Construction and application of exact solutions of the diffusive Lotka–Volterra system: A review and new results
This review summarizes all known results (up to this date) about methods of integration of the classical Lotka–Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. It is the first attempt in this direction. Becau...
Uložené v:
| Vydané v: | Communications in nonlinear science & numerical simulation Ročník 113; s. 106579 |
|---|---|
| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Amsterdam
Elsevier B.V
01.10.2022
Elsevier Science Ltd |
| Predmet: | |
| ISSN: | 1007-5704, 1878-7274 |
| On-line prístup: | Získať plný text |
| Tagy: |
Pridať tag
Žiadne tagy, Buďte prvý, kto otaguje tento záznam!
|
| Abstract | This review summarizes all known results (up to this date) about methods of integration of the classical Lotka–Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. It is the first attempt in this direction. Because the diffusive Lotka–Volterra systems are used for mathematical modeling enormous variety of processes in ecology, biology, medicine, physics and chemistry, the review should be interesting not only for specialists from Applied Mathematics but also those from other branches of Science. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.
•All known results about integration methods of the diffusive Lotka–Volterra systems are summarized.•Wide sets of exact solutions, including traveling fronts, are presented.•Plots and biological interpretation of the most interesting solutions are presented.•Some unsolved problems are highlighted. |
|---|---|
| AbstractList | This review summarizes all known results (up to this date) about methods of integration of the classical Lotka–Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. It is the first attempt in this direction. Because the diffusive Lotka–Volterra systems are used for mathematical modeling enormous variety of processes in ecology, biology, medicine, physics and chemistry, the review should be interesting not only for specialists from Applied Mathematics but also those from other branches of Science. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.
•All known results about integration methods of the diffusive Lotka–Volterra systems are summarized.•Wide sets of exact solutions, including traveling fronts, are presented.•Plots and biological interpretation of the most interesting solutions are presented.•Some unsolved problems are highlighted. This review summarizes all known results (up to this date) about methods of integration of the classical Lotka–Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. It is the first attempt in this direction. Because the diffusive Lotka–Volterra systems are used for mathematical modeling enormous variety of processes in ecology, biology, medicine, physics and chemistry, the review should be interesting not only for specialists from Applied Mathematics but also those from other branches of Science. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems. |
| ArticleNumber | 106579 |
| Author | Davydovych, Vasyl Cherniha, Roman |
| Author_xml | – sequence: 1 givenname: Roman orcidid: 0000-0002-1733-5240 surname: Cherniha fullname: Cherniha, Roman email: r.m.cherniha@gmail.com – sequence: 2 givenname: Vasyl’ surname: Davydovych fullname: Davydovych, Vasyl’ |
| BookMark | eNqFkL9uFDEQhy0UJJLAE9BYot7D9tq7t0gU0Yk_kU6iAVrLGY-Fj419eLyBdLwDb8iT4LujSgHVzPw031j-LthZygkZey7FSgo5vNytIFGilRJKtWQw4_SIncv1uO5GNeqz1gsxdmYU-gm7INqJRk1Gn7OyyYlqWaDGnLhLnrv9fo7gjnMOHH84qJzyvBwSOkT1C3IfQ1go3iHf5vrV_f7563OeK5biON1TxdtX_IoXvIv4_Xg1tVqQlrnSU_Y4uJnw2d96yT69ffNx877bfnh3vbnadtD3snYetBLOB6cMADqPblAezA3K3rteDc5g8CiU1wrgJngRdBBhPSo0OIEw_SV7cbq7L_nbglTtLi8ltSetGiZp5KC1blv9aQtKJioY7L7EW1furRT2INfu7FGuPci1J7mNmh5QEOvRWS0uzv9hX59YbJ9vgooliJgAfSwI1foc_8n_AV6rnZI |
| CitedBy_id | crossref_primary_10_1063_5_0222213 crossref_primary_10_3390_math13020197 crossref_primary_10_3934_dcdss_2025030 crossref_primary_10_1016_j_cnsns_2024_107905 crossref_primary_10_37251_ijome_v2i2_1366 crossref_primary_10_1016_j_chaos_2023_113468 crossref_primary_10_3390_sym14122520 crossref_primary_10_1088_1402_4896_ad8846 crossref_primary_10_1016_j_health_2023_100283 crossref_primary_10_1016_j_cnsns_2022_107068 crossref_primary_10_3390_sym15112025 crossref_primary_10_1016_j_cnsns_2023_107313 crossref_primary_10_3390_math11010160 crossref_primary_10_1017_S095679252200033X crossref_primary_10_1007_s11144_024_02700_3 crossref_primary_10_1007_s41478_023_00690_4 crossref_primary_10_3390_axioms14090655 |
