prey-predator model, stability, prey-taxis, delay, nonhomogeneous Hopf bifurcation

Let $ a_0, a_1, \dots, a_{n-1} $ be real numbers and let $ A = Circ(a_0, a_1, \dots, a_{n-1}) $ be a circulant matrix with $ f(x) = \Sigma ^{n-1}_{j = 0}a_jx^j $. First, we prove that $ Circ(a_0, a_1, \dots, a_{n-1}) $ must be invertible if the sequence $ a_0, a_1, \dots, a_{n-1} $ is a strictly monotonic sequence and $ a_0+a_1+\dots+a_{n-1}\neq 0 $. Next, we reduce the calculation of $ f(\varepsilon ^0)f(\varepsilon)\dots f(\varepsilon ^{n-1}) $ for a prime $ n $ by using the techniques on finite fields, where $ \varepsilon $ is a primitive $ n $-th root of unity. Finally, we provide two examples to explain how to use the obtained results to calculate the determinant of a circulant matrix.