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

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. 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. 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. 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.