| Cites_doi | 10.1515/zpch-1911-7549 10.1017/S0956792520000121 10.1111/j.1469-1809.1937.tb02153.x 10.1016/j.mcm.2011.03.035 10.1088/0031-8949/54/6/003 10.1007/BF02450786 10.1016/j.jmaa.2010.06.026 10.1016/j.jde.2018.12.003 10.1088/1751-8113/43/40/405207 10.1007/s13160-012-0056-2 10.1016/j.nonrwa.2010.12.004 10.1016/0025-5564(77)90077-3 10.1007/BF00275064 10.1016/j.jde.2020.07.006 10.1016/S0092-8240(79)80020-8 10.3934/dcdsb.2012.17.2653 10.1515/zpch-1910-7229 10.1016/S0377-0427(03)00645-9 10.1016/j.nonrwa.2007.07.007 10.1088/1751-8113/46/18/185204 10.1016/j.aml.2021.107731 10.3934/cpaa.2016.15.1451 10.3390/math9161984 10.1006/jdeq.1996.0157 10.3390/sym7041841 10.1021/ja01453a010 10.1007/s11253-005-0142-6 10.3390/bioengineering8030031 10.1016/j.nonrwa.2011.07.002 10.1016/S0167-2789(98)00191-2 10.1137/0133047 10.1007/BF03167410 10.32917/hmj/1206124686 |
| ContentType | Journal Article |
| Copyright | 2022 Elsevier B.V. Copyright Elsevier Science Ltd. Oct 2022 |
| Copyright_xml | – notice: 2022 Elsevier B.V. – notice: Copyright Elsevier Science Ltd. Oct 2022 |
| DBID | AAYXX CITATION |
| DOI | 10.1016/j.cnsns.2022.106579 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Applied Sciences |
| EISSN | 1878-7274 |
| ExternalDocumentID | 10_1016_j_cnsns_2022_106579 S1007570422001861 |
| GroupedDBID | --K --M -01 -0A -0I -0Y -SA -S~ .~1 0R~ 1B1 1RT 1~. 1~5 29F 4.4 457 4G. 5GY 5VR 5VS 7-5 71M 8P~ 92M 9D9 9DA AACTN AAEDT AAEDW AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO ABAOU ABFNM ABJNI ABMAC ABNEU ABXDB ABYKQ ACAZW ACDAQ ACFVG ACGFS ACNNM ACRLP ADBBV ADEZE ADGUI ADMUD ADTZH AEBSH AECPX AEKER AENEX AFKWA AFTJW AFUIB AGHFR AGUBO AGYEJ AHJVU AIEXJ AIGVJ AIKHN AITUG AIVDX AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR ASPBG AVWKF AXJTR AZFZN BJAXD BKOJK BLXMC CAJEA CAJUS CCEZO CCVFK CHBEP CS3 CUBFJ DU5 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 FA0 FDB FEDTE FGOYB FIRID FNPLU FYGXN G-Q GBLVA HVGLF HZ~ IHE J1W JJJVA JUIAU KOM M41 MHUIS MO0 N9A O-L O9- OAUVE OGIMB OZT P-8 P-9 P2P PC. Q-- Q-0 Q38 R-A R-I R2- RIG ROL RPZ RT1 RT9 S.. SDF SDG SES SEW SPC SPCBC SPD SSQ SST SSW SSZ T5K T8Q T8Y U1F U1G U5A U5I U5K UHS ~G- ~LA 9DU AATTM AAXKI AAYWO AAYXX ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO AEIPS AEUPX AFJKZ AFPUW AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP CITATION EFKBS ~HD AFXIZ AGCQF AGRNS BNPGV SSH |
| ID | FETCH-LOGICAL-c331t-dc420adfa25cceadea62dc5be13da326a5efde02d42ccbfd0f4f0f872e5e9c053 |
| ISICitedReferencesCount | 19 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000812961600001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 1007-5704 |
| IngestDate | Fri Jul 25 08:07:52 EDT 2025 Sat Nov 29 07:09:02 EST 2025 Tue Nov 18 22:45:25 EST 2025 Fri Feb 23 02:41:00 EST 2024 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Lie and conditional symmetry Traveling front Population dynamics Diffusive Lotka–Volterra system Exact solution |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c331t-dc420adfa25cceadea62dc5be13da326a5efde02d42ccbfd0f4f0f872e5e9c053 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0002-1733-5240 |
| PQID | 2691516444 |
| PQPubID | 2047477 |
| ParticipantIDs | proquest_journals_2691516444 crossref_primary_10_1016_j_cnsns_2022_106579 crossref_citationtrail_10_1016_j_cnsns_2022_106579 elsevier_sciencedirect_doi_10_1016_j_cnsns_2022_106579 |
| PublicationCentury | 2000 |
| PublicationDate | October 2022 2022-10-00 20221001 |
| PublicationDateYYYYMMDD | 2022-10-01 |
| PublicationDate_xml | – month: 10 year: 2022 text: October 2022 |
| PublicationDecade | 2020 |
| PublicationPlace | Amsterdam |
| PublicationPlace_xml | – name: Amsterdam |
| PublicationTitle | Communications in nonlinear science & numerical simulation |
| PublicationYear | 2022 |
| Publisher | Elsevier B.V Elsevier Science Ltd |
| Publisher_xml | – name: Elsevier B.V – name: Elsevier Science Ltd |
