.3934/dcds.2017272.  Google Scholar

[22]

F. D. Dong, W. T. Li and J. B. Wang, Propagation phenomena for a nonlocal dispersal Lotka-Volterra competition model in shifting habitats, Preprint, 2021. Google Scholar

[23]

F. D. DongJ. ShangW. F. Fagan and B. Li, Persistence and spread of solutions in a two-species Lotka-Volterra competition-diffusion model with a shifting habitat, SIAM J. Appl. Math., 81 (2021), 1600-1622.  doi: 10.1137/20M1341064.  Google Scholar

[24]

Y. Du, Y. Hu and X. Liang, A climate shift model with free boundary: Enhanced invasion, J. Dynam. Differential Equations, 2021. doi: 10.1007/s10884-021-10031-3.  Google Scholar

[25]

Y. DuF. Li and M. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, J. Math. Pures Appl., 154 (2021), 30-66.  doi: 10.1016/j.matpur.2021.08.008.  Google Scholar

[26]

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.  Google Scholar

[27]

Y. Du and W. Ni, Semi-wave, traveling wave and spreading speed for monostable cooperative systems with nonlocal diffusion and free boundaries, arXiv: 2010.01244v1. Google Scholar

[28]

Y. Du and W. Ni, The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, arXiv: 2102.05286v1. Google Scholar

[29]

Y. DuL. Wei and L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, J. Dynam. Differential Equations, 30 (2018), 1389-1426.  doi: 10.1007/s10884-017-9614-2.  Google Scholar

[30]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar

[31]

A. DucrotT. GilettiJ. S. Guo and M. Shimojo, Asymptotic spreading speeds for a predator-prey system with two predators and one prey, Nonlinearity, 34 (2021), 669-704.  doi: 10.1088/1361-6544/abd289.  Google Scholar

[32]

A. DucrotT. Giletti and H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differential Equations, 58 (2019), 137.  doi: 10.1007/s00526-019-1576-2.  Google Scholar

[33]

A. DucrotJ. S. GuoG. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), 146.  doi: 10.1007/s00033-019-1188-x.  Google Scholar

[34]

S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.  Google Scholar

[35]

J. FangY. Lou and J. Wu, Can pathogen spread keep pace with its host invasion?, SIAM J. Appl. Math., 76 (2016), 1633-1657.  doi: 10.1137/15M1029564.  Google Scholar

[36]

J. FangR. Peng and X. Q. Zhao, Propagation dynamics of a reaction-diffusion equation in a time-periodic shifting environment, J. Math. Pures Appl., 147 (2021), 1-28.  doi: 10.1016/j.matpur.2021.01.001.  Google Scholar

[37]

G. Faye, T. Giletti and M. Holzer, Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity, arXiv: 2103.15466v1. Google Scholar

[38]

S. B. Fey and C. M. Herren, Temperature-mediated biotic interactions influence enemy release of nonnative species in warming environments, Ecology, 95 (2014), 2246-2256.  doi: 10.1890/13-1799.1.  Google Scholar

[39]

D. Finkelshtein and P. Tkachov, Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line, Appl. Anal., 98 (2019), 756-780.  doi: 10.1080/00036811.2017.1400537.  Google Scholar

[40]

R. A. Fisher, The wave of advance of advantageous genes, Annu. Eugen., 7 (1937), 353-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[41]

B. G. Freeman and A. M. C. Freeman, Rapid upslope shifts in new guinean birds illustrate strong distributional responses of tropical montane species to global warming, Proc. Natl. Acad. Sci. USA, 111 (2014), 4490-4494.  doi: 10.1073/pnas.1318190111.  Google Scholar

[42]

R. A. Gardner, Existence of travelling wave solutions of predator-prey systems via the connection index, SIAM J. Appl. Math., 44 (1984), 56-79.  doi: 10.1137/0144006.  Google Scholar

[43]

J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.  Google Scholar

[44]

