Pattern formation of a biomass–water reaction–diffusion model
In this paper, we are concerned with a biomass–water reaction–diffusion model subject to the homogeneous Neumann boundary condition. We derive several sufficient conditions on the existence and non-existence of non-constant stationary solutions with respect to large or small diffusion rate, which gi...
Saved in:
| Published in | Applied mathematics letters Vol. 123; p. 107605 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.01.2022
|
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | In this paper, we are concerned with a biomass–water reaction–diffusion model subject to the homogeneous Neumann boundary condition. We derive several sufficient conditions on the existence and non-existence of non-constant stationary solutions with respect to large or small diffusion rate, which give the criteria for the possibility of Turing patterns in this system. Our results confirm the numerical findings of Manor and Shnerb, (2006) and also complement the theoretical results of Wang et al., (2017) for the corresponding ODE model. |
|---|---|
| AbstractList | In this paper, we are concerned with a biomass–water reaction–diffusion model subject to the homogeneous Neumann boundary condition. We derive several sufficient conditions on the existence and non-existence of non-constant stationary solutions with respect to large or small diffusion rate, which give the criteria for the possibility of Turing patterns in this system. Our results confirm the numerical findings of Manor and Shnerb, (2006) and also complement the theoretical results of Wang et al., (2017) for the corresponding ODE model. |
| ArticleNumber | 107605 |
| Author | Zhou, Jialin Zhang, Guanghui Lei, Chengxia |
| Author_xml | – sequence: 1 givenname: Chengxia surname: Lei fullname: Lei, Chengxia email: leichengxia001@163.com organization: School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, China – sequence: 2 givenname: Guanghui orcidid: 0000-0003-0231-2297 surname: Zhang fullname: Zhang, Guanghui email: guanghuizhang@hust.edu.cn organization: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China – sequence: 3 givenname: Jialin surname: Zhou fullname: Zhou, Jialin email: 1584479344@qq.com organization: School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, China |
| BookMark | eNp9kEFOwzAQRS1UJNrCAdjlAil27NiJWFUVUKRKsIC15dhjyVUSI9uA2HEHbshJcCgrFl2NZv680fy_QLPRj4DQJcErggm_2q_U0K8qXJHcC47rEzQnjaBlzepqhua4aWnZ8ro9Q4sY9xhj2tJmjtaPKiUIY2F9GFRyfiy8LVTROT-oGL8_v95V1osASk9qHhhn7WucNgdvoD9Hp1b1ES7-6hI93948bbbl7uHufrPelbpqRSo7BQ3mDHfMUkYaC1i0RAkwhoESmttGEU0oJi2rmW7qCji3kD-2pOuAGLpE5HBXBx9jACtfghtU-JAEyykDuZc5AzllIA8ZZEb8Y7RLvy5TUK4_Sl4fSMiW3hwEGbWDUYNxAXSSxrsj9A9VMnsT |
| CitedBy_id | crossref_primary_10_1371_journal_pone_0314910 crossref_primary_10_3390_math10091500 crossref_primary_10_1142_S0218127422500699 crossref_primary_10_1007_s10440_025_00708_y crossref_primary_10_1017_S0956792523000323 |
| Cites_doi | 10.1016/j.nonrwa.2007.02.005 10.1016/j.jde.2015.03.017 10.1142/S0218202509004005 10.3934/cpaa.2012.11.261 10.1016/j.jtbi.2008.04.012 10.1088/0951-7715/29/12/3777 10.1016/j.mbs.2015.10.015 10.1209/epl/i2005-10580-5 10.1142/S0219199710003968 10.1016/j.nonrwa.2012.07.012 10.1006/jdeq.1996.0157 10.1016/0022-0396(88)90147-7 10.1016/j.na.2008.12.003 10.1088/0951-7715/21/7/006 10.1016/j.jmaa.2009.12.021 10.1016/j.jde.2008.11.007 10.1007/s11425-007-0001-z 10.1016/j.na.2003.09.020 10.1016/j.jmaa.2004.12.026 10.1016/j.nonrwa.2016.11.007 10.1016/j.jde.2016.09.040 10.1016/j.aml.2008.06.032 10.1016/S0022-0396(02)00100-6 10.1016/j.jde.2012.12.009 10.1016/j.jmaa.2020.124860 10.1017/S0308210500000275 10.1016/j.jde.2015.10.036 10.1016/j.aml.2008.02.003 10.1090/S0002-9947-05-04010-9 10.1016/j.mcm.2006.03.001 10.1016/j.jde.2007.06.005 10.1088/0951-7715/21/10/007 10.1137/S003614100343651X 10.1137/05064624X 10.1016/j.nonrwa.2018.06.002 10.1016/j.nonrwa.2014.08.003 10.1007/978-1-4471-5526-3 |
