Stability patterns inside an instability space(Turing instability and Busse Balloon)


Assistant Professor at Ferdowsi University of Mashhad, Mashhad, Iran


     Introduction The
theory of spatial pattern formation via Turing bifurcations plays an important
role in recognizing causes spatial pattern formation in biology, chemistry and
physics. In the past decade, ecogeomorphologists have emphasized that local
interactions between variables in dynamic systems may invoke emergent spatial patterning at larger spatial scales. The complexity of ecogeomorphic systems
make it very difficult to understand how ecosystems will respond to the changes.
In the recent decades, Instability turing theory in ecogeomorphology field
makes possibility of predicting future changes. In this paper, using Rietkerk
model have been indicated that patterned ecosystems (emphasis on semi-arid
region) may respond in a nonlinear way to environmental change, meaning that
gradual changes result in rapid degradation. The result indicate
that patterned arid ecosystems respond in different ways to changes in
rainfall depend on rates, rather than magnitudes of environmental change. Materials
and methods In
the model of Rietkerk three variables state are considered: plant density, soil
water and surface water. The model assumes that rainfall events in arid and
semi-arid ecosystems occur at an intensity exceeding the infiltration capacity
of the soil. Hence, part of the rainwater infiltrates into the soil, while the
remainder produces surface water and runoff routed to other spatial locations. In
arid ecosystems, vegetation cover is often a two-phase mosaic composed of
densely vegetated patches and bare soil areas. The two phases of the mosaic
mainly differ in their infiltration capacity for water. Vegetation improves the
structure of the soil because it stimulates the biological activity in the
soil, its root system forms channels and aerates the soil, and its canopy
intercepts raindrops and prevents crust formation. Thus, infiltration is higher
under vegetation than in bare soil. Thus, after a rain event, water runs off in
bare areas and mainly infiltrates in vegetated patches, which act as sinks of
water. Results
and discussion The
result of this model simulation in works of researchers such as, Kefi,
Rietkerk, HillRislambers, Dakos and Siteur indicate that how scale-dependent
feedback by short-range facilitation and long-range competition between
vegetation and water, induces spatial self-organization, thereby providing a
possible explanation for the observed patterns. The model allows for a
homogeneous equilibrium of plant density, soil water, and surface water. With
decreasing rainfall (R), the homogeneous plant equilibrium decreases until
plants become extinct for R≤1.0. Close to this extinction threshold, the
homogeneous plant equilibrium is unstable against small spatial perturbations.
This is indicative of the principle of pattern formation as first outlined by
Turing: pattern formation can occur if an equilibrium is stable to spatially
homogeneous perturbations but unstable to heterogeneous perturbations. From the
Turing instability points unstable non-homogeneous equilibria originate which
link up to a stable nonhomogeneous equilibrium. This stable non-homogeneous
equilibrium, which is characterized by a single plant peak, exists for a wide
range of rainfall rates, and extends far into the region where homogeneous
plant cover would go extinct (R≤1.0). In general, the pattern formation leads
to higher average plant productivity as compared to the homogeneous situation.  In
this study have been showed that patterned ecosystems systematically respond in
two ways to changing environmental
conditions: by changing vegetation patch biomass (transient spatial pattern
formation in the Turing instability range) or by adapting pattern wavelength
(spatial pattern formation in busse ballon range). In the latter case patches
go extinct or split up and may rearrange. In arid ecosystems, gradual
wavelength adaptation is constrained to conditions of high rainfall, slow
changes in rainfall and high levels of stochastic spatial variation in biomass
(noise). The adaptation process is less gradual under conditions of either low
rainfall, rapid change or low levels of noise. Such conditions do not allow
vegetation patches to rearrange, and facilitate the simultaneous extinction of
half the patches or even a transition to
a degraded state without any patches. Model of Rietkerk shows that an overview
of stable patterned states, the Busse
balloon, is a powerful tool in understanding the response of patterned
ecosystems to changing environmental conditions. If a system is in a
stable patterned state (i.e. in the
Busse balloon), a pattern tends to solely adapt its amplitude, while if the
system leaves the Busse balloon, a pattern adapts its wavenumber. The ability
of patches to rearrange is determined by the period doubling instability. Once
the system surpasses this instability,
patches do not rearrange, leading to extinction of half or all the patches.
this findings suggest that the response of patterned ecosystems to
environmental change does not only
depend on the magnitude of change, but also on the rate with which conditions
change: patterned ecosystems may not be able to respond in a gradual way to
rapid environmental change.  Conclusion This
study highlights that self-organization may influence the flow of resources
through the ecosystem and thereby affects the functioning of ecosystems at
larger spatial scales. Finding similar research about exhibiting regular
vegetation patterns in a variety of ecosystems, such as wetlands, savannas,
mussel beds, coral reefs, ribbon forests, intertidal mudflats, marsh tussocks
and arid ecosystems, highlighted the importance of scale-dependent feedback
mechanisms between organisms and their environment. Besides showing regular
patterns, these systems may exhibit bistability if scale-dependent feedback is
indeed a main driver of their dynamics. Moreover, it might be possible to use
the vegetation patterns themselves, by mimicking the patterns observed in
healthy systems, to restore degraded systems.