| References | Gudkov (b54) 1997; 353 Beteman (b57) 1955 Fife (b11) 1975 Fisher (b53) 1937; 7 Cherniha (b44) 2010; 43 Britton (b6) 2003 Cherniha, Davydovych (b49) 2017; vol. 2196 Aris (b12) 1975 Hung (b35) 2016; 8 Ablowitz, Zeppetella (b23) 1979; 41 Cherniha, Davydovych (b33) 2021; 9 Malfliet (b21) 2004; 164 Lotka (b5) 1910; 72 Zhdanov, Lahno (b56) 1998; 122 Polyanin, Zaitsev (b52) 2018 Rothe (b16) 1976; 3 Hung (b27) 2011; 12 Hirniak (b4) 1911; 75 Cherniha, Davydovych (b34) 2013; 46 Cherniha, Davydovych (b46) 2015; 268 Jorné, Carmi (b15) 1977; 37 Cherniha, Davydovych (b47) 2021; 32 Rodrigo, Mimura (b24) 2001; 18 Okubo, Levin (b9) 2001 Lou, Ni (b55) 1996; 131 Polyanin, Sorokin (b59) 2022; 125 Arrigo, Ekrut, Fliss, Le (b43) 2010; 371 Hou, Leung (b30) 2008; 9 Cherniha, Dutka (b22) 2004; 56 Hung (b25) 2012; 29 Leung, Hou, Feng (b31) 2011; 15 Malfliet, W. (b20) 1996; 54 Gilding, Kersner (b48) 2004 Kudryashov, Zakharchenko (b26) 2015; 254 Chen, Hung, Mimura, Ueyama (b28) 2012; 17 Pliukhin (b36) 2015; 7 Ugalde-Salas, Ramirez, Harm (b58) 2021; 8 Cherniha, Davydovych (b32) 2011; 54 Murray (b8) 2003 Murray (b7) 1989 Hastings (b14) 1978; 6 Conway, Smoller (b13) 1977; 33 Chen, Hung (b29) 2016; 15 Volterra (b2) 1926; 2 Kuang, Nagy, Eikenberry (b10) 2016 Polyanin, Zaitsev (b37) 2012 Cherniha, Serov, Pliukhin (b39) 2018 Lam, Salako, Wu (b18) 2020; 269 Bluman, Cheviakov, Anco (b38) 2010 Fushchych, Shtelen, Serov (b42) 1993 Rodrigo, Mimura (b19) 2000; 30 Bluman, Cole (b40) 1969; 18 Alhasanat, Ou (b17) 2019; 266 Bluman, Anco (b50) 2002 Hirniak (b3) 1908; 12 Fushchych, Serov, Chopyk (b41) 1988; 9 Torrisi, Tracina (b45) 2011; 12 Arrigo (b51) 2015 Lotka (b1) 1920; 42 Cherniha (10.1016/j.cnsns.2022.106579_b22) 2004; 56 Fushchych (10.1016/j.cnsns.2022.106579_b42) 1993 Lotka (10.1016/j.cnsns.2022.106579_b1) 1920; 42 Ugalde-Salas (10.1016/j.cnsns.2022.106579_b58) 2021; 8 Gudkov (10.1016/j.cnsns.2022.106579_b54) 1997; 353 Fisher (10.1016/j.cnsns.2022.106579_b53) 1937; 7 Cherniha (10.1016/j.cnsns.2022.106579_b46) 2015; 268 Cherniha (10.1016/j.cnsns.2022.106579_b44) 2010; 43 Hastings (10.1016/j.cnsns.2022.106579_b14) 1978; 6 Gilding (10.1016/j.cnsns.2022.106579_b48) 2004 Chen (10.1016/j.cnsns.2022.106579_b28) 2012; 17 Fushchych (10.1016/j.cnsns.2022.106579_b41) 1988; 9 Volterra (10.1016/j.cnsns.2022.106579_b2) 1926; 2 Bluman (10.1016/j.cnsns.2022.106579_b50) 2002 Ablowitz (10.1016/j.cnsns.2022.106579_b23) 1979; 41 Polyanin (10.1016/j.cnsns.2022.106579_b52) 2018 Cherniha (10.1016/j.cnsns.2022.106579_b49) 2017; vol. 2196 Kudryashov (10.1016/j.cnsns.2022.106579_b26) 2015; 254 Cherniha (10.1016/j.cnsns.2022.106579_b39) 2018 Malfliet (10.1016/j.cnsns.2022.106579_b21) 2004; 164 Hou (10.1016/j.cnsns.2022.106579_b30) 2008; 9 Murray (10.1016/j.cnsns.2022.106579_b8) 2003 Chen (10.1016/j.cnsns.2022.106579_b29) 2016; 15 Lotka (10.1016/j.cnsns.2022.106579_b5) 1910; 72 Zhdanov (10.1016/j.cnsns.2022.106579_b56) 1998; 122 Pliukhin (10.1016/j.cnsns.2022.106579_b36) 2015; 7 Britton (10.1016/j.cnsns.2022.106579_b6) 2003 Aris (10.1016/j.cnsns.2022.106579_b12) 1975 Cherniha (10.1016/j.cnsns.2022.106579_b34) 2013; 46 Cherniha (10.1016/j.cnsns.2022.106579_b32) 2011; 54 Bluman (10.1016/j.cnsns.2022.106579_b40) 1969; 18 Torrisi (10.1016/j.cnsns.2022.106579_b45) 2011; 12 Rothe (10.1016/j.cnsns.2022.106579_b16) 1976; 3 Rodrigo (10.1016/j.cnsns.2022.106579_b24) 2001; 18 Arrigo (10.1016/j.cnsns.2022.106579_b51) 2015 Hirniak (10.1016/j.cnsns.2022.106579_b3) 1908; 12 Okubo (10.1016/j.cnsns.2022.106579_b9) 2001 Leung (10.1016/j.cnsns.2022.106579_b31) 2011; 15 Hung (10.1016/j.cnsns.2022.106579_b27) 2011; 12 Hirniak (10.1016/j.cnsns.2022.106579_b4) 1911; 75 Cherniha (10.1016/j.cnsns.2022.106579_b33) 2021; 9 Fife (10.1016/j.cnsns.2022.106579_b11) 1975 Rodrigo (10.1016/j.cnsns.2022.106579_b19) 2000; 30 Polyanin (10.1016/j.cnsns.2022.106579_b37) 2012 Cherniha (10.1016/j.cnsns.2022.106579_b47) 2021; 32 Malfliet (10.1016/j.cnsns.2022.106579_b20) 1996; 54 Bluman (10.1016/j.cnsns.2022.106579_b38) 2010 Lam (10.1016/j.cnsns.2022.106579_b18) 2020; 269 Arrigo (10.1016/j.cnsns.2022.106579_b43) 2010; 371 Alhasanat (10.1016/j.cnsns.2022.106579_b17) 2019; 266 Lou (10.1016/j.cnsns.2022.106579_b55) 1996; 131 Murray (10.1016/j.cnsns.2022.106579_b7) 1989 Hung (10.1016/j.cnsns.2022.106579_b35) 2016; 8 Beteman (10.1016/j.cnsns.2022.106579_b57) 1955 Conway (10.1016/j.cnsns.2022.106579_b13) 1977; 33 Polyanin (10.1016/j.cnsns.2022.106579_b59) 2022; 125 Jorné (10.1016/j.cnsns.2022.106579_b15) 1977; 37 Kuang (10.1016/j.cnsns.2022.106579_b10) 2016 Hung (10.1016/j.cnsns.2022.106579_b25) 2012; 29 |
