Part 1.

1.

According to the two commands summary(ants) and ?ants, the ants data consist of 97 marked points.

2.

Yes, the points are marked. The marks indicate whether the nest is from the Cataglyphis bicolor species or the Messor wasmanni species.

3.

Based on the resulting p-value of 0.5417, I would fail to reject the null hypothesis.

4.

Below is the code and resulting histogram of the residuals from mytest.

mytest <- quadrat.test(ants)
## Warning: Some expected counts are small; chi^2 approximation may be inaccurate
residuals(mytest)|>
  hist()

5.

For the Cataglyphis species, the p-value of 0.1575 suggests that if the underlying patter was random, it would be expected that this result would occur 15.75% of the time. As this value is below the set critical p-value of 0.05, I would fail to reject the null hypothesis.

For the Messor species, the p-value of 0.2219 would ultimately lead me to the same conclusion; I fail to reject the null hypothesis in this case.

6.

I would conclude from these results that the graph suggest a random spatial process.

Part 2

7.

The Gcross graph indicates that there is a cross-type repulsion because the observed curve is below the expected curve under the null hypothesis.

The Kcross graph indicates a negative spatial association between hickory and maple because the theoretical line falls below the theoretical line.

The pcfcross graph indicates that hickory and maple are repelling each other at distance r because gij (r) is less than 1.

Each of the graphs do indeed appear to agree; there is some aversion between hickory and maple trees in all cases.

8.

The Gcross graph suggests an independent spatial arrangement between Cataglyphis and Messor ants, as the observed curve follows within the envelope of the theoretical curve.

the Kcross graph suggests spatial independence between Cataglyphis and Messor ants because the observed curve is close to the theoretical curve.

The pcfcross graph also suggests that Cataglyphis and Messor ants are independent of each other at distance r, as gij (r) is close to equal to 1.

All three tests indicate that the two species are randomly dispersed in relation to each other.

9.

There does indeed to be a significant deviation from the null at distances below around 15 or 16 meters. There were also smaller dips between 25-30 meters, and at 34-40 meters.

10.

The deviation suggests a negative association.

11.

This means that trees of different sizes tend to be found closer to one another at distances less than 16 meters. Beyond that threshold, they appear to have a lack of correlation.

12.

The analysis of the anemones data set suggests that anemones of different sizes tend to be closer to each other at distances below about 12 units.

One hypothesis I can think of is that anemones of similar sizes might compete too much for resources with each other, so having a larger spread of size diversity in closer proximities may be more advantageous.