A Predator-Prey Model in the Chemostat with Time Delay

Figure 1

Variation of the largest real part of the eigenvalues as the bifurcation parameter 𝛼 is varied. At 𝛼≈1.15 and 1.5, the largest real part crosses zero and it seems that there is a zero eigenvalue for 𝛼≈1.15 and a pair of pure imaginary eigenvalues for 𝛼≈1.5. The largest real part becomes positive as 𝛼 increases through 1.5. But as 𝛼 increases further, for 𝛼≈17, the largest real part crosses zero again and remains negative thereafter. There is a second Hopf bifurcation at 𝛼≈17. This is consistent with what is observed in Figure 3 when 𝜏=0.5 and 𝛼 varies from 0 to 30. Parameters are 𝐷=Δ=1, 𝑘=24, and 𝜏=0.5.