| References_xml | – volume: 371 start-page: 813 year: 2010 end-page: 820 ident: b43 article-title: Nonclassical symmetries of a class of Burgers’ systems publication-title: J Math Anal Appl – year: 2004 ident: b48 article-title: Travelling Waves in Nonlinear Reaction-Convection–Diffusion – volume: 122 start-page: 178 year: 1998 end-page: 186 ident: b56 article-title: Conditional symmetry of a porous medium equation publication-title: Physica D – volume: 7 start-page: 353 year: 1937 end-page: 369 ident: b53 article-title: The wave of advance of advantageous genes publication-title: Ann Eugenics – volume: 17 start-page: 2653 year: 2012 end-page: 2669 ident: b28 article-title: Exact travelling wave solutions of three-species competition-diffusion systems publication-title: Discrete Contin Dyn Syst Ser B – volume: 37 start-page: 51 year: 1977 end-page: 61 ident: b15 article-title: Liapunov stability of the diffusive Lotka–Volterra equations publication-title: Math Biosci – volume: 15 start-page: 1451 year: 2016 ident: b29 article-title: Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka–Volterra systems of three competing species publication-title: Commun Pure Appl Anal – volume: 56 start-page: 1665 year: 2004 end-page: 1675 ident: b22 article-title: A diffusive Lotka–Volterra system: Lie symmetries, exact and numerical solutions publication-title: Ukr Math J – volume: 42 start-page: 1595 year: 1920 end-page: 1599 ident: b1 article-title: Undamped oscillations derived from the law of mass action publication-title: J Am Chem Soc – volume: 72 start-page: 508 year: 1910 end-page: 511 ident: b5 article-title: Zur Theorie der periodischen Reaktion (in German) publication-title: Z Phys Chem – volume: 54 start-page: 563 year: 1996 end-page: 568 ident: b20 article-title: The tanh method: I. exact solutions of nonlinear evolution and wave equations publication-title: Phys Scripta – volume: 164 start-page: 529 year: 2004 end-page: 541 ident: b21 article-title: The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations publication-title: J Comp Appl Math – volume: 54 start-page: 1238 year: 2011 end-page: 1251 ident: b32 article-title: Conditional symmetries and exact solutions of the diffusive Lotka–Volterra system publication-title: Math Comput Modelling – volume: 125 year: 2022 ident: b59 article-title: Reductions and exact solutions of Lotka–Volterra and more complex reaction–diffusion systems with delays publication-title: Appl Math Lett – volume: 254 start-page: 219 year: 2015 end-page: 228 ident: b26 article-title: Analytical properties and exact solutions of the Lotka–Volterra competition system publication-title: Appl Math Comput – year: 1975 ident: b12 article-title: The mathematical theory of diffusion and reaction in permeable catalysts: the theory of the steady state – volume: 43 year: 2010 ident: b44 article-title: Conditional symmetries for systems of PDEs: new definition and their application for reaction–diffusion systems publication-title: J Phys A Math Theor – volume: 8 start-page: 31 year: 2021 ident: b58 article-title: Microbial interactions as drivers of a nitrification process in a chemostat publication-title: Bioengineering – volume: 9 start-page: 1984 year: 2021 ident: b33 article-title: New conditional symmetries and exact solutions of the diffusive two-component Lotka–Volterra system publication-title: Mathematics – volume: 18 start-page: 657 year: 2001 end-page: 696 ident: b24 article-title: Exact solutions of reaction–diffusion systems and nonlinear wave equations publication-title: Japan J Indu Appl Math – volume: 15 start-page: 171 year: 2011 end-page: 196 ident: b31 article-title: Traveling wave solutions for Lotka–Volterra system re-visited publication-title: Discrete Contin Dyn Syst Ser B – volume: 46 year: 2013 ident: b34 article-title: Lie and conditional symmetries of the three-component diffusive Lotka–Volterra system publication-title: J Phys A Math Theor – volume: 33 start-page: 673 year: 1977 end-page: 686 ident: b13 article-title: Diffusion and the predator–prey interaction publication-title: SIAM J Appl Math – volume: 30 start-page: 257 year: 2000 end-page: 270 ident: b19 article-title: Exact solutions of a competition-diffusion system publication-title: Hiroshima Math J – volume: vol. 2196 year: 2017 ident: b49 publication-title: Nonlinear reaction–diffusion systems — conditional symmetry, exact solutions and their applications in biology – year: 2003 ident: b6 article-title: Essential mathematical biology – year: 1955 ident: b57 article-title: Higher transcendental functions – volume: 75 start-page: 675 year: 1911 end-page: 680 ident: b4 article-title: Zur frage der periodischen reaktionen (in German) publication-title: Z Phys Chem – volume: 131 start-page: 79 year: 1996 end-page: 131 ident: b55 article-title: Diffusion, self-diffusion