# Laub-Loomis Model

The model was initially proposed in a paper from Laub and Loomis , and then studied by several papers in reachability computation such as , . The dynamics of the model is defined by $\left\{ \begin{array}{lcl} \dot{x}_1 & = & 1.4 x_3 - 0.9 x_1 \\ \dot{x}_2 & = & 2.5 x_5 - 1.5 x_2 \\ \dot{x}_3 & = & 0.6 x_7 - 0.8 x_2 x_3 \\ \dot{x}_4 & = & 2 - 1.3 x_3 x_4 \\ \dot{x}_5 & = & 0.7 x_1 - x_4 x_5 \\ \dot{x}_6 & = & 0.3 x_1 - 3.1 x_6 \\ \dot{x}_7 & = & 1.8 x_6 - 1.5 x_2 x_7 \end{array} \right.$

We firstly consider a small initial set which is given by $x1 \in [1.1,1.3], x2 \in [0.95,1.15], x3 \in [1.4,1.6], x4 \in [2.3,2.5],$ $x5 \in [0.9,1.1], x6 \in [0,0.2], x7 \in [0.35,0.55]$
The following figures shows the interval overapproximations of the flowpipes in the time horizon [0,20] computed by Flow* 2.1.0. The total time cost is 18 seconds.  For a larger initial set $x1 \in [1,1.4], x2 \in [0.85,1.25], x3 \in [1.3,1.7], x4 \in [2.2,2.6],$ $x5 \in [0.8,1.2], x6 \in [-0.1,0.3], x7 \in [0.25,0.65]$
over the same time horizon, Flow* spends 60 seconds and generates the following result.  ### References

 M. T. Laub and W. F. Loomis.
A molecular network that produces spontaneous oscillations in excitable cells of dictyostelium.
Molecular Biology of the Cell, 9:3521–3532, 1998.

 R. Testylier and T. Dang.
NLTOOLBOX: A library for reachability computation of nonlinear dynamical systems.
In Proc. of ATVA’13, volume 8172 of LNCS, pages 469–473. Springer, 2013.

 X. Chen and S. Sankaranarayanan.
Decomposed Reachability Analysis for Nonlinear Systems.
In IEEE Real Time Systems Symposium (RTSS), pp. 13-24, 2016.