| Abstract: |
| 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 can be reduced to a Lotka-Volterra (LV) predator-prey type system with two discrete delays. By investigating the distributions of the roots of the characteristic equation, the local stability as well as Hopf-bifurcation are well 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 as the increase of 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. |
|