Exclusion and persistence in deterministic and stochastic chemostat models

Abstract We first introduce and analyze a variant of the deterministic single-substrate chemostat model. In this model, microbe removal and growth rates depend on biomass concentration, with removal terms increasing faster than growth terms. Using a comparison principle we show that persistence of all species is possible in this scenario. Then we turn to modelling the influence of random fluctuations by setting up and analyzing a stochastic differential equation. In particular, we show that random effects may lead to extinction in scenarios where the deterministic model predicts persistence. On the other hand, we also establish some stochastic persistence results.

[1]  L. Imhof.  The long-run behavior of the stochastic replicator dynamics , 2005, math/0503529.

[2]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[3]  Sergei S. Pilyugin,et al.  Persistence Criteria for a Chemostat with Variable Nutrient Input , 2001 .

[4]  D. Herbert,et al.  The continuous culture of bacteria; a theoretical and experimental study. , 1956, Journal of general microbiology.

[5]  Gail S. K. Wolkowicz,et al.  Competition in the Chemostat: A Distributed Delay Model and Its Global Asymptotic Behavior , 1997, SIAM J. Appl. Math..

[6]  Gail S. K. Wolkowicz,et al.  A MATHEMATICAL MODEL OF THE CHEMOSTAT WITH A GENERAL CLASS OF FUNCTIONS DESCRIBING NUTRIENT UPTAKE , 1985 .

[7]  Gregory Stephanopoulos,et al.  A stochastic analysis of the growth of competing microbial populations in a continuous biochemical reactor , 1979 .

[8]  Gail S. K. Wolkowicz,et al.  A System of Resource-Based Growth Models with Two Resources in the Unstirred Chemostat , 2001 .

[9]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[10]  R. McGehee,et al.  Some mathematical problems concerning the ecological principle of competitive exclusion , 1977 .

[11]  Bingtuan Li,et al.  Global Asymptotic Behavior of the Chemostat: General Response Functions and Different Removal Rates , 1998, SIAM J. Appl. Math..

[12]  J. Monod,et al.  Thetechnique of continuous culture. , 1950 .

[13]  S. Hubbell,et al.  Single-nutrient microbial competition: qualitative agreement between experimental and theoretically forecast outcomes. , 1980, Science.

[14]  R. Bhattacharya.  Criteria for Recurrence and Existence of Invariant Measures for Multidimensional Diffusions , 1978 .

[15]  A. Friedman.  Stochastic Differential Equations and Applications , 1975 .

[16]  Arthur E. Humphrey,et al.  Dynamics of a chemostat in which two organisms compete for a common substrate , 1977 .

[17]  Thomas C. Gard.  A new Liapunov function for the simple chemostat , 2002 .

[18]  Horst R. Thieme,et al.  Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations , 1992 .

[19]  R M May,et al.  Harvesting Natural Populations in a Randomly Fluctuating Environment , 1977, Science.

[20]  H. Peyton Young,et al.  Stochastic Evolutionary Game Dynamics , 1990 .

[21]  Sze-Bi Hsu,et al.  Limiting Behavior for Competing Species , 1978 .

[22]  M. Kirkilionis,et al.  On comparison systems for ordinary differential equations , 2004 .

[23]  Sze-Bi Hsu,et al.  A Mathematical Theory for Single-Nutrient Competition in Continuous Cultures of Micro-Organisms , 1977 .

[24]  N. Ikeda,et al.  A comparison theorem for solutions of stochastic differential equations and its applications , 1977 .

[25]  Jacques Monod,et al.  LA TECHNIQUE DE CULTURE CONTINUE THÉORIE ET APPLICATIONS , 1978 .

[26]  D. Fudenberg,et al.  Evolutionary Dynamics with Aggregate Shocks , 1992 .

[27]  D. E. Contois.  Kinetics of bacterial growth: relationship between population density and specific growth rate of continuous cultures. , 1959, Journal of general microbiology.

[28]  A. Novick,et al.  Experiments with the Chemostat on spontaneous mutations of bacteria. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[29]  J. Monod,et al.  Recherches sur la croissance des cultures bactériennes , 1942 .

[30]  S. F. Ellermeyer,et al.  Competition in the Chemostat: Global Asymptotic Behavior of a Model with Delayed Response in Growth , 1994, SIAM J. Appl. Math..