Chapter 11

The competitive Lotka-Volterra model.

Here we consider the non-linear competitive Lotka-Volterra model discussed in this chapter. It models two competing species (or competing business, etc.) with respective population sizes $x_1(t)$ and $x_2(t)$, using the following system of equations:

\begin{align} x_1' &= r_1x_1\Big( 1-\dfrac{x_1+k_{12}x_2}{K_1}\Big), \\ x_2' &= r_2x_2\Big( 1-\dfrac{x_2+k_{21}x_1}{K_2}\Big) \end{align}

where $x_i(t)$ is the population size for species $i$ at time $t$, $r_i>0$ is the “growth rate” that would govern exponential growth if there were no resource constraints, $K_i>0$ is the “carrying capacity” that constrains growth based on the finite resources, and $k_{ij} > 0$ is a measure of the impact of species $j$ on species $i$.

To play with the visulization yourself, change the parameters, and see how the graph changes. Choose your desired initial condition by dragging the blue point on the graph. The stationary points are shown as red dots. Observe that solution curves starting in the first quadrant stay in the first quadrant.

An ODE system with bounded trajectories.

Here we consider the following system of equations:

\begin{align} x_1' &= -(x_1)^3 + ax_1 + bx_2, \\ x_2' &= -(x_2)^3 + cx_1 + dx_2. \end{align}

To play with the visulization yourself, change the parameters, and see how the graph changes. Choose your desired initial condition by dragging the blue point on the graph. Observe that for any choice of parameters, the vector fields point inwards when $|x_1(t)|$ and $|x_2(t)|$ are large. From this you can deduce that no trajectories are going off to infinity.