{"title":"Influence of Boundary Conditions on the Dynamic Properties of the Logistic Equation with Delay and Diffusion","authors":"S.A. Kashchenko, D.O. Loginov","doi":"10.1134/S106192082404006X","DOIUrl":null,"url":null,"abstract":"<p> The logistic equation with delay and diffusion, which is important in mathematical ecology, is considered. It is assumed that the boundary conditions at either end of the interval [0,1] contain parameters. The problem of local dynamics, in a neighborhood of the equilibrium state, of the corresponding boundary value problem is investigated for all values of the boundary condition parameters. Critical cases are identified in the problem of stability of the equilibrium state and normal forms are constructed, which are scalar complex ordinary differential equations of the first order. Their nonlocal dynamics determines the behavior of solutions of the original problem in a small neighborhood of the equilibrium state. The problem of the role of asymptotically small values of the diffusion coefficient in the dynamics of the boundary value problems under consideration is studied separately. In particular, it is shown that boundary layer functions may arise when constructing asymptotic solutions in a neighborhood of the boundary points 0 and 1. </p><p> <b> DOI</b> 10.1134/S106192082404006X </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"31 4","pages":"666 - 681"},"PeriodicalIF":1.7000,"publicationDate":"2025-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S106192082404006X","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}
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Abstract
The logistic equation with delay and diffusion, which is important in mathematical ecology, is considered. It is assumed that the boundary conditions at either end of the interval [0,1] contain parameters. The problem of local dynamics, in a neighborhood of the equilibrium state, of the corresponding boundary value problem is investigated for all values of the boundary condition parameters. Critical cases are identified in the problem of stability of the equilibrium state and normal forms are constructed, which are scalar complex ordinary differential equations of the first order. Their nonlocal dynamics determines the behavior of solutions of the original problem in a small neighborhood of the equilibrium state. The problem of the role of asymptotically small values of the diffusion coefficient in the dynamics of the boundary value problems under consideration is studied separately. In particular, it is shown that boundary layer functions may arise when constructing asymptotic solutions in a neighborhood of the boundary points 0 and 1.
期刊介绍：
Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.
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