| ContentType | Journal Article |
| Copyright | 2021 Elsevier Ltd |
| Copyright_xml | – notice: 2021 Elsevier Ltd |
| DBID | AAYXX CITATION |
| DOI | 10.1016/j.aml.2021.107605 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1873-5452 |
| ExternalDocumentID | 10_1016_j_aml_2021_107605 S0893965921003220 |
| GrantInformation_xml | – fundername: NSF of Jiangsu Province, China grantid: BK20180999 funderid: http://dx.doi.org/10.13039/501100004608 – fundername: NSF of China grantid: 11501225 funderid: http://dx.doi.org/10.13039/501100001809 – fundername: Jiangsu Normal University, China grantid: 17XLR008 funderid: http://dx.doi.org/10.13039/501100004030 – fundername: Fundamental Research Funds for the Central Universities, China grantid: 5003011008) funderid: http://dx.doi.org/10.13039/501100012226 – fundername: NSF of China grantid: 11801232 funderid: http://dx.doi.org/10.13039/501100001809 |
| GroupedDBID | --K --M .~1 0R~ 1B1 1RT 1~. 1~5 23M 4.4 457 4G. 5GY 5VS 6I. 7-5 71M 8P~ 9JN AACTN AAEDT AAEDW AAFTH AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO ABAOU ABEFU ABFNM ABJNI ABMAC ABVKL ABXDB ABYKQ ACAZW ACDAQ ACGFS ACNNM ACRLP ADBBV ADEZE ADMUD ADTZH AEBSH AECPX AEKER AENEX AEXQZ AFFNX AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AHJVU AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR ASPBG AVWKF AXJTR AZFZN BJAXD BKOJK BLXMC CS3 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 F5P FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q GBLVA HVGLF HZ~ IHE IXB J1W JJJVA KOM M26 M41 MHUIS MO0 N9A NCXOZ O-L O9- OAUVE OK1 OZT P-8 P-9 P2P PC. Q38 R2- RIG RNS ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SST SSW SSZ T5K UHS WUQ XPP ZMT ~G- AATTM AAXKI AAYWO AAYXX ABWVN ACRPL ACVFH ADCNI ADNMO ADVLN AEIPS AEUPX AFJKZ AFPUW AGCQF AGQPQ AGRNS AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP BNPGV CITATION SSH |
| ID | FETCH-LOGICAL-c297t-bae80640b4f3418fe0791a7edd4ea7c6f8a1c13019454c852e66fe965f1bbe1d3 |
| IEDL.DBID | .~1 |
| ISSN | 0893-9659 |
| IngestDate | Tue Jul 01 00:38:26 EDT 2025 Thu Apr 24 22:59:50 EDT 2025 Fri Feb 23 02:42:12 EST 2024 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Small/large diffusion Non-constant stationary solution A biomass–water reaction–diffusion model Turing pattern |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c297t-bae80640b4f3418fe0791a7edd4ea7c6f8a1c13019454c852e66fe965f1bbe1d3 |
| ORCID | 0000-0003-0231-2297 |
| ParticipantIDs | crossref_primary_10_1016_j_aml_2021_107605 crossref_citationtrail_10_1016_j_aml_2021_107605 elsevier_sciencedirect_doi_10_1016_j_aml_2021_107605 |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | January 2022 2022-01-00 |
| PublicationDateYYYYMMDD | 2022-01-01 |
| PublicationDate_xml | – month: 01 year: 2022 text: January 2022 |
| PublicationDecade | 2020 |
| PublicationTitle | Applied mathematics letters |
| PublicationYear | 2022 |
| Publisher | Elsevier Ltd |
| Publisher_xml | – name: Elsevier Ltd |
| References | Wang (b14) 2003; 190 Peng, Wang (b39) 2009; 22 Turing (b17) 1952; 237B Peng, Yi, Zhao (b29) 2013; 254 Peng, Wang, Yang (b27) 2006; 44 Abdelmalek, Bendoukha (b31) 2017; 35 Du, Peng, Wang (b10) 2009; 246 Meron (b6) 2016; 271 Ko (b32) 2016; 29 Li, Wang, Zhang (b24) 2018; 44 Peng, Yang (b21) 2009; 71 Li, Peng, Song (b33) 2017; 262 Ni, Tang (b12) 2008; 357 Wang, Wei, Shi (b41) 2016; 260 Marinov, Wang, Yang (b5) 2013; 14 Peng, Yi (b40) 2011; 15 Wei, Winter (b34) 2014 Yi, Wei, Shi (b43) 2009; 22 Peng (b26) 2007; 241 Lieberman (b19) 2005; 36 Manor, Shnerb (b1) 2006; 74 Lin, Ni, Takagi (b9) 1998; 72 