Abstract
A universal feature of open complex systems such as reaction networks in living organisms, ecosystems, and social communities is that their complexity emerges, or at least persists, under successive introductions of new elements. To have a general and simple understanding for the basic condition to let such systems grow, we investigate a simple mathematical process. It is found that the model system either grows infinitely large or stays finite depending upon the model’s unique parameter m, the average number of interactions per element. Comparing to the classical diversity-stability relation based on a linear-stability analysis, the condition for our system to grow is more moderate. The mechanism of this novel universal transition is described in detail. The characteristics of the model, such as lifetime distribution of the elements, and its relevance to the previous works such as SOC models are also discussed.
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Notes
- 1.
m∕2 is an average number of incoming links; therefore, the actual variance of the species with m links is slightly broader.
- 2.
Here we use the relation \(G(\sigma,x) = \left (\frac{\sigma ^{{\prime}}} {\sigma }\right )G\left (\sigma ^{{\prime}}, \frac{\sigma ^{{\prime}}} {\sigma } x\right )\) and \(\quad \int _{0}^{\infty }G(\sigma,\xi )\Phi (\xi )d\xi = \frac{1} {4} + \frac{\arctan (\sigma )} {2\pi }.\)
- 3.
\(\sqrt{\frac{1 +\sigma _{ m=10 }^{2 }} {\sigma _{m=10}^{2}}} = 1.0198,\quad \sqrt{\frac{1 +\sigma _{ m=20 }^{2 }} {\sigma _{m=20}^{2}}} = 1.005\)
- 4.
For the following calculation, we reuse the formulas in 1.3.4.1.
References
Shimada T (2014) A universal transition in the robustness of evolving open systems. Sci Rep 4:4082
Gardner MR, Ashby WR (1970) Connectance of large dynamic (cybernetic) systems: critical values for stability. Nature 228:784–784
May RM (1972) Will a large complex system be stable? Nature 238:413–414
McCann KS (2000) The diversity-stability debate. Nature 405:228–233
Roberts A (1974) The stability of a feasible random ecosystem. Nature 251:607–608
Allesina S, Tang S (2012) Stability criteria for complex ecosystems. Nature 483:205–208
Mougi A, Kondoh M (2012) Diversity of interaction types and ecological community stability. Science 337:349–351
Taylor PJ (1988) Consistent scaling and parameter choice for linear and generalized Lotka-Volterra models used in community ecology. J Theor Biol 135:543–568
Taylor PJ (1988) The construction and turnover of complex community models having generalized Lotka-Volterra dynamics. J Theor Biol 135:569–588
Tokita K, Yasutomi A (1999) Mass extinction in a dynamical system of evolution with variable dimension. Phys Rev E 60:842–847
Shimada T, Yukawa S, Ito N (2002) Self-organization in an ecosystem. Artif Life Robot 6:78–81
Shimada T, Murase Y, Ito N, Aihara K (2007) A simple model of evolving ecosystems. Artif Life Robot 11:153–156
Perotti JI, Billoni OV, Tamarit FA, Dante R, Chialvo DR, Cannas SA (2009) Emergent self-organized complex network topology out of stability constraints. Phys Rev Lett 103:108701
Benincà E, Huisman J, Heerkloss R, Jöhnk KD, Branco P, Van Nes EH, Scheffer M, Ellner SP (2008) Chaos in a long-term experiment with a plankton community. Nature 451:822–825
Pimm SL (1979) Complexity and stability: another look at MacArthur’s original hypothesis. OIKOS 33:351–357
Pimm SL (1980) Food web design and the effect of species deletion. OIKOS 35:139–149
Albert R, Jeong H, Barabási A-L (2000) Error and attack tolerance of complex networks. Nature 406:378–82
Moreira AA, Andrade JS, Herrmann HJ, Joseph OI (2009) How to make a fragile network robust and vice versa. Phys Rev Lett 102:018701
Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S (2010) Catastrophic cascade of failures in interdependent networks. Nature 464:1025–1028
Herrmann HJ, Schneider CM, Moreira AA, Andrade JS, Havlin S (2011) Onion-like network topology enhances robustness against malicious attacks. J Stat Mech 2011:P01027
Albert R, Barabási A-L (2001) Statistical mechanics of complex networks. arXiv:cond-mat/0106096 v1
Bak P, Sneppen K (1993) Punctuated equilibrium and criticality in a simple model of evolution. Phys Rev Lett 71:4083–4086
Solé RV, Bascompte J (1996) Are critical phenomena relevant to large-scale evolution? Proc R Soc Lond B 263:161–168
Murase Y, Shimada T, Ito N (2010) A simple model for skewed species-lifetime distributions. New J Phys 12:063021
Shimada T, Yukawa S, Ito N (2003) Life-span of families in fossil data forms q-exponential distribution. Int J Mod Phys C 14:1267–1271
Murase Y, Shimada T, Ito N, Rikvold PA (2010) Random walk in genome space: a key ingredient of intermittent dynamics of community assembly on evolutionary time scales. J Theor Biol 264:663–672
Mizuno T, Takayasu M (2009) The statistical relationship between product life cycle and repeat purchase behavior in convenience stores. Prog Theor Phys Suppl 179:71–79
Murase Y, Uchitane T (2014) Ito: a tool for parameter-space explorations. N. In: Proceedings of 27th CSP workshop. Phys Proc 57:73–76
Acknowledgments
The author thanks H. Ando, C.-K. Hu, H. Kori, Y. Murase, T. Ohira, Y. Sano, S. Shimada, and C.-P. Zhu for discussion and comments. This work was supported by the JSPS Grant-in-Aid for Young Scientists (B) no. 21740284 MEXT, Japan. The systematic simulations in this study were assisted by OACIS [28].
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Shimada, T. (2015). A Universal Mechanism of Determining the Robustness of Evolving Systems. In: Ohira, T., Uzawa, T. (eds) Mathematical Approaches to Biological Systems. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55444-8_5
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