Zeros of a class of transcendental equation with application to bifurcation of DDE

Zeros of a class of transcendental equation with small parameter ϵ(0 ≤ ϵ ≤ 1) are considered in this paper. There have been many works in the literature considering the distribution of zeros of the transcendental equation by choosing the delay τ as bifurcation parameter. Different from standard consideration, we choose ϵ as bifurcation parameter (not the delay τ) to discuss the distribution of zeros of such transcendental equation. After mathematical analysis, the obtained results are successfully applied to the bifurcation analysis in a biological model in the real word phenomenon. In the real world model, the effect of climate changes can be seen as the small parameter perturbation, which can induce bifurcations and instability. We present two methods to analyze the stability and bifurcations.