P. GonzalezR. P. NeilsonJ. M. Lenihan and R. J. Drapek, Global patterns in the vulnerability of ecosystems to vegetation shifts due to climate change, Glob. Ecol. Biogeogr., 19 (2010), 755-768.  doi: 10.1111/j.1466-8238.2010.00558.x.  Google Scholar

[45]

J. S. GuoY. WangC. H. Wu and C. C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373.  Google Scholar

[46]

J. S. Guo and C. C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.  doi: 10.1016/j.jde.2015.09.036.  Google Scholar

[47]

F. Hamel, Reaction-diffusion problems in cylinders with no invariance by translation. II. Monotone perturbations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 555-596.  doi: 10.1016/S0294-1449(97)80126-6.  Google Scholar

[48] R. J. Hobbs, Invasive species in a changing world, Island Press, London, 2000.   Google Scholar
[49]

M. Holzer and A. Scheel, Accelerated fronts in a two-stage invasion process, SIAM J. Math. Anal., 46 (2014), 397-427.  doi: 10.1137/120887746.  Google Scholar

[50]

C. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations, 259 (2015), 1967-1989.  doi: 10.1016/j.jde.2015.03.025.  Google Scholar

[51]

C. HuJ. Shang and B. Li, Spreading speeds for reaction-diffusion equations with a shifting habitat, J. Dynam. Differential Equations, 32 (2020), 1941-1964.  doi: 10.1007/s10884-019-09796-5.  Google Scholar

[52]

H. HuL. Deng and J. Huang, Traveling wave of a nonlocal dispersal Lotka-Volterra cooperation model under shifting habitat, J. Math. Anal. Appl., 500 (2021), 125100.  doi: 10.1016/j.jmaa.2021.125100.  Google Scholar

[53]

H. HuT. Yi and X. Zou, On spatial-temporal dynamics of a Fisher-KPP equation with a shifting environment, Proc. Amer. Math. Soc., 148 (2020), 213-221.  doi: 10.1090/proc/14659.  Google Scholar

[54]

H. Hu and X. Zou, Existence of an extinction wave in the Fisher equation with a shifting habitat, Proc. Amer. Math. Soc., 145 (2017), 4763-4771.  doi: 10.1090/proc/13687.  Google Scholar

[55]

Y. Hu, X. Hao and Y. Du, Spreading via a free boundary model under shifting climate: Invasion of deteriorated environment, Comm. Contemp. Math., 2021. doi: 10.1142/S0219199720500777.  Google Scholar

[56]

Y. HuX. HaoX. Song and Y. Du, A free boundary problem for spreading under shifting climate, J. Differential Equations, 269 (2020), 5931-5958.  doi: 10.1016/j.jde.2020.04.024.  Google Scholar

[57]

S. F. Iglesias and S. Mirrahimi, Selection and mutation in a shifting and fluctuating environment, Commun. Math. Sci., 19 (2021), 1761-1798.  doi: 10.4310/CMS.2021.v19.n7.a1.  Google Scholar

[58]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[59]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[60]

A. N. KolmogorovI. G. Petrovski 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.   Google Scholar

[61]

K. Y. Lam and X. Yu, Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments, arXiv: 2101.06698v2. Google Scholar

[62]

C. Lei and Y. Du, Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 895-911.  doi: 10.3934/dcdsb.2017045.  Google Scholar

[63]

C. LeiH. NieW. Dong and Y. Du, Spreading of two competing species governed by a free boundary model in a shifting environment, J. Math. Anal. Appl., 462 (2018), 1254-1282.  doi: 10.1016/j.jmaa.2018.02.042.  Google Scholar

[64]

M. A. LewisN. G. Marculis and Z. Shen, Integrodifference equations in the presence of climate change: Persistence criterion, travelling waves and inside dynamics, J. Math. Biol., 77 (2018), 1649-1687.  doi: 10.1007/s00285-018-1206-1.  Google Scholar

[65]

B. LiS. BewickM. R. Barnard and W. F. Fagan, Persistence and spreading speeds of integro-difference equations with an expanding or contracting habitat, Bull. Math. Biol., 78 (2016), 1337-1379.  doi: 10.1007/s11538-016-0180-2.  Google Scholar

