## Fourier series

Jane Ireri, Pokhariyal G, Moindi S. "Chemostat Model with Periodic Nutrient Input Described by Fourier Series." Asian Research Journal of Mathematics. 2020;16(8):16-27. AbstractWebsite

In this paper we present a periodic Chemostat model of two species competing for a single nutrient available in limiting supply. The nutrient input is varied periodically using a Fourier series function to take into account the changing patterns as seasons vary. We show both analytically and numerically that varying the nutrient input using a Fourier Series function results in a better model to describe coexistence of species in natural environments.

Jane Ireri, Pokhariyal G, Moindi S. "Delayed Nutrient Conversion for a Single Species Periodic Chemostat." Journal of Scientific Research and Reports. 2020;26(5):1-9. AbstractWebsite

In this paper we analyze a Chemostat model with periodic nutrient input modelled using Fourier series and incorporate delays in nutrient conversion. We show that both periodicity and delays have complementing influence in the long term behaviour of the species. Numerical results show that periodicity has bigger influence on species density variations for delays below the Hopf Bifurcation point, while for delays above the Bifurcation point,the delay effect is more influential.

Jane Ireri, Pokhariyal G, Moindi S. "Hopf Bifurcation Analysis for a Two Species Periodic Chemostat Model with Discrete Delays." Journal of Advances in Mathematics and Computer Science. 2020;35(3):93-105. AbstractWebsite

In this paper we analyze a Chemostat model of two species competing for a single limiting nutrient input varied periodically using a Fourier series with discrete delays. To understand global aspects of the dynamics we use an extension of the Hopf bifurcation theorem, a method that rigorously establishes existence of a periodic solution. We show that the interior equilibrium point changes its stability and due to the delay parameter it undergoes a Hopf bifurcation.
Numerical results shows that coexistence is possible when delays are introduced and Fourier series produces the required seasonal variations. We also show that for small delays periodic variations of nutrients has more influence on species density variations than the delay.