The Mean-Field Survival Model for Stripe Formation in Zebrafish Exhibits Turing Instability

Abstract: Zebrafish have been used as a model organism in many areas of biology, including the study of pattern formation. The mean-field survival model is a coupled ODE system describing the expected evolution of chromatophores coordinating to form stripes in zebrafish. This paper presents analysis of the model focusing on parameters for the number of cells, length of distant-neighbor interactions, and rates related to birth and death of chromatophores. We derive the conditions on these parameters for a Turing bifurcation to occur and show that the model predicts patterns qualitatively similar to those in nature. In addition to answering questions about this particular model, this paper also serves as a case study for Turing analysis on coupled ODE systems. The qualitative behavior of such coupled ODE models may deviate significantly from continuum limit models. The ability to analyze such systems directly avoids this concern and allows for a more accurate description of the behavior at physically relevant scales.