L. Nirenberg, Courant Institute of Mathematical Sciences, New York, pp.1973-1974. Peng, Shi, Wang (b37) 2007; 67 Li, Wu, Dong (b28) 2015; 259 Peng, Wang (b38) 2007; 50 Zhou, Mu (b22) 2010; 366 Wei, Wu, Guo (b30) 2015; 22 Guo, Yuan (b35) 2009; 19 Wang, Shi, Zhang (b2) 2017; 22 Peng, Shi, Wang (b16) 2008; 21 Davidson, Rynne (b23) 2000; 130 Kletter, von Hardenberg, Meron (b3) 2012; 11 Peng, Wang (b13) 2004; 56 Wang, Shi, Zhang (b7) 2021; 497 Peng, Wang (b11) 2005; 309 Ni (b36) 1998; 45 Marius, Vicentiu (b20) 2010; 12 Ghergu (b18) 2008; 21 Manor, Shnerb (b4) 2008; 253 Li, Zhang (b25) 2017; 10 Lou, Ni (b8) 1996; 131 Yi, Wei, Shi (b42) 2008; 9 Du (10.1016/j.aml.2021.107605_b10) 2009; 246 Ghergu (10.1016/j.aml.2021.107605_b18) 2008; 21 Peng (10.1016/j.aml.2021.107605_b40) 2011; 15 Marinov (10.1016/j.aml.2021.107605_b5) 2013; 14 Ko (10.1016/j.aml.2021.107605_b32) 2016; 29 Peng (10.1016/j.aml.2021.107605_b37) 2007; 67 Turing (10.1016/j.aml.2021.107605_b17) 1952; 237B Wang (10.1016/j.aml.2021.107605_b14) 2003; 190 Abdelmalek (10.1016/j.aml.2021.107605_b31) 2017; 35 Marius (10.1016/j.aml.2021.107605_b20) 2010; 12 Lin (10.1016/j.aml.2021.107605_b9) 1998; 72 Yi (10.1016/j.aml.2021.107605_b42) 2008; 9 Peng (10.1016/j.aml.2021.107605_b13) 2004; 56 Peng (10.1016/j.aml.2021.107605_b26) 2007; 241 Yi (10.1016/j.aml.2021.107605_b43) 2009; 22 Peng (10.1016/j.aml.2021.107605_b11) 2005; 309 Wei (10.1016/j.aml.2021.107605_b34) 2014 Lieberman (10.1016/j.aml.2021.107605_b19) 2005; 36 Wang (10.1016/j.aml.2021.107605_b7) 2021; 497 Wang (10.1016/j.aml.2021.107605_b2) 2017; 22 Ni (10.1016/j.aml.2021.107605_b36) 1998; 45 Manor (10.1016/j.aml.2021.107605_b1) 2006; 74 Peng (10.1016/j.aml.2021.107605_b39) 2009; 22 10.1016/j.aml.2021.107605_b15 Peng (10.1016/j.aml.2021.107605_b27) 2006; 44 Peng (10.1016/j.aml.2021.107605_b21) 2009; 71 Li (10.1016/j.aml.2021.107605_b33) 2017; 262 Wei (10.1016/j.aml.2021.107605_b30) 2015; 22 Li (10.1016/j.aml.2021.107605_b25) 2017; 10 Lou (10.1016/j.aml.2021.107605_b8) 1996; 131 Zhou (10.1016/j.aml.2021.107605_b22) 2010; 366 Ni (10.1016/j.aml.2021.107605_b12) 2008; 357 Peng (10.1016/j.aml.2021.107605_b29) 2013; 254 Li (10.1016/j.aml.2021.107605_b24) 2018; 44 Peng (10.1016/j.aml.2021.107605_b38) 2007; 50 Kletter (10.1016/j.aml.2021.107605_b3) 2012; 11 Wang (10.1016/j.aml.2021.107605_b41) 2016; 260 Peng (10.1016/j.aml.2021.107605_b16) 2008; 21 Davidson (10.1016/j.aml.2021.107605_b23) 2000; 130 Li (10.1016/j.aml.2021.107605_b28) 2015; 259 Manor (10.1016/j.aml.2021.107605_b4) 2008; 253 Meron (10.1016/j.aml.2021.107605_b6) 2016; 271 Guo (10.1016/j.aml.2021.107605_b35) 2009; 19 |
| References_xml | – volume: 21 start-page: 1471 year: 2008 end-page: 1488 ident: b16 article-title: On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law publication-title: Nonlinearity – volume: 45 start-page: 9 year: 1998 end-page: 18 ident: b36 article-title: Diffusion, cross-diffusion, and their spike-layer steady states publication-title: Notices Amer. Math. Soc. – volume: 22 start-page: 155 year: 2015 end-page: 175 ident: b30 article-title: Steady state bifurcations for a glycolysis model in biochemical reaction publication-title: Nonlinear Anal. RWA – volume: 271 start-page: 1 year: 2016 end-page: 18 ident: b6 article-title: Pattern formation-a missing link in the study of ecosystem response to environmental changes publication-title: Math. Biosci. – volume: 22 start-page: 569 year: 2009 end-page: 573 ident: b39 article-title: Some non-existence results for stationary solutions to the Gray-Scott model in a bounded domain publication-title: Appl. Math. Lett. – volume: 44 start-page: 537 year: 2018 end-page: 558 ident: b24 article-title: Analysis on a generalized Sel’kov-Schnakenberg reaction–diffusion system publication-title: Nonlinear