×

A stochastic analysis of the growth of competing microbial populations in a continuous biochemical reactor. (English) Zbl 0449.92018


MSC:

92D25 Population dynamics (general)
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
34A34 Nonlinear ordinary differential equations and systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aris, R.; Humphrey, A. E., Dynamics of a chemostat in which two organisms compete for a common substrate, Biotechn. and Bioeng., 19, 1375 (1977)
[2] Asbj∅rnsen, O. A.; Fjeld, M., Response modes of continuous stirred tank reactors, Chem. Eng. Science, 25, 1627 (1970)
[3] Bharucha Reid, A. T., Elements of the Theory of Markov Processes and Their Applications (1960), McGraw-Hill: McGraw-Hill New York · Zbl 0095.32803
[4] Contois, D. E., Kinetics of bacterial growth: relationship between population density and specific growth rate of continuous culture, J. Gen. Microbial., 21, 40 (1959)
[5] Emerson, S., The growth phase in Neurospora corresponding to the logarithmic phase in unicellular organisms, J. Bacteriol., 60, 221 (1950)
[6] Feller, W., Diffusion processes in genetics, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 227 (1951) · Zbl 0045.09302
[7] Feller, W., Two singular diffusion problems, Ann. Math., 54, 173 (1951) · Zbl 0045.04901
[8] Feller, W., The parabolic differential equations and associated semigroups of transformations, Ann. Math., 55, 468 (1952) · Zbl 0047.09303
[9] Fredrickson, A. G.; Tsuchiya, H. M., Microbial kinetics and dynamics, (Lapidus, L.; Amundson, N. R., Chemical Reactor Theory, A Review (1977), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J), 405-483, Chapter 7
[10] Friedman, Avner, Stochastic Differential Equations and Applications (1975), Academic: Academic New York · Zbl 0323.60056
[11] Ince, E. L., Ordinary Differential Equations (1956), Dover: Dover New York · Zbl 0063.02971
[12] Ito, K., Stochastic integral, Proc. Imp. Acad. Tokyo, 20, 519 (1944) · Zbl 0060.29105
[13] Ito, K., On stochastic differential equations, Mem. Amer. Math. Soc., 4, 1-50 (1951) · Zbl 0054.05803
[14] Jannasch, H. W., Competitive elimination of enterobacteriaceae from seawater, Appl. Microbiol., 16, 10, 1616 (1968)
[15] Jazwinski, A., Nonlinear Filtering Theory (1969), Academic: Academic New York · Zbl 0214.47304
[16] Kimura, M., Processes leading to quasi fixation of genes in natural populations due to random fluctuations of selection intensities, Genetics, 39, 280 (1954)
[17] Kimura, M., Processes leading to quasi fixation of genes in natural populations due to random fluctuations of selection intensities, Genetics, 39, 280 (1954)
[18] King, R. P., An introduction to stochastic differential equations, (Abbreviated Lecture Notes,Rep. No. 1025 (1971), Dept. of Chem. Eng., Univ. of Natal: Dept. of Chem. Eng., Univ. of Natal South Africa)
[19] Marshall, K. C.; Alexander, M., Growth characteristics of fungi and actinomycates, J. Bacteriol., 80, 412 (1960)
[20] Monod, J., Recherches sur la Croissance des Cultures Bactériennes (1942), Hermann: Hermann Paris
[21] Pell, T. M.; Aris, R., Some problems in chemical reactor analysis with stochastic features, Ind. Eng. Chem. Fundamentals, 8, 339 (1969)
[22] Powell, E. O., Criteria for the growth of contaminants and mutants in continuous culture, J. Gen. Microbiol., 18, 259 (1958)
[23] Rao, N. J.; Ramkrishna, D.; Borwanker, J. D., Nonlinear stochastic simulation of stirred tank reactors, Chem. Eng. Sci., 29, 1193 (1974)
[24] Seinfeld, J. H.; Lapidus, L., Mathematical Modelling in Chemical Engineering, (Process Modelling, Estimation and Identification, Vol. 3 (1974), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J)
[25] Soong, T. T., Random Differential Equations in Science and Engineering (1973), Academic: Academic New York · Zbl 0348.60081
[26] G. Stephanopoulos, A.G. Fredrickson and R. Aris, The effect of spatial inhomogeneities on the coexistence of competing microbial populations, Biotech. and Bioeng.; G. Stephanopoulos, A.G. Fredrickson and R. Aris, The effect of spatial inhomogeneities on the coexistence of competing microbial populations, Biotech. and Bioeng. · Zbl 0449.92018
[27] Stephanopoulos, G., Mathematical modelling of the dynamics of interacting microbial populations. Extinction probabilities in a stochastic competition and predation, (Ph.D. Thesis (1978), Univ. of Minnesota: Univ. of Minnesota Minneapolis, Minn)
[28] Stratonovich, R. L., Topics in the Theory of Random Noise (1963), Gordon and Breach: Gordon and Breach New York · Zbl 0119.14502
[29] Weinberger, H. F., A First Course in Partial Differential Equations (1965), Xerox College Publ: Xerox College Publ Lexington, Mass · Zbl 0127.04805
[30] Wong, E.; Zakai, M., On convergence of the solutions of differential equations involving Brownian motion (1965), Electronics Research Lab, Univ. of Calif: Electronics Research Lab, Univ. of Calif Berkeley, Rept. No. 65-5, AF-AFOSR-139-64 · Zbl 0131.16401
[31] Wong, E., Stochastic Processes in Information and Dynamical Systems (1971), McGraw-Hill: McGraw-Hill New York · Zbl 0245.60001
[32] Yang, R. D.; Humphrey, A. E., Dynamic and steady state studies of phenol biodegradation in pure and mixed cultures, Biotech. and Bioeng., 17, 1211 (1975)
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.