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Inspired by the S-procedure and semi-definite programming, we use the sum of squares decomposition based method to ensure the global asymptotic stability of the positive equilibrium through the associated polynomial Lyapunov functions. By exploring the monotonic property of the trace of the Jacobian matrix with respect to [Formula: see text] under the given different conditions, we analytically verify that there is a corresponding unique [Formula: see text] such that the trace is equal to zero and prove the existence of Hopf bifurcation, respectively. <\/jats:p>","DOI":"10.1142\/s0218127417502224","type":"journal-article","created":{"date-parts":[[2018,1,29]],"date-time":"2018-01-29T03:58:05Z","timestamp":1517198285000},"page":"1750222","source":"Crossref","is-referenced-by-count":9,"title":["Bifurcation Analysis of a Predator\u2013Prey System with Ratio-Dependent Functional Response"],"prefix":"10.1142","volume":"27","author":[{"given":"Xin","family":"Jiang","sequence":"first","affiliation":[{"name":"School of Mathematics and Systems Science, Beihang University, Xueyuan Road No. 37, Beijing 100191, P. R. 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