Modules: Discrete Chaos

## Chapter 1: The Stability of One-Dimensional Maps

### Section 1.10: Applications: Fish Population Modeling

Example: Beverton-Holt MAP

In this example we will investigate the stability of fixed points of the map

(1)

modeling population growth where m is the growth rate of the population and K is its carrying capacity.

Figure 1.10.1. Beverton-Holton MAP in Eq.(1) for m = 1.5, K = 4.0. Note that the positive fixed point is asymptotically stable. Two solutions starting near this fixed point move to the fixed point.

Figure 1.10.2. Bifurcation diagram of the Beverton-Holt MAP for m = 1.5 fixed while K is varied in the interval [0, 5.0].

Activities:

• Click on the first picture to load it into your local copy of Phaser.
• Move your mouse cursor (without clicking) and determine the coordinates of the two fixed points. (PhaserTip: Cursor Coordinates) Compare the value of the positive fixed point to the current values of the parameters.
• Set several more initial conditions by clicking the left mouse button near the positive fixed point. Clear and Go. (PhaserTip: Initial Conditions) This fixed point is globally asymptotically stable.
• Set the parameter to m = 0.8. (PhaserTip: Changing Parameters) Clear and Go. What happens to the fixed points?
• Click on the second picture to load it into your local copy of Phaser. What is the dynamical interpretation of this simple bifurcation diagram?
• Set the parameter to m = 1.2. (PhaserTip: Changing Parameters) Clear and Go. What happens to the bifurcation diagram? Can you explain why the bifurcation diagram did not change?

Example: Ricker MAP

In this example, we will examine the MAP

(2)

modeling population growth where r is the growth rate of the population and K is its carrying capacity.

Ricker MAP is perhaps the most significant model of a density-dependent population. Phaser Web site contains a Module on the Ricker MAP with extensive resources. Please visit Phaser Ecology Module: Ricker MAP for further Phaser simulations of Ricker MAP.

Figure 1.10.3. The Ricker MAP. For r = 1.9 (and K = 1), the non-zero fixed point is asymptotically stable.

Figure 1.10.4. Reproduction curves: A Gallery of reproduction curves as the growth rate r is increased while the carrying capacity K is held fixed.

Activities:

• Click on the first picture to load it into your local copy of Phaser.
• Set several more initial conditions by clicking the left mouse button near the fixed points. Clear and Go. (PhaserTip: Initial Conditions)
• Set the parameter to r = 2. (PhaserTip: Changing Parameters) Clear and Go. What happens to stability type of the fixed points?
• Currently the Start Time = 0 and Stop Time = 33; thus the initial conditions are iterated 33 times. Set Stop Time to 333 (PhaserTip: Time). Clear and Go. Does the solution get closer to the positive fixed point? Keep increasing the Stop Time so that visiually the orbit reaches the positive fixed point. Why is it taking so long to get there?
• Set the parameter to r = 2.1. (PhaserTip: Changing Parameters) Clear and Go. What happens to stability type of the fixed points? Do you see a periodic orbit of period 2?
• To make sure that you do have a period-2 orbit, discard the transients by setting, for example, Start plotting = 333 and Stop Plotting = 3333. (PhaserTip: Time).
• Click on the Gallery picture to load it into your local copy of Phaser. Click the Slideshow button on the Gallery and hit play to see the slideshow. (PhaserTip: Slideshow)
• Visit the link to the Phaser Web site Phaser Ecology Module: Ricker MAP to see a bifurcation diagram of the Ricker MAP.

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