Anal. RWA – year: 2014 ident: b34 article-title: Mathematical aspects of pattern formation in biological systems publication-title: Applied Mathematical Sciences, Vol.189 – volume: 11 start-page: 261 year: 2012 end-page: 273 ident: b3 article-title: Ostwald ripening in dryland vegetation publication-title: Commun. Pure Appl. Anal. – volume: 497 year: 2021 ident: b7 article-title: Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction publication-title: J. Math. Anal. Appl. – volume: 262 start-page: 559 year: 2017 end-page: 589 ident: b33 article-title: Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system publication-title: J. Differential Equations – volume: 9 start-page: 1038 year: 2008 end-page: 1051 ident: b42 article-title: Diffusion-driven instability and bifurcation in the Lengyel–Epstein system publication-title: Nonlinear Anal. RWA – volume: 67 start-page: 1479 year: 2007 end-page: 1503 ident: b37 article-title: Stationary pattern of a ratio-dependent food chain model with diffusion publication-title: SIAM J. Appl. Math. – volume: 21 start-page: 2331 year: 2008 end-page: 2345 ident: b18 article-title: Non-constant steady states for the Brusselator type systems publication-title: Nonlinearity – volume: 246 start-page: 3932 year: 2009 end-page: 3956 ident: b10 article-title: Effect of a protection zone in the diffusive Leslie predator–prey model publication-title: J. Differential Equations – volume: 56 start-page: 451 year: 2004 end-page: 464 ident: b13 article-title: Positive steady-state solutions of the Noyes-Field model for Belousov–Zhabotinskii reaction publication-title: Nonlinear Anal. – volume: 254 start-page: 2465 year: 2013 end-page: 2498 ident: b29 article-title: Spatiotemporal patterns in a reaction–diffusion model with the Degn-Harrison reaction scheme publication-title: J. Differential Equations – volume: 190 start-page: 600 year: 2003 end-page: 620 ident: b14 article-title: Non-constant positive steady states of the Sel’kov model publication-title: J. Differential Equations – volume: 366 start-page: 679 year: 2010 end-page: 693 ident: b22 article-title: Pattern formation of a coupled two-cell Brusselator model publication-title: J. Math. Anal. Appl. – volume: 72 start-page: 1 year: 1998 end-page: 27 ident: b9 article-title: Large amplitude stationary solutions to a chemotaxis system publication-title: J. Differential Equations – volume: 71 start-page: 1389 year: 2009 end-page: 1394 ident: b21 article-title: On steady-state solutions of the Brusselator-type system publication-title: Nonlinear Anal. – volume: 259 start-page: 1990 year: 2015 end-page: 2029 ident: b28 article-title: Turing patterns in a reaction–diffusion model with the Degn-Harrison scheme publication-title: J. Differential Equations – reference: L. Nirenberg, Courant Institute of Mathematical Sciences, New York, pp.1973-1974. – volume: 260 start-page: 3495 year: 2016 end-page: 3523 ident: b41 article-title: Global bifurcation analysis and pattern formation in homogeneous diffusive predator–prey systems publication-title: J. Differential Equations – volume: 74 start-page: 837 year: 2006 end-page: 843 ident: b1 article-title: Dynamical failure of Turing patterns publication-title: Europhys. Lett. – volume: 22 start-page: 2971 year: 2017 end-page: 3006 ident: b2 article-title: Interaction between water and plants: rich dynamics in a simple model publication-title: Discrete Contin. Dyn. Syst. Ser. B – volume: 50 start-page: 377 year: 2007 end-page: 386 ident: b38 article-title: On pattern formation in the Gray-Scott model publication-title: Science in China A: Mathematics – volume: 36 start-page: 1400 year: 2005 end-page: 1406 ident: b19 article-title: Bounds for the steady-state Sel’kov model for arbitrary publication-title: SIAM J. Math. Anal. – volume: 131 start-page: 79 year: 1996 end-page: 131 ident: b8 article-title: Self-diffusion and cross-diffusion publication-title: J. Differential Equations – volume: 130 start-page: 507 year: 2000 end-page: 516 ident: b23 article-title: A priori bounds and global existence of solutions of the steady-state Sel’kov model publication-title: Proc. Roy. Soc. Edinburgh, A – volume: 35 start-page: 397 year: 2017 end-page: 413 ident: b31 article-title: On the global asymptotic stability of solutions to a generalised Lengyel-epstein system publication-title: Nonlinear Anal. RWA – volume: 309 start-page: 151 year: 2005 end-page: 166 ident: b11 article-title: Pattern formation in the Brusselator type systems publication-title: J. Math. Anal. Appl. – volume: 12 start-page: 661 year: 2010 end-page: 679 ident: b20 article-title: Turing paterns in general reaction–diffusion systems of Brusselator type publication-title: Commun. Contemp. Math. – volume: 29 start-page: 3777 year: 2016 end-page: 3809 ident: b32 article-title: Bifurcations and asymptotic behavior of positive steady-states of an enzyme-catalyzed reaction–diffusion system publication-title: Nonlinearity – volume: 357 start-page: 3953 year: 2008 end-page: 3969 ident: b12 article-title: Turing patterns in the Lengyel–Epstein system for the CIMA reactions publication-title: Trans. Amer. Math. Soc. – volume: 44 start-page: 945 year: 2006 end-page: 951 ident: b27 article-title: Positive steady-state solutions of the Sel’kov model publication-title: Math. Comput. Model. – volume: 241 start-page: 386 year: 2007 end-page: 398 ident: b26 article-title: Qualitative analysis of steady states to the Sel’kov model publication-title: J. Differential Equations – volume: 22 start-page: 52 year: 2009 end-page: 55 ident: b43 article-title: Global asymptotical behavior of the Lengyel–Epstein reaction–diffusion system publication-title: Appl. Math. Lett. – volume: 253 start-page: 838 year: 2008 end-page: 842 ident: b4 article-title: Facilitation, competition, and vegetation patchiness: from scale free distribution to patterns publication-title: J. Theoret. Biol. – volume: 10 start-page: 1009 year: 2017 end-page: 1023 ident: b25 article-title: Steady states of a Sel’kov-Schnakenberg reaction–diffusion system publication-title: Discrete Contin. Dyn. Syst. Ser. S – volume: 15 start-page: 217 year: 2011 end-page: 230 ident: b40 article-title: On spatiotemporal pattern formation in a diffusive bimolecular model publication-title: Discrete Contin. Dyn. Syst. Ser. B – volume: 19 start-page: 1797 year: 2009 end-page: 1852 ident: b35 article-title: Pattern formation in a ring network with delay publication-title: Math. Models Methods Appl. Sci. – volume: 14 start-page: 507 year: 2013 end-page: 525 ident: b5 article-title: On a vegetation pattern formation model governed by a nonlinear parabolic system publication-title: Nonlinear Anal. RWA – volume: 237B start-page: 37 year: 1952 end-page: 72 ident: b17 article-title: The chemical basis of morphogenesis publication-title: Philos. Trans. Royal Soc. – volume: 9 start-page: 1038 year: 2008 ident: 10.1016/j.aml.2021.107605_b42 article-title: Diffusion-driven instability and bifurcation in the Lengyel–Epstein system publication-title: Nonlinear Anal. RWA doi: 10.1016/j.nonrwa.2007.02.005 – volume: 259 start-page: 1990 year: 2015 ident: 10.1016/j.aml.2021.107605_b28 article-title: Turing patterns in a reaction–diffusion model with the Degn-Harrison scheme publication-title: J. Differential Equations doi: 10.1016/j.jde.2015.03.017 – volume: 19 start-page: 1797 year: 2009 ident: 10.1016/j.aml.2021.107605_b35 article-title: Pattern formation in a ring network with delay publication-title: Math. Models Methods Appl. Sci. doi: 