Critical diapause portion for oscillations: Parametric trigonometric functions and their applications for Hopf bifurcation analyses

An important adaptive mechanism for ticks in respond to variable climate is diapause. Incorporating this physiological mechanism into a tick population dynamics model results in a delay differential system with multiple delays. Here, we consider a mechanistic model that takes into consideration of the development diapause by both larvae and nymph ticks, which share a common set of hosts. We introduce the concept of parametric trigonometric functions (convex combinations of two trigonometric functions with different oscillation frequencies) and explore their qualitative properties to derive an explicit formula of the critical diapause portion for the Hopf bifurcation to take place. Our work shows analytically that diapause can generate complex oscillations even though seasonality is not included.