Dynamical Systems Under Constant Organization II: Homogeneous Growth Functions of Degree $p = 2$
Abstract
Qualitative analysis is presented for a system of differential equations, which play an important role in a theory of molecular self-organization: \[\dot x_i = \left( {\sum\limits_{p = 1}^n {k_{ip} x_p - \sum\limits_p {\sum\limits_q {k_{pq} x_p x_q } } } } \right)x_i ,\quad i = 1, \cdots ,n\]. Besides the general case two simplifications are treated: (1) the norihyperbolic case: $k_{ij} \mathop > \limits_ = 0(k_{ii} = 0)$ and (2) cyclic symmetry: $k_{ij} = k_{i + 1,j + 1} $. Criteria for cooperation and exclusion are derived.
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In homogeneous hypercycle (Ai=0), all members coexist.
“If for all i, Ai = 0, i.e., the replicators cannot replicate on their own, only with the help of another catalyst, then the system is fully cooperative and all members coexist [51,52,53,54,55,56].”
From Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems (2017) · cites this paper as [55]
Cites
Selforganization of matter and the evolution of biological macromolecules
Eigen, M. (1971) · Die Naturwissenschaften · doi:10.1007/bf00623322 · Cited by 3,788 · see citing works
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Group selection of early replicators and the origin of life
Szathmáry, E., & Demeter, L. (1987) · Journal of Theoretical Biology · doi:10.1016/s0022-5193(87)80191-1 · Cited by 407 · see citing works
Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems
Szilágyi, A., Zachar, I., Scheuring, I., Kun, Á., Könnyű, B., & Czárán, T. (2017) · Life · doi:10.3390/life7040048 · Cited by 44 · see citing works
A Difference Equation Model for the Hypercycle
Hofbauer, J. (1984) · SIAM Journal on Applied Mathematics · doi:10.1137/0144054 · Cited by 22 · see citing works
Self-citation
- Authors
- Hofbauer, J., Schuster, P., Sigmund, K., & Wolff, R.
- Venue
- SIAM Journal on Applied Mathematics
- Published
- 1980-04
- Volume
- 38
- Issue
- 2
- Pages
- 282-304
- ISSN
- 0036-1399 (ISSN-L)
- 1095-712X
- OpenAlex
- W1995411255
- Cited by
- 77
- References
- 1
- Citations by year
- 2026: 2 · 2025: 1 · 2023: 1 · 2022: 3 · 2021: 2 · 2020: 3 · 2019: 1 · 2018: 3 · 2017: 2 · 2016: 4 · 2015: 1 · 2014: 1
- FWCI
- 11.07
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- SIAM Journal on Applied Mathematics (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Equations and Dynamical Systems
- Physical Sciences › Computer Science › Computer Networks and Communications › Nonlinear Dynamics and Pattern Formation
- Health Sciences › Medicine › Public Health, Environmental and Occupational Health › Mathematical and Theoretical Epidemiology and Ecology Models
- Keywords
- molecular self-organization, qualitative theory of differential equations, cooperative dynamics, cyclic symmetry, dynamical systems
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