and cross-diffusion publication-title: J Differ Equ – volume: 3 start-page: 319 year: 1976 end-page: 324 ident: b16 article-title: Convergence to the equilibrium state in the Volterra–Lotka diffusion equations publication-title: J Math Biol – volume: 269 start-page: 10758 year: 2020 end-page: 10791 ident: b18 article-title: Entire solutions of diffusive Lotka–Volterra system publication-title: J Differ Equ – volume: 41 start-page: 835 year: 1979 end-page: 840 ident: b23 article-title: Explicit solutions of Fisher’s equation for a special wave speed publication-title: Bull Math Biol – volume: 18 start-page: 1025 year: 1969 end-page: 1042 ident: b40 article-title: The general similarity solution of the heat equation publication-title: J Math Mech – year: 2003 ident: b8 article-title: Mathematical biology II – volume: 2 start-page: 31 year: 1926 end-page: 113 ident: b2 article-title: Variazioni e fluttuazioni del numero d‘individui in specie animali conviventi publication-title: Mem Acad Lincei – year: 1975 ident: b11 article-title: Mathematical aspects of reacting and diffusing systems – year: 2016 ident: b10 article-title: Introduction to mathematical oncology – year: 2012 ident: b37 article-title: Handbook of nonlinear partial differential equations – volume: 32 start-page: 280 year: 2021 end-page: 300 ident: b47 article-title: Conditional symmetries and exact solutions of a nonlinear three-component reaction–diffusion model publication-title: Euro J Appl Math – volume: 268 start-page: 23 year: 2015 end-page: 34 ident: b46 article-title: Nonlinear reaction–diffusion systems with a non-constant diffusivity: conditional symmetries in no-go case publication-title: Appl Math Comput – year: 2010 ident: b38 article-title: Applications of symmetry methods to partial differential equations – year: 2018 ident: b52 article-title: Handbook of ordinary differential equations: exact solutions, methods, and problems – year: 2018 ident: b39 article-title: Nonlinear reaction–diffusion-convection equations: lie and conditional symmetry, exact solutions and their applications – volume: 6 start-page: 163 year: 1978 end-page: 168 ident: b14 article-title: Global stability in Lotka–Volterra systems with diffusion publication-title: J Math Biol – volume: 9 start-page: 17 year: 1988 end-page: 21 ident: b41 article-title: Conditional invariance and nonlinear heat equations (in Ukrainian) publication-title: Proc Acad Sci Ukraine – volume: 12 start-page: 1 year: 1908 end-page: 8 ident: b3 article-title: About periodical chemical reactions (in Ukrainian), Shevchenko Scientific Society in Lviv publication-title: Section Math-Nat-Med – year: 1993 ident: b42 article-title: Symmetry analysis and exact solutions of equations of nonlinear mathematical physics – year: 1989 ident: b7 article-title: Mathematical biology – year: 2001 ident: b9 article-title: Diffusion and ecological problems publication-title: Modern perspectives – volume: 8 start-page: 501 year: 2016 end-page: 520 ident: b35 article-title: Diffusive solutions of the competitive Lotka–Volterra system publication-title: J Difference Equ Appl – year: 2002 ident: b50 article-title: Symmetry and integration methods for differential equations – volume: 12 start-page: 700 year: 2011 end-page: 3691 ident: b27 article-title: Traveling wave solutions of competitive-cooperative Lotka–Volterra systems of three species publication-title: Nonlinear Anal RWA – year: 2015 ident: b51 article-title: Symmetry analysis of differential equations – volume: 353 start-page: 439 year: 1997 end-page: 441 ident: b54 article-title: Exact solutions of the type of propagating waves for certain evolution equations publication-title: Dokl Ros Akad Nauk – volume: 7 start-page: 1841 year: 2015 end-page: 1855 ident: b36 article-title: -Conditional symmetries and exact solutions of nonlinear reaction–diffusion systems publication-title: Symmetry – volume: 29 start-page: 237 year: 2012 end-page: 251 ident: b25 article-title: Exact traveling wave solutions for diffusive Lotka–Volterra systems of two competing species publication-title: Jpn J Indust Appl Math – volume: 12 start-page: 1865 year: 2011 end-page: 1874 ident: b45 article-title: Exact solutions of a reaction–diffusion system for proteus mirabilis bacterial colonies publication-title: Nonlinear Anal RWA – volume: 9 start-page: 213 year: 2008 end-page: 2196 ident: b30 article-title: Traveling wave solutions for a competitive reaction–diffusion