ivp : Time Integration with Euler's Method.

This demo uses Euler's method to locate a stationary solution of the following predator-prey system with harvesting :

$$\begin{array}{cl} u_1 ' &= p_2 u_1 (1 - u_1 ) - u_1 u_2 - p_1 (1-e^{-p_3 u_1}) ,\\ u_2 ' &= -u_2 + p_4 u_1 u_2 ,\\ \end{array}$$ (14.15)

where all problem parameters have a fixed value. The equations are the same as those in demo pp2. The continuation parameter is the independent time variable, namely PAR(14).

Note that Euler time integration is only first order accurate, so that the time step must be sufficiently small to ensure correct results. Indeed, this option has been added only as a convenience, and should generally be used only to locate stationary states. Note that the AUTO -constants DS, DSMIN, and DSMAX control the step size in the space consisting of time, here PAR(14), and the state vector, here $(u_1,u_2)$.

Table 14.22: Commands for running demo ivp.

AUTO -COMMAND	ACTION
! mkdir ivp	create an empty work directory
cd ivp	change directory
demo('ivp')	copy the demo files to the work directory
ld('ivp')	load the problem definition
run(c='ivp.1')	time integration
sv('ivp')	save output-files as b.ivp, s.ivp, d.ivp