Ecosystems are probably being pushed towards critical points due to the current increase of ecological stresses. Tipping points could thus be reached, involving transitions from rich ecosystems (high diversity, high functionality) to poorer ones (i.e. deserts). These critical transitions are typical of nonlinear systems having strong feedbacks given by e.g., cooperative or facilitation processes. In our case, we have studied semiarid ecosystems, which include arid and dry-subhumid systems, spanning more than 40% of extant habitats and containing a similar percentage of the human population. The transition for these ecosystems involves a rapid shift from a vegetated area to a desert. Palaeoclimatic data and theoretical models predict this rapid shift towards the desert state, governed by a rapid and catastrophic collapse of the vegetation. Mathematical models suggest that these catastrophic shifts are governed by saddle-node bifurcations. Interestingly, just after the bifurcation has occurred, the presence of a so-called ghost delays the extinction, making the ecosystem to seem stable even though vegetation extinction is inevitable. Nowadays, there is field data from global survey that supports the existence of this ghost phenomenon, already identified in experiments with nonlinear electronic circuits.
Results/Conclusions
Using a simple mathematical model describing key ecological processes of semiarid ecosystems, we have characterised the saddle-node bifurcation and studied the delays of the ghost in this type of ecosystems. We have seen how the time remaining in the transitory depends on the parametric distance between the critical value of the soil stress causing the bifurcation and further increase of soil stress. This delay times follow the inverse square-root scaling law, in agreement with previous results on bifurcation theory for other dynamical systems. Moreover, we have seen that this behaviour persists even if we apply demographic noise to the simulations. Finally, we have shown that using small interventions (I.e. replanting vegetation) the system can be hold into a permanent transitory state, avoiding the collapse of the ecosystem. The cost of these interventions increase until a limit value, when the replantation rate is higher than the death rate of the plants by the stress.