[66]

B. LiS. BewickJ. Shang and W. F. Fagan, Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math., 74 (2014), 1397-1417.  doi: 10.1137/130938463.  Google Scholar

[67]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[68]

B. Li and J. Wu, Traveling waves in integro-difference equations with a shifting habitat, J. Differentail Equations, 268 (2020), 4059-4078.  doi: 10.1016/j.jde.2019.10.018.  Google Scholar

[69]

F. LiJ. Coville and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., 37 (2017), 879-903.  doi: 10.3934/dcds.2017036.  Google Scholar

[70]

W. T. LiJ. B. Wang and X. Q. Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, J. Nonlinear Sci., 28 (2018), 1189-1219.  doi: 10.1007/s00332-018-9445-2.  Google Scholar

[71]

G. LiuT. Xu and J. Yin, Forced waves of reaction-diffusion model with density-dependent dispersal in shifting environments, J. Differential Equations, 282 (2021), 127-147.  doi: 10.1016/j.jde.2021.02.027.  Google Scholar

[72]

Y. MengZ. Yu and S. Zhang, Spatial dynamics of the lattice Lotka-Volterra competition system in a shifting habitat, Nonlinear Anal. Real World Appl., 60 (2021), 103287.  doi: 10.1016/j.nonrwa.2020.103287.  Google Scholar

[73]

G. F. Midgley and W. J. Bond, Future of african terrestrial biodiversity and ecosystems under anthropogenic climate change, Nat. Clim. Change, 5 (2015), 823-829.  doi: 10.1038/nclimate2753.  Google Scholar

[74]

D. L. Murray, M. J. L. Peers, Y. N. Majchrzak, M. Wehtje, C. Ferreira, R. S. A. Pickles, J. R. Row and D. H. Thornton, Continental divide: Predicting climate-mediated fragmentation and biodiversity loss in the boreal forest, PLoS ONE, 12 (2017), e0176706. doi: 10.1371/journal.pone.0176706.  Google Scholar

[75]

K. J. Painter, Mathematical models for chemotaxisand their applications in self-organisation phenomena, J. Theor. Biol., 481 (2019), 162-182.  doi: 10.1016/j.jtbi.2018.06.019.  Google Scholar

[76]

Y. PanY. Su and J. Wei, Accelerating propagation in a recursive system arising from seasonal population models with nonlocal dispersal, J. Differential Equations, 267 (2019), 150-179.  doi: 10.1016/j.jde.2019.01.009.  Google Scholar

[77]

L. Y. Pang and S. L. Wu, Propagation dynamics for lattice differential equations in a time-periodic shifting habitat, Z. Angew. Math. Phys., 72 (2021), 93.  doi: 10.1007/s00033-021-01522-w.  Google Scholar

[78]

C. L. ParrE. F. Gray and W. J. Bond, Cascading biodiversity and functional consequences of a global change-induced biome switch, Divers. Distrib., 18 (2012), 493-503.  doi: 10.1111/j.1472-4642.2012.00882.x.  Google Scholar

[79]

J. J. PolovinaJ. P. DunneP. A. Woodworth and E. A. Howell, Projected expansion of the subtropical biome and contraction of the temperate and equatorial upwelling biomes in the North Pacific under global warming, ICES J. Mar. Sci., 68 (2011), 986-995.  doi: 10.1093/icesjms/fsq198.  Google Scholar

[80]

A. B. Potapov and M. A. Lewis, Climate and competition: The effect of moving range boundaries on habitat invasibility, Bull. Math. Biol., 66 (2004), 975-1008.  doi: 10.1016/j.bulm.2003.10.010.  Google Scholar

[81]

S. X. Qiao, W. T. Li and J. B. Wang, Asymptotic propagations of a nonlocal dispersal population model with shifting habitats, European J. Appl. Math., (2021), 1–28. doi: 10.1017/S095679252100019X.  Google Scholar

[82]

