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The Effect of the Substrate Concentration of the Feed on the Steady States and Dynamics of a Chemostat Meng-Cheng Cheng (鄭孟晟) , Chung-Min Lien (連崇閔) , Hau Lin (林浩) Department of Chemical and Materials Engineering, Southern Taiwan University 南台科技大學化學工程與材料工程系 Abstract A study was conducted to analyze the effect of the substrate concentration of the feed on the steady states and dynamics of the prey-predator interaction in a chemostat. The specific growth rates of Substrate Inhibition model and Monod model were used for the prey and predator respectively. The dynamic equations were derived by assuming that the reaction was occurring in a perfectly mixed flow reactor (chemostat). There were three types of steady states for this system and the three types of steady states and stability were analyzed in detail. The dynamic equations of this system were solved by the numerical method and the dynamic analysis was performed by computer graphs. The graphs of concentrations of predator, prey and substrate versus reaction time and the graphs of the concentration of prey versus concentration of predator, concentration of predator versus concentration of substrate, and concentration of substrate versus concentration of prey were plotted for dynamic analysis. The results showed that the dynamic behavior of this system consisted of stable steady states and limit cycles. When the parameters μm = 0.56hr–1, νm = 0.1hr–1, Ki = 16 mg/L , Kp = 10000 mg/L, L = 6.1 mg/L , X = 0.73, Y = 0.428 , D = 0.0595hr–1 , k1 = 0hr–1, k2 = 0hr–1, the dynamic behavior of the the substrate concentrations of the feed sf = 0.5 mg/L, sf = 2 mg/L, sf = 50 mg/L and sf = 250 mg/L showed the first type of stable steady state, the second type of stable steady state, the third type of stable steady state and the limit cycle respectively. Fig.3 s versus t； sf = 50mg/L Third Type of Stable Steady State Introduction Fig.4 s versus t； sf = 250mg/L Limit Cycle behavior The prey-predator interaction exists in the rivers frequently and a common interaction between two organisms inhabiting the same environments involves one organism (predator) deriving its nourishment by capturing and ingesting the other organism (prey). A study was conducted to analyze the steady state and dynamics of the prey-predator interaction in a chemostat. There are three types of steady states for this system and the three types of steady states are analyzed in detail. The stability was analyzed by calculating the eigenvalues of this chemostat system. The dynamic equations of this system are solved by the numerical method and the dynamic analysis is performed by computer graphs. In this study, the graphs of concentrations of predator, prey and substrate versus reaction time are plotted for dynamic analysis. Research Methods The mathematical methods and numerical method are used in this study. For this chemostat system, steady state equations are solved by mathematical methods. For dynamic analysis, because of the complexity of the dynamic equations, Runge-Kutta numerical analysis method is used to solve the dynamic equations, and the dynamic analysis was performed by computer graphs. Results and Discussion The dynamic equations for the prey-predator interaction in a chemostat are as follows dp Dp ν (b)p k 1 p dt Table1 The steady states, eigenvalues and stability for substrate concentration of feed sf = 0.5mg/L Table2 The steady states, eigenvalues and stability for substrate concentration of feed sf = 2mg/L db 1 Db μ (s)b ν (b)p k 2 b dt X ds 1 D(s f s) μ (s)b dt Y where p = concentration of the predator (mg/L), b = concentration of the prey (mg/L), s = concentration of the substrate (mg/L), sf = feed concentration of the substrate (mg/L), X= yield coefficient for production of the predator, Y = yield coefficient for production of the prey, F = flow rate(L/hr), V = reactor volume (L), D = F / V = dilution rate (hr–1), k1＝death rate coefficient for the predator (hr–1)，k2＝death rate coefficient for the prey (hr–1). Substrate Inhibition model model is used for the specific growth rate of the prey and Monod model is used for the specific growth rate of the predator Table3 The steady states, eigenvalues and stability for substrate concentration of feed sf = 50 mg/L μ ms μ (s) Ki s s2 / K p Table4 The steady states, eigenvalues and stability for substrate concentration of feed sf = 250 mg/L ν mb ν ( b) Lb whereμm = constant (hr–1) ; Ki, Kp = constants (mg/L) ; m = maximum specific growth rates (hr–1) ; L = saturation constant (mg/L). For the situation of this this system, three types of steady state solutions are possible (1) Washout of both prey and predator： p＝0，b＝0，s＝sf (2) Washout of the predator only： p＝0，b＞0，sf＞s＞0 (3) Coexistance of both prey and predator： p＞0，b＞0，sf＞s＞0 For dynamic analysis, because of the complexity of the dynamic equations, Runge-Kutta numerical analysis method was used to solve the dynamic equations and the dynamic analysis was performed by computer graphs. Fig.1-4 show the graphs of s versus time for different parameters. Tables 1-4 show the steady states, eigenvalues and stability for different substrate concentrations of feed. The initial conditions for Fig.1-4 are p = 5.0 mg/L, b = 25.0 mg/L, s = 10.0 mg/L . The parameters for Fig.1- 4 are μm=0.56hr–1, νm =0.1hr–1， Ki = 16 mg/L , Kp = 10000 mg/L, L=6.1mg/L, X=0.73, Y=0.428, D = 0.0595hr–1 , k1=0 hr-1, k2=0 hr-1 . Fig.1-4 show the First Type of Stable Steady State behavior ; the Second Type of Stable Steady State behavior ;, the Third Type of Stable Steady State behavior and the Limit Cycle behavior respectively. Fig.1 s versus t； sf = 0.5mg/L First Type of Stable Steady State Fig.2 s versus t； sf = 2mg/L Second Type of Stable Steady State Conclusions For the prey-predator interaction in a chemostat, there are three types of steady state solutions. The steady state equations were solved by mathematical methods. The stability was analyzed by calculating the eigenvalues of this chemostat system. The dynamic equations of this system were solved by the numerical method and the dynamic analysis was performed by computer graphs. The graphs of concentrations of predator, prey and substrate versus reaction time were plotted for dynamic analysis. The results show that the dynamic behavior of this system consists of three types of stable steady states and limit cycles. References [1] Saunders, P. T. , Bazin, M. J. “On the Stability of Food Chains, ” J. Theor. Biol., Vol.52, pp.121142 (1975). [2] Hastings, A. , “Multiple Limit Cycles in Predator-Prey Models,” J. Math. Biol., Vol.11, pp.51-63 (1981). [3] Pavlou, S., “Dynamics of a Chemostat in Which One Microbial Population Feeds on Another,” Biotechnol. and Bioeng., Vol. 27, pp.1525-1532 (1985). [4] Weber, A. E., San, K-Y, “Population Dynamics of a Recombinant Culture in a Chemostat under Prolonged Cultivation ,” Biotechnol. and Bioeng. 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