1. Chemostat Model

2. dx/dt = rate produced – rate out dS/dt = rate in – rate out – rate consumed

4. g(S)=mS/(a+S), m a Assume g(S)=mS/(a+S), a Monod or Michaelis-Menton saturation function, which means that the organism is limited in its ability to consume nutrient. m is the maximal growth rate (units 1/t) and a is the half-saturation constant (units mass/vol). g(a)=m/2

6. Model Analysis

7. m>1λ 1 and λ<1, the equilibrium (0,1) is a saddle and E=(1-λ,λ) is globally stable. If m=3 and a=1 then E=(.5,.5).

8. Competition

9. Competition

10. Competition λ 1 =.5, λ 2 =.667λ 1 =.5, λ 2 =.4 λ 1 =.5, λ 2 =.5 x 1, x 2 on Σ

11. Prey-Predator

12. Prey-Predator a 1 =0.3

13. References L. Edelstein-Kesket, Mathematical Models in Biology, Random House, New York, 1988. L. Edelstein-Kesket, Mathematical Models in Biology, Random House, New York, 1988. S.B. Hsu, S.P. Hubbell and P. Waltman, A mathematical theory for single nutrient competition in continuous cultures of micro- organisms, SIAM Jour. Appl. Math. 32, 366-383, 1977. S.B. Hsu, S.P. Hubbell and P. Waltman, A mathematical theory for single nutrient competition in continuous cultures of micro- organisms, SIAM Jour. Appl. Math. 32, 366-383, 1977. H.L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995. H.L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.