In classical ecology there is the idea of a species area curve. Basically this means that the more area (volume) the greater the amount of species that can coexist. So, we chose this variable as the first to investigate this semester.

It was found that with the lattice turned on that we got an ecological paradox. As volume increased it seemed that the chance at coexistence was reduced. The green basin corresponding to survival of both species got smaller. This suggests that coexistence is more probable at low volumes. Watch as volume increases and the basin pulls in to the spoon shape. Volume Movie

Eventually, as volume tends towards infinity, we expect the lattice model to look like the continuos model. However, our question was whether it expanded or contracted to the spoon shape seen in the continuous representation of the attractor. After watching the contraction we saw a couple of interesting phenomena. First, the initial large change from almost a green square to an expanded spoon shape occurs in two jumps, with the top half of the graph dropping of suddenly and then the bottom half disappearing at a higher volume. Watch the first jump.

It turns out that the jumping is the result of the lattice. The lattice is formed by rounding values to the nearest integer. Therefore, if a trajectory gets to a point where it does not move far enough away and it gets rounded back to the same point it gets hung up. What happens then, in the case of the basin jump, is that trajectories become clustered and then as volume is increased the cluster moves towards the axis, eventually crossing into the extinction region. In the LPA competition model many trajectories end up on the same point and when that is swallowed, a whole region disappears. This is shown in the following pictures. These pictures were generated with an offshoot of the lpaplot.m program. Instead of looking at the whole region we focused in on a smaller area and actually looked at the trajectories. The yellow lines are lattice lines and the red area the extinction zone. Red points are the last points of a trajectory and blue the ones leading up to it. Darker points are from trajectories that have already gone extinct.

In the first picture a series of points are held up short of extinction.

Now volume increases a little bit and they move closer

The points reach the cutoff in another volume step.

And now they are fully into the extinction region.

Upon further investigation it turned out for the jump in the upper left portion that the trajectories got attached to a particular lattice line. Then as volume increased the extinction region covered more lattice lines. Eventually it would get to the forty-filth line, and the jump would occur. Here is a table of values that show this:

Jump 1 |
Jump 2 |
||||
---|---|---|---|---|---|

E value |
Volume |
E * V |
E value |
Volume |
E * V |

3 |
15 |
45 |
3 |
36.6667 |
110 |

4 |
11.25 |
45 |
4 |
30 |
120 |

4.5 |
10 |
45 |
4.5 |
26.2222 |
118 |

5 |
9 |
45 |
5 |
23.6 |
118 |

5.5 |
8.181818 |
45 |
5.5 |
21.27273 |
117 |

6 |
7.5 |
45 |
6 |
19.5 |
117 |

6.5 |
6.92308 |
45 |
6.5 |
18 |
117 |

7 |
6.4286 |
45 |
7 |
16.58934 |
116.13 |

8 |
5.625 |
45 |
8 |
14.5 |
116 |

Notice that the for the first jump the e value times the volume always equals forty-five. At these volumes the trajectories travel until they hit the forty-fifth lattice line and then stop. The next lattice line never gets close enough for there to be a jump. However, the second jump occurs at higher volume and therefore, could lead to the possibility of interesting dynamics. Notice the result of multiplying the numbers for the second jump. These turn out to be variable. At these volumes, lattice lines are close enough together that the possibility of a second jumping phenomenon exist. Here we observed that a trajectories finishing point can actually change from one lattice line to the next as the volume increases. For example, at an extinction cutoff of seven, as trajectories approach death as volume increases the final point jumps from lattice line to lattice line. We believe this is what led to the noninteger number found. This is contrary to being fixed on one line. The following movie shows how the jumping back and forth between lattice lines affects the basinplot. Watch the large region in the bottom right flicker on and off. The final point is jumping between the 116 and 117 lattice lines, in and out of the extinction region.

Now onto rounding