S. X. QiaoW. T. Li and J. B. Wang, Multi-type forced waves in nonlocal dispersal KPP equations with shifting habitats, J. Math. Anal. Appl., 505 (2022), 125504.  doi: 10.1016/j.jmaa.2021.125504.  Google Scholar

[83]

S. X. QiaoJ. L. Zhu and J. B. Wang, Asymptotic behaviors of forced waves for the lattice Lotka-Volterra competition system with shifting habitats, Appl. Math. Lett., 118 (2021), 107168.  doi: 10.1016/j.aml.2021.107168.  Google Scholar

[84]

F. J. Rahel and J. D. Olden, Assessing the effects of climate change on aquatic invasive species, Conserv. Biol., 22 (2008), 521-533.  doi: 10.1111/j.1523-1739.2008.00950.x.  Google Scholar

[85]

N. RawalW. Shen and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst., 35 (2015), 1609-1640.  doi: 10.3934/dcds.2015.35.1609.  Google Scholar

[86]

L. RoquesA. RoquesH. Berestycki and A. Kretzschmar, A population facing climate change: Joint influences of Allee effects and environmental boundary geometry, Popul. Ecol., 50 (2008), 215-225.  doi: 10.1007/s10144-007-0073-1.  Google Scholar

[87]

M. SchefferM. HirotaM. HolmgrenE. H. Van Nes and F. S. Chapin, Thresholds for boreal biome transitions, Proc. Natl. Acad. Sci. USA, 109 (2012), 21384-21389.  doi: 10.1073/pnas.1219844110.  Google Scholar

[88]

W. Shen and S. Xue, Persistence and spreading speeds of parabolic-elliptic Keller-Segel models in shifting environments, J. Differential Equations, 269 (2020), 6236-6268.  doi: 10.1016/j.jde.2020.04.040.  Google Scholar

[89]

W. Shen and S. Xue, Forced waves of parabolic-elliptic Keller-Segel models in shifting environments, J. Dynam. Differential Equations, 2021. doi: 10.1007/s10884-020-09924-6.  Google Scholar

[90]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[91]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101.   Google Scholar

[92]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar

[93]

H. H. Vo, Persistence versus extinction under a climate change in mixed environments, J. Differential Equations, 259 (2015), 4947-4988.  doi: 10.1016/j.jde.2015.06.014.  Google Scholar

[94]

G. R. WaltherE. PostP. ConveyA. MenzelC. ParmesanT. J. C. BeebeeJ. M. FromentinO. Hoegh-Guldberg and F. Bairlein, Ecological responses to recent climate change, Nature, 416 (2002), 389-395.  doi: 10.1038/416389a.  Google Scholar

[95]

H. WangC. Pan and C. Ou, Existence of forced waves and gap formations for the lattice Lotka-Volterra competition system in a shifting environment, Appl. Math. Lett., 106 (2020), 106349.  doi: 10.1016/j.aml.2020.106349.  Google Scholar

[96]

H. Wang, C. Pan and C. Ou, Existence, uniqueness and stability of forced waves to the Lotka-Volterra competition system in a shifting environment, Stud. Appl. Math., 2021. doi: 10.1111/sapm.12438.  Google Scholar

[97]

J. B. Wang and W. T. Li, Wave propagation for a cooperative model with nonlocal dispersal under worsening habitats, Z. Angew. Math. Phys., 71 (2020), 147.  doi: 10.1007/s00033-020-01374-w.  Google Scholar

[98]

J. B. Wang and C. Wu, Forced waves and gap formations for a Lotka-Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Anal. Real World Appl., 58 (2021), 103208.  doi: 10.1016/j.nonrwa.2020.103208.  Google Scholar

[99]

J. B. Wang and X. Q. Zhao, Uniqueness and global stability of forced waves in a shifting environment, Proc. Amer. Math. Soc., 147 (2019), 1467-1481.  doi: 10.1090/proc/14235.  Google Scholar

[100]

L. Wei, G. Zhang and M. Zhou, Long time behavior for solutions of the diffusive logistic equation with advection and free boundary, Calc. Var. Partial Differ. Equ., 55 (2016), 34 pp. doi: 10.1007/s00526-016-1039-y.  Google Scholar

