20 References In Complexity Theory and Biology
I’m more confident in dynamic mode decomposition as a general tool for modelling biological systems.
This winter, I tried to learn as much as I can about a range of biological systems which I think are amenable for modelling with DMD. Here are all the systems I’ve read about, as well as other papers that try to dynamic mode decomposition on biological systems.
Broadly, I’ve split them into ecology, chemical reaction networks, and soil microbiology ideas. I’ll be writing more detailed reviews soon!
Ecological & Population Dynamics
1. Generalized Lotka-Volterra
Description: Multi-species predator-prey/competition dynamics with antisymmetric interaction matrix ensuring conservation properties. Supports arbitrary (even) number of species.
Key Features: Equilibrium populations explicitly specified, conservation laws, oscillatory dynamics
References:
Volterra, V. (1926). “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi.” Memorie della R. Accademia Nazionale dei Lincei, 6(2), 31-113.
May, Robert M. (1972). “Will a large complex system be stable?” Nature, 238(5364), 413-414.
Gardner, M. R., & Ashby, W. R. (1970). “Connectivity of large dynamical (cybernetic) systems: Critical values for stability.” Nature, 228(5273), 784.
Goh, B. S. (1976). “Global stability in many-species systems.” The American Naturalist, 110(973), 135-143.
Goh, B. S. (1979). “Stability in models of mutualism.” The American Naturalist, 113(2), 261-275.
Rieger, H., et al. Lotka-Volterra stability analysis.
2. Volterra-Lotka with Equivalence Numbers
Description: Extended model with intrinsic growth rates and species-specific equivalence numbers that scale interaction strength
3. Generalized Lotka-Volterra (No Equilibrium Assumption)
Description: Most general form allowing asymmetric interactions, competition, mutualism, and predation without requiring equilibrium
Use Case: General ecological systems without conservation constraints
4. Two-Species Predator-Prey
Description: Classic rabbit-fox style predator-prey oscillations with closed periodic orbits
References:
Volterra, V. (1926). “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi.” Memorie della R. Accademia Nazionale dei Lincei, 6(2), 31-113.
5. Ros et al. Generalized Lotka-Volterra Models
Description: Multi-species model with random non-reciprocal interactions and correlation structure
References:
Ros, V., Roy, F., Biroli, G., Bunin, G., & Turner, A. M. (2023). “Generalized Lotka-Volterra equations with random, non-reciprocal interactions: the typical number of equilibria.” arXiv: 2212.01837v2 [cond-mat.dis-nn].
Chemical Reaction Networks (4 systems)
7. Complex Stoichiometric Chemistry (8-Species, 10-Reaction Networks)
Description: Biochemical pathway with enzyme catalysis, substrate competition, and feedback regulation
References:
Horowitz, Jordan M. & England, Jeremy L. (2017). “Spontaneous fine-tuning to environment in many-species chemical reaction networks.” PNAS (Proceedings of the National Academy of Sciences).
8. Horowitz-England Self-Organization Models
Description: 25-species random chemical network demonstrating spontaneous fine-tuning to environmental energy sources
References:
Horowitz, Jordan M. & England, Jeremy L. (2017). “Spontaneous fine-tuning to environment in many-species chemical reaction networks.” PNAS (Proceedings of the National Academy of Sciences).
9. Glycolysis Models
Description: Two-variable model of glycolytic oscillations with allosteric feedback
References:
Wüstner, Daniel, Gundestrup, Henrik Helge, & Thaysen, Katja (2025). “Dynamic mode decomposition for analysis and prediction of metabolic oscillations from time-lapse imaging of cellular autofluorescence.” Scientific Reports. DOI: 10.1038/s41598-025-07255-4.
10. General Koopman Operator Theory for Chemical Reaction Networks
Description: Framework connecting chemical master equations to Koopman theory
References:
Koopman, B. O. (1931). “Hamiltonian systems and transformation in Hilbert space.” Proc. Natl. Acad. Sci., 17, 315-318.
Mezić, Igor (2005). “Spectral properties of dynamical systems, model reduction and decompositions.” Nonlinear Dyn., 41, 309-325.
Williams, M. O., Kevrekidis, I. G., & Rowley, C. W. (2015). “A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition.” J. Nonlinear Sci., 25, 1307-1346.
Gupta, Ankit & Khammash, Mustafa (2025). “A Spectral Koopman Approximation Framework for Stochastic Reaction Networks.” arXiv: 2511.23114.
Gupta, Ankit & Khammash, Mustafa (2025). “Sparse Spectral Estimation of the Koopman Operator Reveals Key Insights into the Dynamics of Stochastic Reaction Networks.” Joint Meeting of Asian Conference for Mathematical Biology & Japanese Society for Mathematical Biology.
Narasingam, Abhinav & Kwon, Joseph Sang-Il (2019). “Koopman Lyapunov-based Model Predictive Control of Nonlinear Chemical Process Systems.” AIChE Journal.
Narasingam, Abhinav & Kwon, Joseph Sang-Il (2020). “Application of Koopman Operator for Model-Based Control of Fracture Propagation and Proppant Transport in Hydraulic Fracturing Operation.”
Microbial Ecology Models (3 systems)
11. Bacteria-Phage Predator-Prey Systems
Description: E. coli K-12 + bacteriophage (T4, lambda) dynamics
Applications: Home laboratory experiments for studying predator-prey oscillations
Protocols: Culture methods, plaque assays, parameter estimation
12. Soil Bacterial Communities
Description: Multi-species bacterial communities from soil samples
Common Species: Bacillus, Pseudomonas, Streptomyces, Arthrobacter
Applications: Study of competition, diversity, resource competition, community assembly
13. Cyanobacteria Circadian Oscillations
Description: Synechococcus elongatus colonies with circadian clock synchronization
References:
Schmitt, Matthew S., Koch-Janusz, Maciej, Fruchart, Michel, Seara, Daniel S., Rust, Michael, & Vitelli, Vincenzo (2025). “Information theory for data-driven model reduction in physics and biology.” arXiv: 2312.06608v3.