Phase lag-driven indirect tipping in a seasonally forced coupled ecological system

Tipping phenomena represent critical transitions to alternative states, often irreversible, arising from parameter changes over time. Indirect tipping is a special kind of cascading dynamics observed in coupled systems which occurs when a downstream system collapses despite the upstream driver system remaining almost undisturbed in spite of being perturbed externally. We investigate this phenomenon in a seasonally forced, unidirectionally coupled ecological model consisting of two competing toxin-producing phytoplankton and a fish population with an Allee effect. The phytoplankton growth rates vary periodically with a phase lag ϕ, capturing distinct seasonal niches and both single and multiple bloom cycles. Although the phytoplankton system remains oscillatory and stable for all ϕ, variations in phase lag can induce tipping in fish population leading to a state of extinction. In particular, synchronous periodic growth of the two driving species ( ϕ=0) leads to complete tipping, whereas anti-phase variation of growth ( ϕ=π) results in partial tipping in fish population. These results demonstrate that seasonal variation of two competing phytoplankton bloom with a phase lag and bloom periodicity critically govern indirect tipping in aquatic ecosystems. Furthermore, we apply the Fourier spectrum-based early warning indicators that reliably signal the impending transition in this periodically forced system.