[101]

P. Weng and X. Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[102]

C. WuY. Wang and X. Zou, Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment, J. Differential Equations, 267 (2019), 4890-4921.  doi: 10.1016/j.jde.2019.05.019.  Google Scholar

[103]

C. WuY. Yang and Z. Wu, Existence and uniqueness of forced waves in a delayed reaction–diffusion equation in a shifting environment, Nonlinear Anal. Real World Appl., 57 (2021), 103198.  doi: 10.1016/j.nonrwa.2020.103198.  Google Scholar

[104]

W. B. XuW. T. Li and G. Lin, Nonlocal dispersal cooperative systems: Acceleration propagation among species, J. Differential Equations, 268 (2020), 1081-1105.  doi: 10.1016/j.jde.2019.08.039.  Google Scholar

[105]

W. B. XuW. T. Li and S. Ruan, Fast propagation for reaction-diffusion cooperative systems, J. Differential Equations, 265 (2018), 645-670.  doi: 10.1016/j.jde.2018.03.004.  Google Scholar

[106]

W. B. XuW. T. Li and S. Ruan, Spatial propagation in nonlocal dispersal Fisher-KPP equations, J. Funct. Anal., 280 (2021), 108957.  doi: 10.1016/j.jfa.2021.108957.  Google Scholar

[107]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953.  doi: 10.2977/prims/1260476648.  Google Scholar

[108]

Y. YangC. Wu and Z. Li, Forced waves and their asymptotics in a Lotka-Volterra cooperative model under climate change, Appl. Math. Comput., 353 (2019), 254-264.  doi: 10.1016/j.amc.2019.01.058.  Google Scholar

[109]

T. YiY. Chen and J. Wu, Asymptotic propagations of asymptotical monostable type equations with shifting habitats, J. Differential Equations, 269 (2020), 5900-5930.  doi: 10.1016/j.jde.2020.04.025.  Google Scholar

[110]

T. Yi and X. Q. Zhao, Propagation dynamics for monotone evolution systems without spatial translation invariance, J. Funct. Anal., 279 (2020), 108722.  doi: 10.1016/j.jfa.2020.108722.  Google Scholar

[111]

Y. YuanY. Wang and X. Zou, Spatial dynamics of a Lotka-Volterra model with a shifting habitat, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5633-5671.  doi: 10.3934/dcdsb.2019076.  Google Scholar

[112]

Y. Yuan and X. Zou, Spatial-temporal dynamics of a diffusive Lotka–Volterra competition model with a shifting habitat II: Case of faster diffuser being a weaker copetitor, J. Dynam. Differential Equations, 33 (2021), 2091-2132.  doi: 10.1007/s10884-020-09885-w.  Google Scholar

[113]

G. B. Zhang and X. Q. Zhao, Propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat, J. Differential Equations, 268 (2020), 2852-2885.  doi: 10.1016/j.jde.2019.09.044.  Google Scholar

[114]

Z. ZhangW. Wang and J. Yang, Persistence versus extinction for two competing species under a climate change, Nonlinear Anal. Model. Control, 22 (2017), 285-302.  doi: 10.15388/NA.2017.3.1.  Google Scholar

[115]

M. ZhaoY. ZhangW. 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.  Google Scholar

[116]

Y. Zhou and M. Kot, Discrete-time growth-dispersal models with shifting species ranges, Theor. Ecol., 4 (2011), 13-25.  doi: 10.1007/s12080-010-0071-3.  Google Scholar

[117]

Y. Zhou and M. Kot, Life on the move: Modeling the effects of climate-driven range shifts with integrodifference equations, Dispersal, Individual Movement and Spatial Ecology, 2071 (2013), 263-292.  doi: 10.1007/978-3-642-35497-7_9.  Google Scholar

[118]

J. L. Zhu, J. B. Wang and F. D. Dong, Existence of multi-type forced waves in the lattice competition system under shifting habitats, Preprint, 2021. Google Scholar

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