Two-proteins-x-and-y-control-each-other-through-mutual-repression-The-dynamic-model-for-the-system-consists-of-the-following-system-of-ODEs-calculate-the-steady-state-values-of-these-two-protein

Question Number 207151 by Wuji last updated on 07/May/24

Two proteins (x and y) control each

other through mutual repression . The

dynamic model for the system consists

of the following system of ODEs .

calculate the steady state values of

these two proteins . comment on the

stability of the steady state and the type of

phase portrait expected for this system .

consider x, y ⩾ 0

\frac{dx}{dt} = \frac{y}{1 + y} - 2 \frac{dy}{dt} = \frac{x}{1 + x} - y

Answered by Berbere last updated on 07/May/24

\frac{d x}{d t} = \frac{y}{1 + y} - 2 x = f (x, y); \frac{d y}{d t} = \frac{x}{1 + x} - y = g (x, y)

f (x, y) = g (x, y) = 0 \Leftrightarrow {\begin{cases} \frac{y}{1 + y} - 2 x = 0 \\ \frac{x}{1 + x} - y = 0 \end{cases} \Leftrightarrow {\begin{cases} \frac{y}{1 + y} - 2 x = 0 \\ x (1 - y) = y \end{cases}

\Rightarrow \frac{y}{1 + y} - 2 \frac{y}{1 - y} = 0 \Rightarrow y (1 - y) - 2 y (1 + y) = 0

- 3 y^{2} - y \Rightarrow y (3 y + 1) = 0 \Rightarrow y = 0; y ⩾ 0

y = 0 \Rightarrow x = 0

J a c o b i e n M a t r i X J = (\begin{matrix} \frac{\partial f}{\partial x} \frac{\partial g}{\partial x} \\ \frac{\partial f}{\partial y} \frac{\partial g}{\partial y} \end{matrix}) (x, y) = (0, 0)

J = (\begin{matrix} - 2 1 \\ 1 - 1 \end{matrix})

d e t (X - {z I}_{2}) = 0 \Rightarrow X^{2} + X - 3 = 0 X \in {\frac{- 1 - \sqrt{13}}{2}, \frac{- 1 + \sqrt{13}}{2}}

R e (\frac{- 1 + \sqrt{13}}{2}) > 0 N o s t a b i l i t y o f E q u i l i b r e

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