Journal on Policy and Complex Systems
According to Jumadinova , Matache , and Dasgupta ( 2011 ), this map does not show any complex behavior , which is possible to see when analyzing its fixed points — that are obtained by evaluating the conditions where p r
( t + 1 ) = p r
( t ). However , if z is different for every agent ( in the case where heterogeneity is introduced ), the derived map is not valid anymore .
Hence , rather than trying to derive ( if possible ) a map for such condition , it is already possible to check in the previous simulations that complexity emerges , but still , depending on the degree of heterogeneity introduced in the system , the system floats around the original fixed points zero and one present in the derived map above . When a large heterogeneity is introduced over the parameter z , such behavior completely disappears , giving rise to a full complex behavior over p r
( t ).
Consequently , it is possible to show that , depending on the degree of heterogeneity introduced , the system can behave deterministically ( as in the derived map ), probabilistically , around two main fixed points , and a very complex behavior with long memory components , as in the third case .
Thus , a new and unexpected behavior rises .
Cellular Automata as Dynamic Stochastic Systems
Cellular Automata are one of the
simplest dynamic systems capable of showing complex behavior , which , on top of a reduced set of rules , can be investigated using the standard analytical framework .
Moreover , Cellular Automata have been used to model a wide range of different problems , where it is not trivial to derive mean-field equations based on the establishment of ( partial ) differential equations , or when it is hard to model local and spatial-dependent interactions . A few examples of what kind of modeling this technique encompasses are :
• spatial-dependent predator – prey interactions as in Vilcarromero , Jafelice , and Barros ( 2010 );
• epidemics dynamics as in White , Del Rey , and Sánchez ( 2007 ) and Pfeifer et al . ( 2008 );
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