The Existence Of Positive Steady-state Solutions To A Kind Of Food-chain Chemostat Model

Abstract The Chemostat is an apparatus usually used in the chemical engineering, it also serves as a reactor used in microorganisms culture, waste treatment, biology pharmacy, and food processing etc. The Chemostat is a well controlled and well observed apparatus, we can get our expected goals by controlling some microorganisms concentration or adjusting some parameters in the system. Obviously, studies on chemostat have became very necessary. Depending on mathematical methods to model, analyze, control, and optimize the system, which have very important significance in the reactor designing and the cost of production decreasing.This paper deals with an un-stirred chemostat food - chain model with a single nutrient, one predator and one prey populations, where the prey concentration is changing with the concentration of nutrient and predator. The Chemostat model can be depicted in mathematically as followings:with boundary conditionsand nonnegative initial conditionsHere S(t),u₁(t),U₂(t) are the concentrations of the nutrient, prey, predator at time t respectively, Ω is a bounded domain in Rⁿ(n ≥ 1) with sufficiently smooth boundary Ω, d₀ is the diffusive coefficient for the nutrient S, d_i is the random motility coefficient of microbial population u_i with death rates k₁, f₁ and f₂ are the growth rate of S and u₁ respectively, a and 6 are the maximum growth rates of u₁ and u₂ respectively. From the equations we can get that u₁ preys on S and u₂ preys on u₁ There are not direct relationship between u2 and S, therefor, the system forms a simple food-chain.