Mutual inhibition, competition, and periodicity in a two species Chemostat-like System
Kimathi, Eliphas G
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In this study, we developed a model that addresses competition in a periodic chemostat-like system with mutual growth inhibition. A periodic Kolmogorov system of ordinary differen tial equations that describe mutual inhibition for two-species competing for a single, essential nutrient in a chemostat-like system was developed, and its dynamics studied. We then pre sented a model of the general chemostat with mutual inhibition, where global behavior of the solutions of this model were discussed and we showed that for two species competing for a single nutrient available in limiting supply, at most one species would survive. Mutual inhibition in the periodic chemostat, where the operating parameters including the nutri ent uptake function, washout rate, and nutrient concentration were allowed to be periodic functions of time, with commensurate periods was also studied. It was assumed that the chemostat was spatially homogeneous, but all the parameters in the model were periodic. The species specific nutrient uptake was assumed to be a monotone increasing function of the nutrient concentration, but allowed to be periodic as a function of time with its period being commensurate with that of the other parameters. We took a Holing Type II function for the nutrient uptake, that is, the function followed Michaelis-Menten kinetics. As long as both species were present, inhibition was also present. Each species did not inhibit its own growth, and that overall inhibition effect of a species decreased as its biomass increased and vice-versa. In addition, inhibition to the growth of a given species was increased by the pres ence and growth of its competitor. We described periodic nutrient input and dilution rates that make persistence of both species possible in the system without inhibition and then showed that this persistence did not hold in the mutually inhibiting periodic chemostat-like system. We introduced a function that accounted for mutual inhibition effects of the species involved in the competition. First, we considered mutual inhibition in the ordinary chemostat and used stability theories to study the global behavior of the solutions of the model. For two species competing for a single nutrient available in limiting supply, we showed that at most one species survived. To account for realistic periodic variations, we used a cosine Fourier series to introduce periodic parameters in the ordinary chemostat then applied a threshold result on the global dynamics of the scalar asymptotically periodic Kolmogorov equation to a growth model of two species. The theory of asymptotically periodic semiflows and comparison methods were used to demonstrate uniform persistence of all the species and that the full dynamical system admitted at least one positive, periodic solution. Applying comparison theorems, we showed that with mutual inhibition, competitive exclusion always holds in models that would allow coexistence without inhibition. Furthermore, we demonstrated that initial conditions playa Xll crucial role in determining which species survived. Using Matlab, we got numerical results that confirmed the predictions of the models. The thesis is organized as follows: In Chapter I, we introduced the chemostat, its equa tions, effects of shading, mutual inhibition and periodic semifiows. In Chapter 2, we gave an overview of the progress made regarding chemostat models and identified gaps that existed in the current models. we also gave some important results that were useful in the proof of our theorems in chapters that followed. In Chapter 3, we presented a model of the general chemostat with mutual inhibition and discussed the global behavior of this chemostat. We also gave some numerical results for the model of that section. In Chapter 4, we used Fourier series to describe the nutrient input concentration and washout (dilution) rates. This in troduced periodicity in the chemostat and we applied the theory of asymptotically periodic semifiows and the comparison method was used to determine criteria that guaranteed the existence of a positive solution for the full system. In Chapter 5, we discussed a model for competition in the periodic chemostat with mutual inhibition. We found that with mutual inhibition, coexistence was not possible. This finding differed with other findings of the periodic chemostat in the sense that without mutual inhibition, coexistence was possible. It is in Chapter 6 that we gave conclusions and discussed our findings. Suggestions for further study and open questions that arose out of the thesis are indicated. Further open questions that have not been addressed in our study have been given.
CitationPhilosophy In Applied Mathematics
University of Nairobi