Poland, Douglas The effect of clustering on the Lotka-Volterra model. (English) Zbl 0671.34048 Physica D 35, No. 1-2, 148-166 (1989). Summary: Using the Lotka-Volterra model as a reference for an oscillating system, the effects of nonideality, excluded volume and clustering, are investigated. It is found that these two different types of interaction have opposite effects on the stability of the system and that for a range of parameters stable limit cycles can result. Even limited clustering, such as dimer formation, is sufficient to give rise to limit-cycle behavior. Lattice models are also investigated where clustering is treated using the Bethe approximation. Cited in 2 Documents MSC: 34D20 Stability of solutions to ordinary differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 92D25 Population dynamics (general) Keywords:Lotka-Volterra model; oscillating system; clustering; limit cycles; Bethe approximation PDFBibTeX XMLCite \textit{D. Poland}, Physica D 35, No. 1--2, 148--166 (1989; Zbl 0671.34048) Full Text: DOI References: [1] PNAS, 6, 420-425 (1920) [2] Volterra, V., Leçon sur la théorie mathématique de la Lutte pour la vie (1931), Gauthier-Villars: Gauthier-Villars Paris · JFM 57.0466.02 [3] Bray, W. C., J. Am. Chem. Soc., 43, 1262-1267 (1921) [4] Jordan, D. W.; Smith, P., Nonlinear Ordinary Differential Equations (1979), Oxford University Press: Oxford University Press Oxford, ch. 2 · Zbl 0417.34002 [5] McQuarrie, D. A., Statistical Mechanics (1976), Harper and Row: Harper and Row New York, ch. 12 [6] Peschel, M.; Mende, W., The Predator-Prey Model, (Do We Live in a Volterra World? (1986), Springer: Springer New York) · Zbl 0601.92023 [7] Schuster, H. G., Deterministic Chaos (1984), Physik: Physik Weinheim, ch. 3 [8] Farkas, H.; Noszticzius, Z., J. Chem. Soc. Faraday Trans., 281, 1487 (1985), Similar models have been treated in an open system: [9] Kshirsagar, G.; Field, R. J.; Gyorgyi, L., J. Phys. Chem., 92, 2472 (1988) [10] Hill, Terrell L., Linear Aggregation Theory in Cell-Biology (1987), Springer: Springer New York [11] Nicolis, G.; Prigogine, I., Self-Organization in Nonequilibrium Systems, ((1977), Wiley-Interscience: Wiley-Interscience New York), 165 · Zbl 0363.93005 [12] Tyson, J. J.; Light, J. C., J. Chem. Phys., 59, 4164-4173 (1973) [13] Nicolis, G.; Prigogine, I., Self-Organization in Nonequilibrium Systems, ((1977), Wiley-Interscience: Wiley-Interscience New York), 193 · Zbl 0363.93005 [14] Field, R. J.; Noyes, R. M., J. Chem. Phys., 60, 1877 (1974) [15] Noszticzius, Z.; Farkas, H.; Schelly, Z. A., J. Chem. Phys., 80, 6062 (1984) [16] Hill, Terrell L., Introduction to Statistical Thermodynamics (1960), Addison-Wesley: Addison-Wesley New York, ch. 14 [17] Springgate, M. W.; Poland, D., J. Chem. Phys., 62, 680-689 (1975), See, for example [18] Sykes, M. F.; Gaunt, D. S.; Martin, J. L.; Mattingly, S. R.; Essam, J. W., J. Math. Phys., 14, 1071-1074 (1973) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.