Awakened oscillations in coupled consumer-resource pairs. (English) Zbl 1442.92137

Summary: The paper concerns two interacting consumer-resource pairs based on chemostat-like equations under the assumption that the dynamics of the resource is considerably slower than that of the consumer. The presence of two different time scales enables to carry out a fairly complete analysis of the problem. This is done by treating consumers and resources in the coupled system as fast-scale and slow-scale variables, respectively, and subsequently considering developments in phase planes of these variables, fast and slow, as if they are independent. When uncoupled, each pair has unique asymptotically stable steady state and no self-sustained oscillatory behavior (although damped oscillations about the equilibrium are admitted). When the consumer-resource pairs are weakly coupled through direct reciprocal inhibition of consumers, the whole system exhibits self-sustained relaxation oscillations with a period that can be significantly longer than intrinsic relaxation time of either pair. It is shown that the model equations adequately describe locally linked consumer-resource systems of quite different nature: living populations under interspecific interference competition and lasers coupled via their cavity losses.


92D25 Population dynamics (general)
Full Text: DOI arXiv


[1] Balanov, A.; Janson, N.; Postnov, D.; Sosnovtseva, O., Synchronization: From Simple to Complex (2009), Berlin, Germany: Springer, Berlin, Germany · Zbl 1163.34001
[2] Hoppensteadt, F. C.; Izhikevich, E. M., Weakly Connected Neural Networks. Weakly Connected Neural Networks, Applied Mathematical Sciences, 126 (1997), New York, NY, USA: Springer, New York, NY, USA · Zbl 0887.92003
[3] Pikovsky, A.; Rosenblum, M.; Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences (2001), Cambridge, Mass, USA: Cambridge University Press, Cambridge, Mass, USA · Zbl 0993.37002
[4] Strogatz, S., Sync: The Emerging Science of Spontaneous Order (2003), New York, NY, USA: Hyperion, New York, NY, USA
[5] Vandermeer, J., Oscillating populations and biodiversity maintenance, BioScience, 56, 12, 967-975 (2006)
[6] Smale, S.; Cowan, J. D., A mathematical model of two cells via Turing’s equation, Some Mathematical Questions in Biology V. Some Mathematical Questions in Biology V, Lectures on Mathematics in the Life Sciences, 6, 15-26 (1974), Providence, RI, USA: American Mathematical Society, Providence, RI, USA
[7] Loewenstein, Y.; Yarom, Y.; Sompolinsky, H., The generation of oscillations in networks of electrically coupled cells, Proceedings of the National Academy of Sciences of the United States of America, 98, 14, 8095-8100 (2001)
[8] Gomez-Marin, A.; Garcia-Ojalvo, J.; Sancho, J. M., Self-sustained spatiotemporal oscillations induced by membrane-bulk coupling, Physical Review Letters, 98, 16 (2007)
[9] Szatmári, I.; Chua, L. O., Awakening dynamics via passive coupling and synchronization mechanism in oscillatory cellular neural/nonlinear networks, International Journal of Circuit Theory and Applications, 36, 5-6, 525-553 (2008) · Zbl 1191.94166
[10] Murdoch, W. W.; Briggs, C. J.; Nisbet, R. M., Consumer-Resource Dynamics (2003), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA
[11] Murray, J. D., Mathematical Biology: I. An Introduction (2002), New York, NY, USA: Springer, New York, NY, USA
[12] Agrawal, G. P.; Dutta, N. K., Semiconductor Lasers (1993), New York, NY, USA: Van Nostrand Reinhold, New York, NY, USA
[13] Zhang, W.-B., Synergetic Economics: Time and Change inNonlinear Economics. Synergetic Economics: Time and Change inNonlinear Economics, Springer Series in Synergetics, 53 (1991), Berlin , Germany: Springer, Berlin , Germany · Zbl 0728.90001
[14] Volkenstein, M. V., General Biophysics (1983), New York, NY, USA: Academic Press, New York, NY, USA
[15] Chernavskii, D. S.; Palamarchuk, E. K.; Polezhaev, A. A.; Solyanik, G. I.; Burlakova, E. B., A mathematical model of periodic processes in membranes (with application to cell cycle regulation), BioSystems, 9, 4, 187-193 (1977)
[16] Gause, G. F.; Witt, A. A., Behavior of mixed populations and the problem of natural selection, The American Naturalist, 69, 725, 596-609 (1935)
[17] MacArthur, R., Species packing and competitive equilibrium for many species, Theoretical Population Biology, 1, 1, 1-11 (1970)
[18] Baer, T., Large-amplitude fluctuations due to longitudinal mode coupling in diode-pumped intracavity-doubled Nd :YAG lasers, Journal of the Optical Society of America B: Optical Physics, 3, 9, 1175-1180 (1986)
[19] Erneux, T.; Mandel, P., Minimal equations for antiphase dynamics in multimode lasers, Physical Review A, 52, 5, 4137-4144 (1995)
[20] Tikhonov, A. N., Systems of differential equations containing small parameters in the derivatives, Matematicheskii Sbornik, 31(73), 3, 575-586 (1952)