10.1142/S0218202509004005 – volume: 11 start-page: 261 year: 2012 ident: 10.1016/j.aml.2021.107605_b3 article-title: Ostwald ripening in dryland vegetation publication-title: Commun. Pure Appl. Anal. doi: 10.3934/cpaa.2012.11.261 – volume: 253 start-page: 838 year: 2008 ident: 10.1016/j.aml.2021.107605_b4 article-title: Facilitation, competition, and vegetation patchiness: from scale free distribution to patterns publication-title: J. Theoret. Biol. doi: 10.1016/j.jtbi.2008.04.012 – volume: 29 start-page: 3777 year: 2016 ident: 10.1016/j.aml.2021.107605_b32 article-title: Bifurcations and asymptotic behavior of positive steady-states of an enzyme-catalyzed reaction–diffusion system publication-title: Nonlinearity doi: 10.1088/0951-7715/29/12/3777 – volume: 271 start-page: 1 year: 2016 ident: 10.1016/j.aml.2021.107605_b6 article-title: Pattern formation-a missing link in the study of ecosystem response to environmental changes publication-title: Math. Biosci. doi: 10.1016/j.mbs.2015.10.015 – volume: 74 start-page: 837 year: 2006 ident: 10.1016/j.aml.2021.107605_b1 article-title: Dynamical failure of Turing patterns publication-title: Europhys. Lett. doi: 10.1209/epl/i2005-10580-5 – volume: 15 start-page: 217 year: 2011 ident: 10.1016/j.aml.2021.107605_b40 article-title: On spatiotemporal pattern formation in a diffusive bimolecular model publication-title: Discrete Contin. Dyn. Syst. Ser. B – ident: 10.1016/j.aml.2021.107605_b15 – volume: 12 start-page: 661 year: 2010 ident: 10.1016/j.aml.2021.107605_b20 article-title: Turing paterns in general reaction–diffusion systems of Brusselator type publication-title: Commun. Contemp. Math. doi: 10.1142/S0219199710003968 – volume: 14 start-page: 507 year: 2013 ident: 10.1016/j.aml.2021.107605_b5 article-title: On a vegetation pattern formation model governed by a nonlinear parabolic system publication-title: Nonlinear Anal. RWA doi: 10.1016/j.nonrwa.2012.07.012 – volume: 131 start-page: 79 year: 1996 ident: 10.1016/j.aml.2021.107605_b8 article-title: Self-diffusion and cross-diffusion publication-title: J. Differential Equations doi: 10.1006/jdeq.1996.0157 – volume: 72 start-page: 1 year: 1998 ident: 10.1016/j.aml.2021.107605_b9 article-title: Large amplitude stationary solutions to a chemotaxis system publication-title: J. Differential Equations doi: 10.1016/0022-0396(88)90147-7 – volume: 71 start-page: 1389 year: 2009 ident: 10.1016/j.aml.2021.107605_b21 article-title: On steady-state solutions of the Brusselator-type system publication-title: Nonlinear Anal. doi: 10.1016/j.na.2008.12.003 – volume: 21 start-page: 1471 year: 2008 ident: 10.1016/j.aml.2021.107605_b16 article-title: On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law publication-title: Nonlinearity doi: 10.1088/0951-7715/21/7/006 – volume: 366 start-page: 679 year: 2010 ident: 10.1016/j.aml.2021.107605_b22 article-title: Pattern formation of a coupled two-cell Brusselator model publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2009.12.021 – volume: 246 start-page: 3932 year: 2009 ident: 10.1016/j.aml.2021.107605_b10 article-title: Effect of a protection zone in the diffusive Leslie predator–prey model publication-title: J. Differential Equations doi: 10.1016/j.jde.2008.11.007 – volume: 50 start-page: 377 year: 2007 ident: 10.1016/j.aml.2021.107605_b38 article-title: On pattern formation in the Gray-Scott model publication-title: Science in China A: Mathematics doi: 10.1007/s11425-007-0001-z – volume: 22 start-page: 2971 year: 2017 ident: 10.1016/j.aml.2021.107605_b2 article-title: Interaction between water and plants: rich dynamics in a simple model publication-title: Discrete Contin. Dyn. Syst. Ser. B – volume: 56 start-page: 451 year: 2004 ident: 10.1016/j.aml.2021.107605_b13 article-title: