Competitive Exclusion In Delayed Chemostat Modelswith General Response Functions

Chemostat is a device in an artificial laboratory culture of microorganisms by controlling the input-output ratio of the device to understand the interaction of microbes, and then study the dynamic behavior of microorganisms under different nutritional conditions of the growth. Since the model parameters obtained experimentally easy to control, adjustment, and therefore can be seen as essentially a chemostat study artificial lakes, marine ecosystems experimental device, chemostat model for understanding and awareness of the natural evolution of ecosystems is an important reference value.This paper contains n Chemostat populations have different removal and its growth depends in some nutrition, functional response of the n populations are nonlinear continuous function; at the same time, chemostat model we consider also reflects the biological populations consuming nutrients to be converted into biomass itself experienced during different types of time lag factors, including the length of the delay fixed discrete time-delay factor and the delay time length is not fixed but subject to a certain range within a certain probability distribution continuous Time Delay(or distributed delays) in two categories; herein by constructing Lyapunov functional and the use of global asymptotic behavior analysis model La Salle invariance principle, and has been functional response under certain conditions, competitive exclusion principle premise model establishment, that competition results depend on the size of the critical loss, only a minimal loss of critical population may eventually survive competition from other species will be repelled extinction. The main contents are as follows:The first part, the main research with discrete delays removal of n different populations- single nutrient chemostat model, by constructing Lyapunov functional application La Salle invariance principle is proved when the model functional response when certain conditions are met, only species with minimum loss threshold of it may ultimately survive, while other species will be excluded to the extinction of the species, the so-called competitive exclusion principle will be established. The second part, the first part of finite distributed delays and corresponding model chemostat model for the more general chemostat model, we use a similar method to establish a similar competitive exclusion principle established theorem. The third part, proposed in the literature for different functions of the reaction function is applied pre-competitive exclusion principle results in two parts, proved that such functional response type are applicable to the first two parts of competitive exclusion principle outcome, and verify the corresponding numerical simulation the theoretical results.