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Technical report, Max-Planck-Institut f\u00fcr Mathematik in den Naturwissenschaften."},{"issue":"3","key":"ref22","doi-asserted-by":"crossref","first-page":"e1003359","DOI":"10.1371\/journal.pcbi.1003359","article-title":"Direct solution of the chemical master equation using quantized tensor trains","volume":"10","author":"V Kaazev","year":"2014","journal-title":"PLOS Computational Biology"},{"issue":"4","key":"ref23","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/0021-9991(76)90041-3","article-title":"A general method for numerically simulating the stochastic time evolution of coupled chemical reactions","volume":"22","author":"DT Gillespie","year":"1976","journal-title":"Journal of Computational Physics"},{"issue":"25","key":"ref24","doi-asserted-by":"crossref","first-page":"2340","DOI":"10.1021\/j100540a008","article-title":"Exact stochastic simulation of coupled chemical reactions","volume":"81","author":"DT Gillespie","year":"1977","journal-title":"The Journal of Physical Chemistry"},{"key":"ref25","doi-asserted-by":"crossref","first-page":"1876","DOI":"10.1021\/jp993732q","article-title":"Efficient exact stochastic simulation of chemical systems with many species and many channels","volume":"104","author":"MA Gibson","year":"2000","journal-title":"The Journal of Physical Chemistry A"},{"key":"ref26","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1017\/S002190020003535X","article-title":"Limit theorems for sequences of jump Markov processes approximating ordinary differential processes","volume":"8","author":"TG Kurtz","year":"1971","journal-title":"J Appl Probability"},{"key":"ref27","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1139\/p61-056","article-title":"A power series expansion of the master equation","volume":"39","author":"NG van Kampen","year":"1961","journal-title":"Canad J Phys"},{"key":"ref28","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/BFb0120765","article-title":"Limit theorems and diffusion approximations for density dependent Markov chains","author":"TG Kurtz","year":"1976","journal-title":"Math Programming Stud"},{"key":"ref29","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.physrep.2013.03.004","article-title":"Markovian dynamics on complex reaction networks","volume":"529","author":"J Goutsias","year":"2013","journal-title":"Physics Reports"},{"key":"ref30","unstructured":"Khalil HK (1992) Nonlinear systems. 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St. Julian&apos;s, Malta, pp. 142\u2013147.","DOI":"10.1109\/ISCCSP.2008.4537208"},{"key":"ref55","doi-asserted-by":"crossref","first-page":"042720","DOI":"10.1103\/PhysRevE.87.042720","article-title":"Exact protein distributions for stochastic models of gene expression using partitioning of poisson processes","volume":"87","author":"H Pendar","year":"2013","journal-title":"Physical Review E"},{"key":"ref56","first-page":"1","article-title":"Dynamical properties of discrete reaction networks","author":"L Paulev\u00e9","year":"2013","journal-title":"Journal of Mathematical Biology"},{"key":"ref57","article-title":"Determining the long-term behavior of cell populations: A new procedure for detecting ergodicity in large stochastic reaction networks","author":"A Gupta","year":"2013","journal-title":"Technical Report arXiv"},{"key":"ref58","unstructured":"Norris JR (1998) Markov chains. Cambridge: Cambridge University Press, xvi+237 pp."},{"key":"ref59","doi-asserted-by":"crossref","first-page":"518","DOI":"10.2307\/1427522","article-title":"Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes","volume":"25","author":"SP Meyn","year":"1993","journal-title":"Adv in Appl Probab"},{"key":"ref60","doi-asserted-by":"crossref","unstructured":"Boyd S, Vandenberghe L (2004) Convex Optimization. Cambridge, MA, USA: Cambridge University Press.","DOI":"10.1017\/CBO9780511804441"},{"key":"ref61","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1080\/10556789908805766","article-title":"Using SEDUMI 1.02, a Matlab Toolbox for Optimization Over Symmetric Cones","volume":"11","author":"JF Sturm","year":"2001","journal-title":"Optimization Methods and Software"},{"key":"ref62","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s10107-002-0347-5","article-title":"Solving semidefinite-quadratic-linear programs using SDPT3","volume":"95","author":"RH T\u00fct\u00fcnc\u00fc","year":"2003","journal-title":"Mathematical Programming Ser B"},{"issue":"22","key":"ref63","doi-asserted-by":"crossref","first-page":"8372","DOI":"10.1073\/pnas.0507818103","article-title":"Stochastic models for regulatory networks of the genetic toggle switch","volume":"103","author":"T Tian","year":"2006","journal-title":"Proc Natl Acad Sci"},{"key":"ref64","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1016\/j.jtbi.2004.11.033","article-title":"Stochastic modeling of nonlinear epidemiology","volume":"234","author":"WY Chen","year":"2005","journal-title":"J theor Biol"},{"issue":"9","key":"ref65","doi-asserted-by":"crossref","first-page":"5988","DOI":"10.1073\/pnas.092133899","article-title":"Mechanisms of noise-resistance in genetic oscillator","volume":"99","author":"JMG Vilar","year":"2002","journal-title":"Proc Natl Acad Sci"},{"key":"ref66","unstructured":"L\u00f6fberg J (2004) Yalmip : A toolbox for modeling and optimization in MATLAB. <ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" ext-link-type=\"uri\" xlink:href=\"http:\/\/control.ee.ethz.ch\/joloef\/yalmip.php\" xlink:type=\"simple\">http:\/\/control.ee.ethz.ch\/joloef\/yalmip.php<\/ext-link>"},{"key":"ref67","doi-asserted-by":"crossref","first-page":"2006.0033","DOI":"10.1038\/msb4100068","article-title":"Oscillations and variability in the p53 system","volume":"2","author":"N Geva-Zatorsky","year":"2006","journal-title":"Molecular Systems Biology"},{"key":"ref68","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/BF00277744","article-title":"Global asymptotic stability in volterra&apos;s population systems","volume":"19","author":"K Gopalsamy","year":"1984","journal-title":"Journal of Mathematical Biology"},{"issue":"23\u201324","key":"ref69","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1016\/j.crma.2010.11.001","article-title":"Convergence to equilibrium in competitive Lotka-Volterra and chemostat systems","volume":"348","author":"N Champagnat","year":"2010","journal-title":"Compte Rendus de l'Acad\u00e9mie des Sciences - 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