Timing and propagule size of invasion determine its success by a time-varying threshold of demographic regime shift.

Theory of invasion ecology indicates that the number of invading individuals (propagule size) and the timing of invasion are important for invasion success. Propagule size affects establishment success due to an Allee effect and the effect of demographic stochasticity, whereas the timing of invasion does so via niche opportunity produced by fluctuating predation pressure and resource abundance. We propose a synthesis of these two mechanisms by a time-varying dose-response curve where the dose is propagule size and the response is establishment probability. We show an example of the synthesis in a simple predator-prey model where successful invasion occurs as a demographic regime shift because of the bistability of the system. The two mechanisms are not independent, but simultaneously determine invasion success in our model. We found that positive growth rate of an invading species does not ensure its establishment, especially when its propagule size is small or when its growth rate is in a decreasing trend. We suggest the difficulty of understanding invasion process based on a dose-response curve of propagule size as no unique curve can be determined due to the effects of invasion timing (i.e., the threshold of demographic regime shift is time varied). The results of our model analysis also have an implication on the phase relationship between population cycles of predators and prey.