Bifurcations and chaos of a discrete-time model in genetic regulatory networks

Stability and bifurcation analysis of a gene expression model with small RNAs and mixed delays

“…It complies with the Routh-Hurwitz criterion that the equilibrium point of system (18) without delays is asymptotically stable if (H3) holds. We give the following hypothesis:…”

“…(ii) When τ > τ 0 , model (18) shows an oscillatory dynamic. Furthermore, a Hopf bifurcation takes place at the equilibrium point (x * , y * , z * , u * , v * ) when τ = τ 0 .…”

“…We assume (x * , y * , z * , u * , v * ) is one equilibrium point of system (18), and let y 1 = xx * ,…”

“…The gene regulatory network systematically studies the function and behavior of genes in the highly connected cell environment [16,17], and it regards the gene as a whole organizational structure. With the development of information technology and computer science progress in recent years, the gene expression model has been widely addressed [18][19][20]. On the basis of a comprehensive interpretation of cell metabolism, it has played a great role in exploring the mechanism of life activities, the cause and treatment of the disease.…”

“…Then the linearized system of(18) at (x * , y * , z * , u * , v * ) is as follows:⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ẏ 1 (t) = -(c + dy * )y 1 (t)dx * y 2 (t) + g (v * )y 4 (t), y 2 (t) = -dy * y 1 (t) -(f + dx * )y 2 (t), y 3 (t) = -by 3 (t) + ay 1 (tτ ), y 4 (t) = σ y 3 (t)σ y 4 (t), y 5 (t) = σ y 4 (t)σ y 5 (t).…”