system and their asymptotics publication-title: Nonlinear Anal RWA – volume: 266 start-page: 7357 year: 2019 end-page: 7378 ident: b17 article-title: Minimal-speed selection of travelling waves to the Lotka–Volterra competition model publication-title: J Differ Equ – year: 1975 ident: 10.1016/j.cnsns.2022.106579_b11 – volume: 75 start-page: 675 year: 1911 ident: 10.1016/j.cnsns.2022.106579_b4 article-title: Zur frage der periodischen reaktionen (in German) publication-title: Z Phys Chem doi: 10.1515/zpch-1911-7549 – year: 2003 ident: 10.1016/j.cnsns.2022.106579_b8 – volume: 32 start-page: 280 year: 2021 ident: 10.1016/j.cnsns.2022.106579_b47 article-title: Conditional symmetries and exact solutions of a nonlinear three-component reaction–diffusion model publication-title: Euro J Appl Math doi: 10.1017/S0956792520000121 – year: 2004 ident: 10.1016/j.cnsns.2022.106579_b48 – volume: 7 start-page: 353 year: 1937 ident: 10.1016/j.cnsns.2022.106579_b53 article-title: The wave of advance of advantageous genes publication-title: Ann Eugenics doi: 10.1111/j.1469-1809.1937.tb02153.x – year: 1989 ident: 10.1016/j.cnsns.2022.106579_b7 – volume: vol. 2196 year: 2017 ident: 10.1016/j.cnsns.2022.106579_b49 – year: 1993 ident: 10.1016/j.cnsns.2022.106579_b42 – volume: 8 start-page: 501 year: 2016 ident: 10.1016/j.cnsns.2022.106579_b35 article-title: Diffusive solutions of the competitive Lotka–Volterra system publication-title: J Difference Equ Appl – volume: 54 start-page: 1238 year: 2011 ident: 10.1016/j.cnsns.2022.106579_b32 article-title: Conditional symmetries and exact solutions of the diffusive Lotka–Volterra system publication-title: Math Comput Modelling doi: 10.1016/j.mcm.2011.03.035 – volume: 54 start-page: 563 year: 1996 ident: 10.1016/j.cnsns.2022.106579_b20 article-title: The tanh method: I. exact solutions of nonlinear evolution and wave equations publication-title: Phys Scripta doi: 10.1088/0031-8949/54/6/003 – year: 2001 ident: 10.1016/j.cnsns.2022.106579_b9 article-title: Diffusion and ecological problems – volume: 6 start-page: 163 year: 1978 ident: 10.1016/j.cnsns.2022.106579_b14 article-title: Global stability in Lotka–Volterra systems with diffusion publication-title: J Math Biol doi: 10.1007/BF02450786 – volume: 353 start-page: 439 year: 1997 ident: 10.1016/j.cnsns.2022.106579_b54 article-title: Exact solutions of the type of propagating waves for certain evolution equations publication-title: Dokl Ros Akad Nauk – volume: 254 start-page: 219 year: 2015 ident: 10.1016/j.cnsns.2022.106579_b26 article-title: Analytical properties and exact solutions of the Lotka–Volterra competition system publication-title: Appl Math Comput – year: 2003 ident: 10.1016/j.cnsns.2022.106579_b6 – volume: 371 start-page: 813 year: 2010 ident: 10.1016/j.cnsns.2022.106579_b43 article-title: Nonclassical symmetries of a class of Burgers’ systems publication-title: J Math Anal Appl doi: 10.1016/j.jmaa.2010.06.026 – volume: 266 start-page: 7357 year: 2019 ident: 10.1016/j.cnsns.2022.106579_b17 article-title: Minimal-speed selection of travelling waves to the Lotka–Volterra competition model publication-title: J Differ Equ doi: 10.1016/j.jde.2018.12.003 – year: 2002 ident: 10.1016/j.cnsns.2022.106579_b50 – volume: 43 year: 2010 ident: 10.1016/j.cnsns.2022.106579_b44 article-title: Conditional symmetries for systems of PDEs: new definition and their application for reaction–diffusion systems publication-title: J Phys A Math Theor doi: 10.1088/1751-8113/43/40/405207 – volume: 29 start-page: 237 year: 2012 ident: 10.1016/j.cnsns.2022.106579_b25 article-title: Exact traveling wave solutions for diffusive Lotka–Volterra systems of two competing species publication-title: Jpn J Indust Appl Math doi: 10.1007/s13160-012-0056-2 – volume: 12 start-page: 1865 year: 2011 ident: 10.1016/j.cnsns.2022.106579_b45 article-title: Exact solutions of a reaction–diffusion system for proteus mirabilis bacterial colonies publication-title: Nonlinear Anal RWA doi: 10.1016/j.nonrwa.2010.12.004 – year: 2016 ident: 10.1016/j.cnsns.2022.106579_b10 – volume: 37 start-page: 51 year: 1977 ident: 10.1016/j.cnsns.2022.106579_b15 article-title: Liapunov stability of the diffusive Lotka–Volterra equations publication-title: Math Biosci doi: 10.1016/0025-5564(77)90077-3 – volume: 9 start-page: 17 year: 1988 ident: 10.1016/j.cnsns.2022.106579_b41 article-title: Conditional invariance and nonlinear heat equations (in Ukrainian) publication-title: Proc Acad