Malchow, Horst; Petrovskii, S. V. Dynamical stabilization of an unstable equilibrium in chemical and biological systems. (English) Zbl 1021.92026 Math. Comput. Modelling 36, No. 3, 307-319 (2002). Summary: The dynamics of two-component diffusion-reaction systems is considered. Using well-known models from population dynamics and chemical physics, it is shown that for certain parameter values the systems exhibit a rather unusual behaviour: a locally unstable equilibrium may become stable during a certain transition process. Both the analytical and numerical investigations of this phenomenon are presented in one and two spatial dimensions. Cited in 24 Documents MSC: 92D25 Population dynamics (general) 92E20 Classical flows, reactions, etc. in chemistry 92D40 Ecology 35K57 Reaction-diffusion equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences 37N25 Dynamical systems in biology Keywords:reaction-diffusion systems; wave propagation; predator-prey model; Gray-Scott model PDFBibTeX XMLCite \textit{H. Malchow} and \textit{S. V. Petrovskii}, Math. Comput. Modelling 36, No. 3, 307--319 (2002; Zbl 1021.92026) Full Text: DOI References: [1] Gray, P.; Scott, S. K., Chemical Oscillations and Instabilities (1990), Oxford University Press: Oxford University Press Oxford · Zbl 0706.92025 [2] Murray, J. D., Mathematical Biology (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0682.92001 [3] Okubo, A., Diffusion and Ecological Problems: Mathematical Models (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0422.92025 [4] Shigesada, N.; Kawasaki, K., Biological Invasions: Theory and Practice (1997), Oxford University Press: Oxford University Press Oxford [5] Haken, H., Advanced Synergetics, (Springer Series in Synergetics, Volume 20 (1983), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0447.58032 [6] Nicolis, G.; Prigogine, I., Self-Organization in Non-Equilibrium Systems (1977), Wiley: Wiley New York · Zbl 0363.93005 [7] Turing, A. M., On the chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond., B 237, 37-72 (1952) · Zbl 1403.92034 [8] Segel, L. A.; Jackson, J. L., Dissipative structure: An explanation and an ecological example, J. Theor. Biol., 37, 545-559 (1972) [9] (Kaneko, K., Theory and Applications of Coupled Map Lattices (1993), Wiley & Sons: Wiley & Sons Chichester) · Zbl 0777.00014 [10] Schimansky-Geyer, L.; Mieth, M.; Rosé, H.; Malchow, H., Structure formation by active Brownian particles, Phys. Lett., A 207, 140-146 (1995) · Zbl 1020.82613 [11] Schweitzer, F.; Ebeling, W.; Tilch, B., Complex motion of Brownian particles with energy deposit, Phys. Rev. Lett., 80, 5044-5047 (1998) [12] Skellam, J. G., Random dispersal in theoretical populations, Biometrika, 38, 196-218 (1951) · Zbl 0043.14401 [13] Dubois, D., A model of patchiness for prey-predator plankton populations, Ecological Modelling, 1, 67-80 (1975) [14] Wroblewski, J. S.; O’Brien, J. J., A spatial model of plankton patchiness, Marine Biology, 35, 161-176 (1976) [15] Malchow, H., Spatio-temporal pattern formation in nonlinear non-equilibrium plankton dynamics, (Proc. R. Soc. Lond. (1993)), 103-109, B251 [16] Petrovskii, S. V.; Malchow, H., A minimal model of pattern formation in a prey-predator system, Mathl. Comput. Modelling, 29, 8, 49-63 (1999) · Zbl 0990.92040 [17] Petrovskii, S. V.; Malchow, H., Critical phenomena in plankton communities: KISS model revisited, Non-linear Analysis: Real World Applications, 1, 37-51 (2000) · Zbl 0996.92037 [18] Petrovskii, S. V.; Vinogradov, M. E.; Morozov, A. Yu., Spatial-temporal dynamics of a localized populational “burst” in a distributed prey-predator system, Oceanology, 38, 881-890 (1998) [19] Dunbar, S. R., Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17, 11-32 (1983) · Zbl 0509.92024 [20] Sherratt, J. A., Invadive wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave, Physica D, 117, 145-166 (1998) · Zbl 0940.35111 [21] Sherratt, J. A.; Lewis, M. A.; Fowler, A. C., Ecological chaos in the wake of invasion, (Proc. Natl. Acad. Sci. USA (1995)), 2524-2528, 92 · Zbl 0819.92024 [22] Petrovskii, S. V.; Malchow, H., Wave of chaos: A new mechanism of pattern formation in a prey-predator system, Theoretical Population Biology, 59, 157-174 (2001) · Zbl 1035.92046 [23] Pearson, J. E., Complex patterns in simple systems, Science, 261, 189-192 (1993) [24] Rasmussen, K. E.; Mazin, W.; Mosekilde, E.; Dewel, G.; Borckmans, P., Wave-splitting in the bistable Gray-Scott model, Int. J. of Bifurcations and Chaos, 6, 1077-1092 (1996) · Zbl 0881.92038 [25] Davidson, F., Chaotic wakes and other wave-induced behavior in a system of reaction-diffusion equations, Int. J. of Bifurcation and Chaos, 8, 1303-1313 (1998) · Zbl 0935.35068 [26] Dunbar, S. R., Travelling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math., 46, 1057-1078 (1986) · Zbl 0617.92020 [27] Murdoch, W. W.; McCauley, E., Three distinct types of dynamic behaviour shown by a single planktonic system, Nature, 316, 628-630 (1985) [28] Ranta, E.; Kaitala, V.; Lundberg, P., The spatial dimension in population fluctuations, Science, 278, 1621-1623 (1997) 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.