Figure 2. Characteristics of the delay transitions. The left panel, inverse square-root scaling law
dependence on extinction transients as parameter d increases above the critical degradation rate.
The right plot illustrate some examples of transients to the desert state (time is in log scale).
ORDINARY DIFFERENTIAL EQUATIONS MODEL
The mathematical model given by the following equations describes the time dynamics of
the vegetation and the fertile soil. In this model, we assume that the whole ecosystem is
constant and conformed by desert regions, fertile soil and vegetated areas. For this reason,
D = 1 - V - S (see diagram 1) .
The model is a simplified version of the model by Kéfi 2007 . The transition between a
desert patch to fertile soil is driven by facilitation due to the interaction vegetation-
microbes (term 𝑓𝑉). Plants colonize fertile soil at a rate 𝑉(𝑏 − 𝑐𝑉). Finally, the soil can
be degraded (at a rate 𝑑) and plants can die at rate 𝑚due to the climate or to grazing. See
figure 1 for the bifurcation diagram depending on the aridity parameter 𝑑.
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•Nowadays, some semiarid ecosystems could persists as a ghost state, and the vegetation could suffer an unexpected, sudden collapse.
•If we continue pushing ecosystems towards the critical point the possibilities of ghost ecosystems and further collapses increase.
•We propose an intervention method: by re-planting vegetation at fixed time intervals, we could sustain the ecosystem above the critical value,
maintaining it inside the dynamical (transient) ghost.
•The cost of the interventions rises as the ecosystem degradation increases.
•We have numerically shown that, even under the presence of stochasticity, delayed transitions can be sustained with the proposed intervention method
The tipping point identified in this model is governed by a saddle-node bifurcation
(caused by the collision of a saddle and a stable node). Below the critical value, vegetation
reach a steady state (about 27%of the area is vegetated, dashed line of the figure 2). Just
after the tipping point, all the trajectories go to extinction in a slow manner. The time to
extinction a follows the power-law with exponent -0.5.
Diagram 1. Schematic diagram of the transitions between the
vegetation (V), fertile Soil (S) and desert (D)states. The lower panel
is a detailed photograph of the soil crust interacting with a plant that
Background Dynamical system: semiarid ecosystems
Characterizing ghosts in semiarid ecosystems
Arid and semi-arid areas are those where the
evaporation of water and its evapotranspiration
(the fraction used by the plants and other
organisms) divided by total amount of incoming
water is lower than one. This means that there are
regions where there is a remarkable lack of water.
In the case of semiarid ecosystems, this factor is
lower than 50%. This kind of ecosystems span
more than a third of the total planet area .
Almost the same fraction of the human population
lives there. In the face of climate change (one of
the major Anthropocene effects), these ecosystems
are increasing in extension [4-5]. With the global
warming there are more regions where water is
evaporating faster, leading to more arid
ecosystems. Because of the aridity of these regions
the plants need to adapt and self-organize. The
balance between their grow and the competition
for water gives rise to spectacular large-scale
spatial patterns (such as the named fairy circles).
From a dynamical systems point of view, these
ecosystems are bistable and are often endangered
by potential catastrophic shifts [4,6]. Under
bistability, in the same environmental conditions,
there is coexistence of vegetation and desert
regions [2-3]. A typical example is the Egypt-
Israel border. Depending on the management
policies, vegetated regions can became a desert.
Because of the intrinsic feedbacks that make
vegetation to self-sustain, such as the facilitation
 in water uptake dynamics becomes nonlinear
and transients  can play a key role. This is
specially relevant if we analyze paleologic records,
where one can observe that Sahara was once green
and became a full desert in few decades . This
rapid collapse could be what in dynamical systems
is called a saddle-node ghost or a delayed
Exploiting delay-transitions to sustain the ecosystem
Figure 1. Bifurcation diagram of the model. In this picture the grey
region represents where the ghost exist. The spherical cellular automata
illustrate how the vegetation is distributed spatially depending on the
Photograph of a semiarid ecosystem from central Spain.
The zone is an experimental station at Aranjuez from
Fernando Maestre`s Lab (Alicante University)
Figure 3. The upper panels are
from the mean field version of
the model. The red trajectory is
the time dynamics without any
intervention, while the black one
is an example of a successful
intervention condition. The last
figure shows the minimal cost of
depending on the degradation
rate. The lower panels are
obtained from stochastic
simulations of the system. The
effect of the intervention is
shown overlapped: intervention
time-series (black) and the
original one (red). Finally, the
last image shows the survival
probability depending on the
intervention performed for a fix
value of aridity (rate d).
-- For more information: Vidiella B, Sardanyés J, Solé R. 2018 Exploiting delayed transitions to sustain semiarid ecosystems after catastrophic shifts. J. R. Soc. Interface 15: 20180083 --