Sci Ukraine – year: 1975 ident: 10.1016/j.cnsns.2022.106579_b12 – volume: 12 start-page: 1 year: 1908 ident: 10.1016/j.cnsns.2022.106579_b3 article-title: About periodical chemical reactions (in Ukrainian), Shevchenko Scientific Society in Lviv publication-title: Section Math-Nat-Med – volume: 3 start-page: 319 year: 1976 ident: 10.1016/j.cnsns.2022.106579_b16 article-title: Convergence to the equilibrium state in the Volterra–Lotka diffusion equations publication-title: J Math Biol doi: 10.1007/BF00275064 – year: 2012 ident: 10.1016/j.cnsns.2022.106579_b37 – volume: 269 start-page: 10758 year: 2020 ident: 10.1016/j.cnsns.2022.106579_b18 article-title: Entire solutions of diffusive Lotka–Volterra system publication-title: J Differ Equ doi: 10.1016/j.jde.2020.07.006 – volume: 18 start-page: 1025 year: 1969 ident: 10.1016/j.cnsns.2022.106579_b40 article-title: The general similarity solution of the heat equation publication-title: J Math Mech – volume: 41 start-page: 835 year: 1979 ident: 10.1016/j.cnsns.2022.106579_b23 article-title: Explicit solutions of Fisher’s equation for a special wave speed publication-title: Bull Math Biol doi: 10.1016/S0092-8240(79)80020-8 – volume: 17 start-page: 2653 year: 2012 ident: 10.1016/j.cnsns.2022.106579_b28 article-title: Exact travelling wave solutions of three-species competition-diffusion systems publication-title: Discrete Contin Dyn Syst Ser B doi: 10.3934/dcdsb.2012.17.2653 – volume: 72 start-page: 508 year: 1910 ident: 10.1016/j.cnsns.2022.106579_b5 article-title: Zur Theorie der periodischen Reaktion (in German) publication-title: Z Phys Chem doi: 10.1515/zpch-1910-7229 – volume: 164 start-page: 529 year: 2004 ident: 10.1016/j.cnsns.2022.106579_b21 article-title: The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations publication-title: J Comp Appl Math doi: 10.1016/S0377-0427(03)00645-9 – volume: 9 start-page: 213 year: 2008 ident: 10.1016/j.cnsns.2022.106579_b30 article-title: Traveling wave solutions for a competitive reaction–diffusion system and their asymptotics publication-title: Nonlinear Anal RWA doi: 10.1016/j.nonrwa.2007.07.007 – volume: 46 year: 2013 ident: 10.1016/j.cnsns.2022.106579_b34 article-title: Lie and conditional symmetries of the three-component diffusive Lotka–Volterra system publication-title: J Phys A Math Theor doi: 10.1088/1751-8113/46/18/185204 – volume: 125 year: 2022 ident: 10.1016/j.cnsns.2022.106579_b59 article-title: Reductions and exact solutions of Lotka–Volterra and more complex reaction–diffusion systems with delays publication-title: Appl Math Lett doi: 10.1016/j.aml.2021.107731 – volume: 268 start-page: 23 year: 2015 ident: 10.1016/j.cnsns.2022.106579_b46 article-title: Nonlinear reaction–diffusion systems with a non-constant diffusivity: conditional symmetries in no-go case publication-title: Appl Math Comput – volume: 15 start-page: 1451 year: 2016 ident: 10.1016/j.cnsns.2022.106579_b29 article-title: Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka–Volterra systems of three competing species publication-title: Commun Pure Appl Anal doi: 10.3934/cpaa.2016.15.1451 – volume: 9 start-page: 1984 year: 2021 ident: 10.1016/j.cnsns.2022.106579_b33 article-title: New conditional symmetries and exact solutions of the diffusive two-component Lotka–Volterra system publication-title: Mathematics doi: 10.3390/math9161984 – volume: 131 start-page: 79 year: 1996 ident: 10.1016/j.cnsns.2022.106579_b55 article-title: Diffusion, self-diffusion and cross-diffusion publication-title: J Differ Equ doi: 10.1006/jdeq.1996.0157 – volume: 7 start-page: 1841 year: 2015 ident: 10.1016/j.cnsns.2022.106579_b36 article-title: Q-Conditional symmetries and exact solutions of nonlinear reaction–diffusion systems publication-title: Symmetry doi: 10.3390/sym7041841 – volume: 42 start-page: 1595 year: 1920 ident: 10.1016/j.cnsns.2022.106579_b1 article-title: Undamped oscillations derived from the law of mass action publication-title: J Am Chem Soc doi: 10.1021/ja01453a010 – volume: 56 start-page: 1665 year: 2004 ident: 10.1016/j.cnsns.2022.106579_b22 article-title: A diffusive Lotka–Volterra system: Lie symmetries, exact and numerical solutions publication-title: Ukr Math J doi: 10.1007/s11253-005-0142-6 – volume: 15 start-page: 171 year: 2011 ident: 10.1016/j.cnsns.2022.106579_b31 article-title: Traveling wave solutions for Lotka–Volterra system re-visited publication-title: Discrete Contin Dyn Syst Ser B – year: 2018 ident: 