Modeling and Analysis of Gene Regulatory Networks - Modeling in Computational Biology and Biomedicine - page 49

all coordinates are either zero or positive (to be called the positive orthant ). Define
R + =[0 , + ∞ ) and the two-dimensional positive orthant as R 2 + = R + × R + .The

evolution of variable x i along time is governed by a known function f i : R 2 + → R + ,
which depends on both variables. Given initial values x 0 =( x 10 ,x 20 ), solutions
x i ( t ; x 10 ,x 20 ) for i =1 , 2 may be found by solving the initial value problem :

x 1 = f 1 ( x 1 ,x 2 ) , 1 (0) = x 10 ,
x 2 = f 2 ( x 1 ,x 2 ) , 2 (0) = x 20 .

A sufficient condition to guarantee that this problem has a unique solution is that
the functions f 1 and f 2 are continuous and have bounded, continuous derivatives,
with respect to both variables. The positive orthant is invariant for this system if:
whenever x i 0 (0) ≥ 0,then x i ( t ; x 10 ,x 20 ) ≥ 0 for all t

≥ 0 ( i =1 , 2). The
following condition guarantees invariance of the positive orthant :

x i =0 ⇒ f i ( x 1 ,x 2 ) ≥ 0 ,i =1 , 2 , (2.1)

which means that, at the boundary of the positive orthant, the vector field is either
zero or points towards the interior of the orthant, thus preventing the variables to
decrease to negative numbers. From now on, it will be assumed that functions f i
satisfy the required conditions, and that solutions of the initial value problem exist,
are unique, and non-negative.
For most systems the f i are nonlinear functions, and it is not possible to obtain
closed form solutions of the initial value problem. However, qualitative analysis of
the phase space can give a very good idea of the general behavior of the solutions.
The signs of the vector field ( f 1 ( z ) ,f 2 ( z )) at each point z ∈ R 2 + indicate the

direction of the solution at that point: for example, if f 1 ( z ) < 0 and f 2 ( z ) > 0,
then the variable x 1 will decrease and x 2 will increase whenever a solution passes
through the point z .The nullclines are curves that delimit regions of the plane where
the sign of the vector fields is constant:

∈ R 2 + : f i ( x )=0 } Nullcline i : Γ i = { x .

For an example see Fig. 2.5 . The points of intersection of the nullclines are called
the equilibria or steady states of the system:

x ∗ =( x 1 ,x 2 ) ∈ R 2 + : f 1 ( x 1 ,x 2 )=0 and f 2 ( x 1 ,x 2 )=0 .

A steady state is a configuration of the system where both variables remain constant,
and may be stable or unstable. To characterize this stability property suppose a small
perturbation is applied to the initial condition, when x (0) = x ∗ . If the solution
always returns back to x ∗ after a while, then the steady state x ∗ is stable; if the
solution moves away from x ∗ without returning to the point, then the steady state
x ∗ is unstable. The basin of attraction of x ∗ is the set of points x 0 ∈ R 2 + such