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Tipping in an ecological model under multiple impulsive forces
Syamal K. Dana
Centre for Mathematical Biology and Ecology, Department of Mathematics
Jadavpur University, Kolkata 700032, India
We explore tipping or critical transitions near two saddle-node bifurcation points of a population model against the time-varying carrying capacity of the system. The autonomous system exhibits bistability with a coexisting refuge state of low population density and an outbreak state of large population. If the carrying capacity is varied at a linear rate, the system does not show sharp transitions as expected immediately at the bifurcation points but tips to the alternate states after an elapse of time. This delay in tipping decreases with faster rate of change of the carrying capacity. The impact of environmental shocks is then considered which is modelled by a triangular shape impulse. The external impulse varies the carrying capacity at a constant rate, however, it is withdrawn at an identical or different constant rate after reaching a minimum. Delayed tipping occurs in such a situation of external shock modelled by a single triangular impulse but shows a dependence on the falling and rising rates of the impulse that pushes the carrying capacity to cross the bifurcation points. The extended range of the rate of rising and falling of the impulse has thus been identified in a phase diagram that clearly delineates the rate parameter zones of tipping and no tipping. The active time window of the external impulse on the carrying capacity called as exceedance time plays a decisive role on the occurrence of tipping. We analytically derive the critical value of the exceedance time that depends upon the rate parameters (rising and falling rates) and the depth of the impulsive. In addition, we apply a second impulse, in case the first impulse is not large enough in strength that fails to induce any tipping to the desirable low population state from an outbreak state. The second impulse is assumed to be even weaker in strength compared to the first one. In such a situation of multiple impulses, tipping occurs as a consequence of a past effect of the first impulse. The role of the rate parameters and strength of the impulses, and most importantly, the time interval of the impulses is considered in detail to delineate the tipping zones in parameter space. We demonstrate the scenarios with numerical experiments, and the process of tipping using the dynamical change in the potential of the system.