[21] Sanders, J.; Verhulst, F.; Murdock, J., Averaging Methods in Nonlinear Dynamical Systems. Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, 59 (2007), New York, NY, USA: Springer, New York, NY, USA · Zbl 1128.34001
[22] Verhulst, F., Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics. Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Texts in Applied Mathematics, 50 (2005), New York, NY, USA: Springer, New York, NY, USA · Zbl 1148.35006
[23] Kirlinger, G., Permanence in Lotka-Volterra equations: linked prey-predator systems, Mathematical Biosciences, 82, 2, 165-191 (1986) · Zbl 0607.92022
[24] May, R. M.; Leonard, W. J., Nonlinear aspects of competition between three species, SIAM Journal on Applied Mathematics, 29, 2, 243-253 (1975) · Zbl 0314.92008
[25] Vandermeer, J., Intransitive loops in ecosystem models: from stable foci to heteroclinic cycles, Ecological Complexity, 8, 1, 92-97 (2011)
[26] Chesson, P., MacArthur’s consumer-resource model, Theoretical Population Biology, 37, 1, 26-38 (1990) · Zbl 0689.92016
[27] Mirrahimi, S.; Perthame, B.; Wakano, J. Y., Direct competition results from strong competition for limited resource, Journal of Mathematical Biology, 68, 4, 931-949 (2014) · Zbl 1293.35346
[28] Devetter, M.; Sedâ, J., The relative role of interference competition in regulation of a Rotifer community during spring development in a eutrophic reservoir, International Review of Hydrobiology, 93, 1, 31-43 (2008)
[29] Bazykin, A. D., Nonlinear Dynamics of Interacting Populations. Nonlinear Dynamics of Interacting Populations, World Scientific Series on Nonlinear Science, Series A, 11 (1998), River Edge, NJ, USA: World Scientific, River Edge, NJ, USA
[30] Kuang, Y.; Fagan, W. F.; Loladze, I., Biodiversity, habitat area, resource growth rate and interference competition, Bulletin of Mathematical Biology, 65, 3, 497-518 (2003) · Zbl 1334.92349
[31] Stewart, F. M.; Levin, B. R., Partitioning of resources and the outcome of interspecific competition: a model and some general considerations, American Naturalist, 107, 954, 171-198 (1973)
[32] Herbert, D.; Elsworth, R.; Telling, R. C., The continuous culture of bacteria; a theoretical and experimental study, Journal of General Microbiology, 14, 3, 601-622 (1956)
[33] Lynch, M., Complex interactions between natural coexploiters—Daphnia and Ceriodaphnia, Ecology, 59, 3, 552-564 (1978)
[34] Xiang, Z.; Song, X., Extinction and permanence of a two-prey two-predator system with impulsive on the predator, Chaos, Solitons & Fractals, 29, 5, 1121-1136 (2006) · Zbl 1142.34306
[35] Tilman, D., Resource Competition and Community Structure (1982), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA
[36] Statz, H.; deMars, G. A.; Townes, C. H., Transients and oscillation pulses in masers, Quantum Electronics, 530-537 (1960), New York, NY, USA: Columbia University Press, New York, NY, USA
[37] Ghosh, A.; Goswami, B. K.; Vijaya, R., Nonlinear resonance phenomena of a doped fibre laser under cavity-loss modulation: experimental demonstrations, Pramana-Journal of Physics, 75, 5, 915-921 (2010)
[38] Tian, J.; Yao, Y.; Xiao, J. J.; Yu, X.; Chen, D., Tunable multiwavelength erbium-doped fiber laser based on intensity-dependent loss and intra-cavity loss modulation, Optics Communications, 285, 9, 2426-2429 (2012)
[39] Hofelich-Abate, E.; Hofelich, F., Phenomenological theory of damped and undamped intensity oscillations in lasers, Zeitschrift für Physik, 221, 4, 362-378 (1969)
[40] Kennedy, C. J.; Barry, J. D., Stability of an intracavity frequency-doubled Nd:YAG laser, IEEE Journal of Quantum Electronics, 10, 8, 596-599 (1974)
[41] Kellou, A.; Aït-Ameur, K.; Sanchez, F., Suppression of undamped oscillations in erbium-doped fibre lasers with intensity-dependent losses, Journal of Modern Optics, 45, 9, 1951-1956 (1998)
[42] Li, F.; Kutz, J. N.; Wai, P. K. A., Characterizing bifurcations and chaos in multiwavelength lasers with intensity-dependent loss and saturable homogeneous gain, Optics Communications, 285, 8, 2144-2153 (2012)
[43] El Amili, A.; Kervella, G.; Alouini, M., Experimental evidence and theoretical modeling of two-photon absorption dynamics in the reduction of intensity noise of solid-state Er:Yb lasers, Optics Express, 21, 7, 8773-8780 (2013)
[44] Nguyen, B. A.; Mandel, P., Stability of two loss-coupled lasers, Journal of Optics B: Quantum and Semiclassical Optics, 1, 3, 320-324 (1999)
[45] Mustafin, A., Two mutually loss-coupled lasers featuring astable multivibrator, Physica D: Nonlinear Phenomena, 218, 2, 167-176 (2006) · Zbl 1122.34034
[46] Mandel, P., Theoretical Problems in Cavity Nonlinear Optics (1997), Cambridge, Mass, USA: Cambridge University Press, Cambridge, Mass, USA
[47] Erneux, T.; Glorieux, P., Laser Dynamics (2010), Cambridge, Mass, USA: Cambridge University Press, Cambridge, Mass, USA
[48] Wiesenfeld, K.; Bracikowski, C.; James, G.; Roy, R., Observation of antiphase states in a multimode laser, Physical Review Letters, 65, 14, 1749-1752 (1990)
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