pp2 : A 2D Predator-Prey Model.

This demo illustrates the computation of families of stationary solutions, including bifurcating stationary families, as well as the detection of a Hopf bifurcation. In the second run the family of periodic solutions that emanates from the Hopf bifurcation is computed. This family terminates in a heteroclinic orbit.

The equations, which model a predator-prey system with harvesting, are

$$\begin{array}{cl} u_1 ' &= p_2 u_1 (1 - u_1 ) - u_1 u_2 - p_1 (1-e^{-p_3 u_1}) ,\\ u_2 ' &= -u_2 + p_4 u_1 u_2 .\end{array}$$ (14.3)

Here $p_1$ is the principal continuation parameter, while $p_2=p_4=3$ and $p_3=5$, are fixed The use of PLAUT is also illustrated. The saved plots are shown in Figure 14.1 and Figure 14.2.

Table 14.9: Commands for running demo pp2.

COMMAND	ACTION
mkdir pp2	create an empty work directory
cd pp2	change directory
@dm pp2	copy the demo files to the work directory
@R pp2 1	1st run, using constants-file c.pp2.1; stationary solutions
@sv pp2	save output-files as b.pp2, s.pp2, d.pp2
@R pp2 2	2nd run, using constants-file c.pp2.2; periodic solutions
@ap pp2	append output-files to b.pp2, s.pp2, d.pp2

Table 14.10: Plotting commands for demo pp2.

AUTO-COMMAND	ACTION
@p pp2	run PLAUT to graph the contents of b.pp2 and s.pp2;
PLAUT-COMMAND	ACTION
d2	set convenient defaults
ax	select axes
1 3	select real columns 1 and 3 in b.pp2
bd0	plot the bifurcation diagram; $max~u_1$ versus $p_1$
d1	choose other default settings
bd	get blow-up of current bifurcation diagram
0  1 -0.25  1	enter diagram limits
sav	save plot (see Figure 14.1)
fig.1	upon prompt, enter a new file name, e.g., fig.1
cl	clear the screen
2d	enter 2D mode, for plotting labeled solutions
11 15 19 23	select these labeled orbits in s.pp2
d	default orbit display; $u_1$ versus time
1 3	select columns 1 and 3 in s.pp2
d	display the orbits; $u_2$ versus time
2 3	select columns 2 and 3 in s.pp2
d	phase plane display; $u_2$ versus $u_1$
sav	save plot (see Figure 14.2)
fig.2	upon prompt, enter a new file name
ex	exit from 2D mode
end	exit from PLAUT

Figure 14.1: The bifurcation diagram of demo pp2.

Figure 14.2: The phase plot of solutions 11, 15, 19, and 23 in demo pp2.