Vector Field Plots

A drawback of the above phase-plane plots is that they illustrate how the variables change only along certain trajectories (e.g., starting at n,(0) = n2(0) = 10). A more complete visual representation of the dynamics is possible by

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Figure 4.15: Vector-field plot for the two-species Lotka-Volterra model of competition. The change over one generation is shown from a series of different starting positions. All of the parameters are the same as in Figure 4,14 with a12 set to 0. The filled circle represents an equilibrium of this model (n, = 1000; n2 = 500).

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drawing arrows specifying the direction of change from a variety of different positions. Such a phase-plane diagram is known as a vector-field plot. Figure 4.15 is an example of a vector-field plot for the discrete-time model of competition, where each arrow represents the change over a time step in the numbers of species 1 (along the horizontal axis) and species 2 (along the vertical axis). To see how the arrows are drawn, consider the top left arrow. The base of this arrow is at {50,1200), which represents the starting population sizes of the two species. Plugging in these initial conditions as well as the parameters (r, = r2 = 0.5, an = 0, a,-: = 0-5, and = K2 = 1000) into equation (3.14) predicts the numbers of species in the next time step:

Thus, the tip of the arrow is placed at {73.75,1065}. The same procedure is followed for as many arrows as desired (100 in this plot). It should be noted that sometimes the change per time step is too short or too long to plot with an arrow in this way. Thus, the arrows are often rescaled by a factor to make their lengths more visually appealing.

Vector-field plots are a great way to visualize interactions between two variables. In this example, the arrows help us to see that the variables tend to converge to the equilibrium (arrows point toward the filled circle), suggesting

A vector-field plot places arrows on a phase-plane diagram to indicate the direction of change of the system.

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