{"title":"Heterogeneous mean-field analysis of the generalized Lotka–Volterra model on a network","authors":"Fabián Aguirre-López","doi":"10.1088/1751-8121/ad6ab2","DOIUrl":null,"url":null,"abstract":"We study the dynamics of the generalized Lotka–Volterra model with a network structure. Performing a high connectivity expansion for graphs, we write down a mean-field dynamical theory that incorporates degree heterogeneity. This allows us to describe the fixed points of the model in terms of a few simple order parameters. We extend the analysis even for diverging abundances, using a mapping to the replicator model. With this we present a unified approach for both cooperative and competitive systems that display complementary behaviors. In particular we show the central role of an order parameter called the critical degree, <inline-formula>\n<tex-math><?CDATA $g_\\mathrm{c}$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">c</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href=\"aad6ab2ieqn1.gif\"></inline-graphic></inline-formula>. In the competitive regime <inline-formula>\n<tex-math><?CDATA $g_\\mathrm{c}$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">c</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href=\"aad6ab2ieqn2.gif\"></inline-graphic></inline-formula> serves to distinguish high degree nodes that are more likely to go extinct, while in the cooperative regime it has the reverse role, it will determine the low degree nodes that tend to go relatively extinct.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"161 1","pages":""},"PeriodicalIF":2.0000,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Physics A: Mathematical and Theoretical","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1751-8121/ad6ab2","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We study the dynamics of the generalized Lotka–Volterra model with a network structure. Performing a high connectivity expansion for graphs, we write down a mean-field dynamical theory that incorporates degree heterogeneity. This allows us to describe the fixed points of the model in terms of a few simple order parameters. We extend the analysis even for diverging abundances, using a mapping to the replicator model. With this we present a unified approach for both cooperative and competitive systems that display complementary behaviors. In particular we show the central role of an order parameter called the critical degree, gc. In the competitive regime gc serves to distinguish high degree nodes that are more likely to go extinct, while in the cooperative regime it has the reverse role, it will determine the low degree nodes that tend to go relatively extinct.
期刊介绍:
Publishing 50 issues a year, Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.