Delayed feedback induced complex dynamics in an Escherichia coli and Tetrahymena system

Abstract

Abundances of bacterial species such as Escherichia coli in a given environment are partly regulated by predation by protozoa species. Interactions among prey and predator species are not simple since a prey species has acquired defense mechanism against predation. A variety of defense mechanisms are known, for instance, production of toxin as a negative feedback effect to suppress predator species. On the other hand, negative feedback effects can be generated by a predator species to the prey species. The resulting population dynamics under the presence of negative feedback effects would be significantly affected by the strength of feedback control and time delay which naturally occurs until negative feedback effects become effective. In this paper, we develop a mathematical model to investigate the interaction between Shiga-toxin producing Escherichia coli and Tetrahymena with delayed feedback controls by production of Shiga-toxin and recruitment of neutrophils. By applying the quasi-steady-state approximation, the proposed model is reduced to a Lotka–Volterra type predator–prey system with two discrete delays. By investigating the distribution of the roots of the characteristic equation, the local stability as well as Hopf-bifurcation are studied. We provide a clear classification framework to detect the possibility of Hopf-bifurcation when two delays are present. Numerical simulations are carried out to verify the analytical results. Our findings reveal that the instability regions of coexistence equilibrium in two delay parameters plane always enlarge with the increase in negative feedback control coefficients, and especially the feedback controls on Tetrahymena population play a dominant role in the destabilization of coexistence equilibrium. Besides, we observe some interesting phenomena such as peak-adding bifurcation, quasi-periodic oscillation and chaos.