10.1016/j.cnsns.2022.106579_b52 – volume: 8 start-page: 31 year: 2021 ident: 10.1016/j.cnsns.2022.106579_b58 article-title: Microbial interactions as drivers of a nitrification process in a chemostat publication-title: Bioengineering doi: 10.3390/bioengineering8030031 – year: 2018 ident: 10.1016/j.cnsns.2022.106579_b39 – year: 2015 ident: 10.1016/j.cnsns.2022.106579_b51 – year: 1955 ident: 10.1016/j.cnsns.2022.106579_b57 – year: 2010 ident: 10.1016/j.cnsns.2022.106579_b38 – volume: 12 start-page: 700 year: 2011 ident: 10.1016/j.cnsns.2022.106579_b27 article-title: Traveling wave solutions of competitive-cooperative Lotka–Volterra systems of three species publication-title: Nonlinear Anal RWA doi: 10.1016/j.nonrwa.2011.07.002 – volume: 122 start-page: 178 year: 1998 ident: 10.1016/j.cnsns.2022.106579_b56 article-title: Conditional symmetry of a porous medium equation publication-title: Physica D doi: 10.1016/S0167-2789(98)00191-2 – volume: 33 start-page: 673 year: 1977 ident: 10.1016/j.cnsns.2022.106579_b13 article-title: Diffusion and the predator–prey interaction publication-title: SIAM J Appl Math doi: 10.1137/0133047 – volume: 2 start-page: 31 year: 1926 ident: 10.1016/j.cnsns.2022.106579_b2 article-title: Variazioni e fluttuazioni del numero d‘individui in specie animali conviventi publication-title: Mem Acad Lincei – volume: 18 start-page: 657 year: 2001 ident: 10.1016/j.cnsns.2022.106579_b24 article-title: Exact solutions of reaction–diffusion systems and nonlinear wave equations publication-title: Japan J Indu Appl Math doi: 10.1007/BF03167410 – volume: 30 start-page: 257 year: 2000 ident: 10.1016/j.cnsns.2022.106579_b19 article-title: Exact solutions of a competition-diffusion system publication-title: Hiroshima Math J doi: 10.32917/hmj/1206124686 |
| SSID | ssj0016954 |
| Score | 2.470189 |
| SecondaryResourceType | review_article |
| Snippet | This review summarizes all known results (up to this date) about methods of integration of the classical Lotka–Volterra systems with diffusion and presents a... |
| SourceID | proquest crossref elsevier |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 106579 |
| SubjectTerms | Applications of mathematics Approximation Boundary value problems Diffusive Lotka–Volterra system Exact solution Exact solutions Lie and conditional symmetry Mathematical models Numerical analysis Numerical methods Population dynamics Traveling front |
| Title | Construction and application of exact solutions of the diffusive Lotka–Volterra system: A review and new results |
| URI | https://dx.doi.org/10.1016/j.cnsns.2022.106579 https://www.proquest.com/docview/2691516444 |
| Volume | 113 |
| WOSCitedRecordID | wos000812961600001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1878-7274 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0016954 issn: 1007-5704 databaseCode: AIEXJ dateStart: 19960101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Lb9NAEF6FlAMX3ohCQXvgVlzFazu2uUVQBKiqECpVbtZmH1KKa1d2EiW3_oee-XP8EmafNq2I6IGLlWzs0Srz7Xhmd-YbhN5koZCUMBnwOBVBzMMoyBkXAVFka4kUJOFa00fp8XE2neZfB4OfrhZmVaZVla3X-cV_VTWMgbJV6ewt1O2FwgB8BqXDFdQO139SvGrB6UhhDRVrd0atXEOxVmWRfg4uSUB1SlnqXPajevGDuiSI6LRW5-kNtZzPppLdFrwo6eCWw9d2WRpKKM960C880Tm3lSHloM2-KyVSqKuW5sio3G_n57aVWJdyIJpqbk6kvtXnHY4_0NWG16uNaWN1SttN6eab97cxIAJ2CXHO8qo90yQ1vYi9aQ6jnnGF6DUxnWdu2H2zBXF2wKq2UiTshBx0d__Jsn3t7edzEl2621mhhRRKSGGE3EE7gN08G6KdyefD6Rd_TDXOdZs9P3dHa6UTCG_M5W-uzzUnQHs2Jw_RfRuS4ImB0iM0ENVj9MCGJ9ga__YJavrIwqB73EMWriXWyMIeWWoIkIU9srBG1q_LK4cpbDD1Dk-wQZSWCojCFlFP0fePhyfvPwW2ZUfAoihcBJzFZEQ5LP-EMZWLT8eEs2QmwohTiBRoIiQXI8JjwthM8pGM5UhmKRGJyBm8EJ6hIcBRPEcY4vKYkVCGNI5iPqKZykCkjGVslkcJo7uIuH-zYJbPXrVVKYstmtxFb_1DF4bOZfvtY6emwq4N42kWALztD-45pRbWNsDv4xz8awhA4he3m8ZLdK9bMntoCKoWr9BdtlrM2-a1BeVvXba-sg |
| linkProvider | Elsevier |
| 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=Construction+and+application+of+exact+solutions+of+the+diffusive+Lotka%E2%80%93Volterra+system%3A+A+review+and+new+results&rft.jtitle=Communications+in+nonlinear+science+%26+numerical+simulation&rft.au=Cherniha%2C+Roman&rft.au=Davydovych%2C+Vasyl%E2%80%99&rft.date=2022-10-01&rft.issn=1007-5704&rft.volume=113&rft.spage=106579&rft_id=info:doi/10.1016%2Fj.cnsns.2022.106579&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_cnsns_2022_106579 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1007-5704&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1007-5704&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1007-5704&client=summon |