Narrow Results

We consider the dynamic behaviors of a mathematical chemostat model with state dependent impulsive perturbations. By using the Poincaré map and analogue of Poincaré's criterion, some conditions for the existence and stability of positive periodic solution are obtained. Moreover, we show that there is no periodic solution with order larger than or equal to three. Numerical simulation are carried out to illustrate the feasibility of our main results, thus implying that the presence of pulses makes the dynamic behavior more complex.

In this paper, we consider a chemostat model with pulsed input. We find a critical value of the period of pulses. If the period is more than the critical value, the microorganism-free periodic solution is globally asymptotically stable. If less, the system is permanent. Moreover, the nutrient and the microorganism can co-exist on a periodic solution of period τ. Finally, by comparing the corresponding continuous system, we find that the periodically pulsed input destroys the equilibria of the continuous system and initiates periodic solutions. Our results are valuable for the manufacture of products by genetically altered organisms.

In this paper, we aim to propose a new chemostat model with continuous microbial culture and harvest, and to investigate the dynamics of the model. Different to the conventional ones, our model includes a constant periodic flocculant transmission. For the proposed system, by using theory of impulsive differential equations, we show that the microbe-extinction periodic solution is globally asymptotically stable when a threshold value is less than 1, and system is permanent when a certain threshold value is greater than 1. Then, according to the threshold associated with microbial extinction or existence, the control strategy for microbial continuous cultivation and harvest is discussed. Under such control strategy, continuous microbial culture and harvest can be achieved by adjusting input time, input amount or concentration of the flocculant. Finally, an example with numerical simulations is given to illustrate our theoretical conclusions.

In this paper, we study the existence of numerical solution and stability of a chemostat model under fractal-fractional order derivative. First, we investigate the positivity and roundedness of the solution of the considered system. Second, we find the existence of a solution of the considered system by employing the Banach and Schauder fixed-point theorems. Furthermore, we obtain a sufficient condition that allows the existence of the stabling of solutions by using the numerical-functional analysis. We find that the proposed system exists as a unique positive solution that obeys the criteria of Ulam–Hyers (U-H) and generalized U-H stability. We also establish a numerical analysis for the proposed system by using a numerical scheme based on the Lagrange interpolation procedure. Finally, we provide two numerical examples to verify the correctness of the theoretical results. We remark that the structure described by the considered model is also sometimes called side capacity or cross-flow model. The structure considered here can be also seen as a limiting case of the pattern chemostats in parallel with diffusion connection. Moreover, the said model forms in natural and engineered systems and can significantly affect the hydrodynamics in porous media. Fractal calculus is an excellent tool to discuss fractal characteristics of porous media and the characteristic method of the porous media.

We introduce and study a chemostat model with plasmid-bearing, plasmid-free competition and impulsive effect. According to the stability analysis of the boundary periodic solution, we obtain the invasion threshold of the plasmid-free organism and plasmid-bearing organism. Furthermore, by using standard techniques of bifurcation theory, we prove the system has a positive τ-periodic solution, which shows that the impulsive effect destroys the equilibria of the unforced continuous system and initiates the periodic solution. Our results can be applied to control bioreactors aimed at producing commercial products through genetically altered organisms.

In this paper, we consider two chemostat models with random perturbation, in which single species depends on two perfectly substitutable resources for growth. For the autonomous system, we first prove that the solution of the system is positive and global. Then we establish sufficient conditions for the existence of an ergodic stationary distribution by constructing appropriate Lyapunov functions. For the non-autonomous system, by using Has’minskii theory on periodic Markov processes, we derive it admits a nontrivial positive periodic solution. Finally, numerical simulations are carried out to illustrate our main results.