Question

Consider an animal that forages in a field, which we will model as a flat 2-dimensional space. The movement dynamics of the animal are given by the following linear system of ODEs

dx/dt = Ax where A=[ 1 −2
1 −1]
(a) Find the nullclines of this system.

Answer 1

The nullclines for the given system of ODEs are obtained by setting the rate of change of each variable to zero and solving for the other variable. The nullclines are x1 = 2x2 and x2 = x1, representing the points in the phase plane where the growth rate of each variable is zero.

Step-by-step explanation:

The student's question involves solving a system of ordinary differential equations (ODEs) and finding the nullclines of the system. A nullcline is the set of points in the phase plane where the rate of change of one of the variables is zero. Given the matrix A for the system of ODEs dx/dt = Ax, where A is a 2x2 matrix [1 -2; 1 -1], we can find the nullclines by setting each derivative in our system to zero separately.

For dx1/dt = 0, we have 1*x1 - 2*x2 = 0. Solving for x1, we get x1 = 2x2. This is the nullcline for the first variable. For dx2/dt = 0, we have x1 - x2 = 0. Solving for x2, we get x2 = x1. This is the nullcline for the second variable.

These nullclines can be plotted on a phase plane to better understand the behavior of the system. When analyzing such systems, similar methods can be used to understand and visualize the dynamics of the motion, such as projecting motions onto perpendicular components and analyzing them independently.