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In this manuscript, we prove the existence of slow and fast traveling wave solutions in the original Gatenby–Gawlinski model. We prove the existence of a slow traveling wave solution with an interstitial gap. This interstitial gap has previously been observed experimentally, and here we derive its origin from a mathematical perspective. We give a geometric interpretation of the formal asymptotic analysis of the interstitial gap and show that it is determined by the distance between a layer transition of the tumor and a dynamical transcritical bifurcation of two components of the critical manifold. This distance depends, in a nonlinear fashion, on the destructive influence of the acid and the rate at which the acid is being pumped. [ABSTRACT FROM AUTHOR]

One of the deficits observed in autism spectrum disorder (ASD) is impaired imaginative play. One form of imaginative play common in many typically developing (TD) children is having an imaginary companion (IC). The occurrence of ICs has not been investigated extensively in children with ASD. We examined differences in parent report of IC between TD and ASD populations in 215 (111 with ASD) gender-matched children aged between 2 and 8 years. Findings indicate that significantly fewer children with ASD created ICs, although there were many between-group similarities in IC forms and functions. Results are discussed in terms of qualitative differences in play, social attributions, and how children with ASD conceptualize their ICs’ minds. [ABSTRACT FROM AUTHOR]