Positive steady-state solutions of the Noyes-Field model for Belousov–Zhabotinskii reaction publication-title: Nonlinear Anal. doi: 10.1016/j.na.2003.09.020 – volume: 237B start-page: 37 year: 1952 ident: 10.1016/j.aml.2021.107605_b17 article-title: The chemical basis of morphogenesis publication-title: Philos. Trans. Royal Soc. – volume: 309 start-page: 151 year: 2005 ident: 10.1016/j.aml.2021.107605_b11 article-title: Pattern formation in the Brusselator type systems publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2004.12.026 – volume: 35 start-page: 397 year: 2017 ident: 10.1016/j.aml.2021.107605_b31 article-title: On the global asymptotic stability of solutions to a generalised Lengyel-epstein system publication-title: Nonlinear Anal. RWA doi: 10.1016/j.nonrwa.2016.11.007 – volume: 262 start-page: 559 year: 2017 ident: 10.1016/j.aml.2021.107605_b33 article-title: Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system publication-title: J. Differential Equations doi: 10.1016/j.jde.2016.09.040 – volume: 22 start-page: 569 year: 2009 ident: 10.1016/j.aml.2021.107605_b39 article-title: Some non-existence results for stationary solutions to the Gray-Scott model in a bounded domain publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2008.06.032 – volume: 190 start-page: 600 year: 2003 ident: 10.1016/j.aml.2021.107605_b14 article-title: Non-constant positive steady states of the Sel’kov model publication-title: J. Differential Equations doi: 10.1016/S0022-0396(02)00100-6 – volume: 254 start-page: 2465 year: 2013 ident: 10.1016/j.aml.2021.107605_b29 article-title: Spatiotemporal patterns in a reaction–diffusion model with the Degn-Harrison reaction scheme publication-title: J. Differential Equations doi: 10.1016/j.jde.2012.12.009 – volume: 497 issue: 1 year: 2021 ident: 10.1016/j.aml.2021.107605_b7 article-title: Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2020.124860 – volume: 130 start-page: 507 year: 2000 ident: 10.1016/j.aml.2021.107605_b23 article-title: A priori bounds and global existence of solutions of the steady-state Sel’kov model publication-title: Proc. Roy. Soc. Edinburgh, A doi: 10.1017/S0308210500000275 – volume: 10 start-page: 1009 year: 2017 ident: 10.1016/j.aml.2021.107605_b25 article-title: Steady states of a Sel’kov-Schnakenberg reaction–diffusion system publication-title: Discrete Contin. Dyn. Syst. Ser. S – volume: 260 start-page: 3495 year: 2016 ident: 10.1016/j.aml.2021.107605_b41 article-title: Global bifurcation analysis and pattern formation in homogeneous diffusive predator–prey systems publication-title: J. Differential Equations doi: 10.1016/j.jde.2015.10.036 – volume: 22 start-page: 52 year: 2009 ident: 10.1016/j.aml.2021.107605_b43 article-title: Global asymptotical behavior of the Lengyel–Epstein reaction–diffusion system publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2008.02.003 – volume: 357 start-page: 3953 year: 2008 ident: 10.1016/j.aml.2021.107605_b12 article-title: Turing patterns in the Lengyel–Epstein system for the CIMA reactions publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-05-04010-9 – volume: 44 start-page: 945 year: 2006 ident: 10.1016/j.aml.2021.107605_b27 article-title: Positive steady-state solutions of the Sel’kov model publication-title: Math. Comput. Model. doi: 10.1016/j.mcm.2006.03.001 – volume: 241 start-page: 386 year: 2007 ident: 10.1016/j.aml.2021.107605_b26 article-title: Qualitative analysis of steady states to the Sel’kov model publication-title: J. Differential Equations doi: 10.1016/j.jde.2007.06.005 – volume: 21 start-page: 2331 year: 2008 ident: 10.1016/j.aml.2021.107605_b18 article-title: Non-constant steady states for the Brusselator type systems publication-title: Nonlinearity doi: 10.1088/0951-7715/21/10/007 – volume: 36 start-page: 1400 year: 2005 ident: 10.1016/j.aml.2021.107605_b19 article-title: Bounds for the steady-state Sel’kov model for arbitrary p in any number of dimensions publication-title: SIAM J. Math. Anal. doi: 10.1137/S003614100343651X – volume: 67 start-page: 1479 year: 2007 ident: 10.1016/j.aml.2021.107605_b37 article-title: Stationary pattern of a ratio-dependent food chain model with diffusion publication-title: SIAM J. Appl. Math. doi: 10.1137/05064624X – volume: 44 start-page: 537 year: 2018 ident: 10.1016/j.aml.2021.107605_b24 article-title: Analysis on a generalized Sel’kov-Schnakenberg reaction–diffusion system publication-title: Nonlinear Anal. RWA doi: 10.1016/j.nonrwa.2018.06.002 – volume: 22 start-page: 155 year: 2015 ident: 10.1016/j.aml.2021.107605_b30 article-title: Steady state bifurcations for a glycolysis model in biochemical reaction publication-title: Nonlinear Anal. RWA doi: 10.1016/j.nonrwa.2014.08.003 – year: 2014 ident: 10.1016/j.aml.2021.107605_b34 article-title: Mathematical aspects of pattern formation in biological systems doi: 10.1007/978-1-4471-5526-3 – volume: 45 start-page: 9 year: 1998 ident: 10.1016/j.aml.2021.107605_b36 article-title: Diffusion, cross-diffusion, and their spike-layer steady states publication-title: Notices Amer. Math. Soc. |
| SSID | ssj0003938 |
| Score | 2.3543477 |
| Snippet | In this paper, we are concerned with a biomass–water reaction–diffusion model subject to the homogeneous Neumann boundary condition. We derive several... |
| SourceID | crossref elsevier |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 107605 |
| SubjectTerms | A biomass–water reaction–diffusion model Non-constant stationary solution Small/large diffusion Turing pattern |
| Title | Pattern formation of a biomass–water reaction–diffusion model |
| URI | https://dx.doi.org/10.1016/j.aml.2021.107605 |
| Volume | 123 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3NS8MwFA9jXvQgfuL8GDl4EuLSLm2a4xiOqWwIOtitJGkCE-3G3PAm_g_-h_4lvqTtVFAPHhveo-0vj_eRvA-ETjnNMhZmijDGDWGSCiIzEZFQaEktNwn15WKDYdwfsatxNK6hblUL49IqS91f6HSvrcuVVolmazaZtG4pmFrXDg-CFgpi6eJ2eKGT8vOXzzSPtvDTrB0xcdTVzabP8ZKP7vYhDOCZx26C3U-26Yu96W2hzdJRxJ3iW7ZRzeQ7aGOw6rL6tIs6N745Zo5XFYh4arHErqQefOL317dn8CTnGPxCX70AC24eytIdkGE_AmcPjXoXd90-KUciEB0KviBKOvQYVcyC-UmsoVwEkhsA3EiuY5vIQINZCgSLmE6i0MSxNfDfNlDKBFl7H9XzaW4OEA6DOIHYTElp2qydUWWotkyLxDCZge5pIFqBkeqyX7gbW_GQVolh9ynglzr80gK_BjpbscyKZhl_EbMK4fTbjqegzH9nO_wf2xFaD13hgj88OUb1xXxpTsCdWKiml5cmWutcXveHH16YytM |
| linkProvider | Elsevier |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LS8NAEB5Ke1AP4hPrMwdPQugm2Tz2WIoltQ8EW-gtbDa7UNG01Bav_gf_ob_E2U1SFNSDxyw7kHxZvpndnZkP4DokWUbdLLUpDaVNOWE2z5hvu0xwokIZEVMuNhwF8YTeTf1pDTpVLYxOqyy5v-B0w9blSKtEs7WYzVoPBF2tboeHmxaCyxL37Q3dncqvQ6Pd68ejDSF7zAha6_m2NqguN02aF3_WFxCug89hoEXsfnJPX1xOdw92y1jRahevsw81mR_AznDTaPXlENr3pj9mbm2KEK25srilq-oxLP54e3_FYHJpYWhoChhwQEuirPUZmWVUcI5g0r0dd2K7VEWwhcvClZ1yDSAlKVXogSIlScgcHkrEXPJQBCrijkDP5DBEQ0S-K4NASfxu5aSpdDLvGOr5PJcnYLlOEOH2LOVcetTLSCqJUFSwSFKeIf00gVRgJKJsGa6VK56SKjfsMUH8Eo1fUuDXhJuNyaLol_HXZFohnHz76Qny-e9mp_8zu4KteDwcJIPeqH8G266uYzBnKedQXy3X8gKji1V6Wa6eT2NHzYQ |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Pattern+formation+of+a+biomass%E2%80%93water+reaction%E2%80%93diffusion+model&rft.jtitle=Applied+mathematics+letters&rft.au=Lei%2C+Chengxia&rft.au=Zhang%2C+Guanghui&rft.au=Zhou%2C+Jialin&rft.date=2022-01-01&rft.issn=0893-9659&rft.volume=123&rft.spage=107605&rft_id=info:doi/10.1016%2Fj.aml.2021.107605&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_aml_2021_107605 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0893-9659&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0893-9659&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0893-9659&client=summon |