Bifurcation Analysis of a Chemostat Model of Plasmid-Bearing and Plasmid-Free Competition with Pulsed Input

A chemostat model of plasmid-bearing and plasmid-free competition with pulsed input is proposed. The invasion threshold of the plasmid-bearing and plasmid-free organisms is obtained according to the stability of the boundary periodic solution. By use of standard techniques of bifurcation theory, the periodic oscillations in substrate, plasmid-bearing, and plasmid-free organisms are shown when some conditions are satisfied. Our results can be applied to control bioreactor aimed at producing commercial producers through genetically altered organisms.


Introduction
Biofermentation has become an active area of research on the continuous cultivation of microorganism in recent years [1][2][3].The chemostat is an important laboratory apparatus used to continuously culturing microorganisms [2][3][4][5][6][7][8].It can be used to investigate microbial growth because the parameters are measurable, the experiments are reasonable, and the mathematics is tractable [9].
Fermentations using genetically modified (recombinant) microorganisms typically contain two kinds of cellsrecombinant cells and wild-type cells.The former contains a genetically inserted plasmid (a foreign DNA molecule that can exist independent of the host chromosome and can replicate autonomously) which is responsible for the coding functions that result in the synthesis of a desired protein.Wild-type or plasmid-free cells do not contain this plasmid and therefore cannot generate the protein.Nevertheless, they consume nutrients, grow, and multiply.From this perspective, plasmid-free cells may thus be considered undesirable, and different methods are employed to check their proliferation.As recombinant (or plasmid-bearing) cells have to support a larger metabolic load than plasmid-free cells, their growth rates are smaller.In addition, these cells lose their plasmids during the fermentation process.
With the scientific technology, the importance of the genetically altered technology is widely recognized.Therefore, it is necessary to understand the dynamic behavior of the fermentation process.Xiang and Song [10] analyze a simple chemostat model for plasmid-bearing and plasmid-free organisms with the pulsed substrate and linear functional response.They prove that system is permanent if the impulsive period is less than some critical value.Shi et al. [11] consider a new Monod type chemostat model with delayed growth response and pulsed input in the polluted environment.Normally, the velocity of the enzyme reaction increases with the increase in substrate concentration.Some enzymes, however, display the phenomenon of excess substrate inhibition, which means that large amounts of substrate can have the adverse effect and actually slow the reaction down.The Monod function does not account for any inhibitory effect at high substrate concentration.Therefore, it is crucial to choose a response function showing the excess substrate inhibition.Pal et al. [12] introduce the Monod-Haldene functional responses into a three-tier model of phytoplankton, zooplankton, and nutrient in order to investigate the phenomenon of excess substrate inhibition.Therefore, we introduce the Monod-Haldene functional where Δ = ( + /) − (/), Δ is the concentration of plasmid-free organism at time . 1 and  2 are the uptake constant of the microorganism. is the constant yield (It is reasonable to assume that the yield constants for two organisms are the same since they are the same organism just with or without the plasmid). 0 represents the input concentration of the nutrient each time, and the probability that a plasmid is lost in reproduction is represented by  (0 <  < 1). (0 <  < 1) is the washout proportion of the chemostat each time./ is the period of the pulse.The variables in the above system may be rescaled by measuring () ≡ ()/ 0 , () ≡  1 ()/ 0 , () ≡  2 ()/ 0 , and  = , and then we have the following system: where

The Behavior of the Substrate and Plasmid-Free Organism Subsystem
In the absence of the plasmid-bearing organism, system (2) is reduced to This nonlinear system has a simple periodic solution.For our purpose, we present the solution in this section.

The Bifurcation of the System
In order to investigate the properties of system (2), we add the first, second, and third equations of system (2) and take variable change  =  +  + , and the following lemma is obvious.with (0) > 0, and we obtain the following.
Proof.The proof of ( 1) is easy; we want to prove (2) and (3).The local stability of periodic solution (  (), 0,   ()) may be determined by considering the behavior of small amplitude of the solution.Define
The eigenvalues of the matrix Φ() are  If , the boundary periodic solution (  (), 0,   ()) of system ( 2) is unstable.The proof is completed.