Applying assignment to understand invasions

We'll use Structure to assess whether cultivated genotypes of Berberis thunbergii contribute to ongoing invasions in Connecticut and Massachusetts [1]. The first problem is to determine what $K$ to use, because $K$ doesn't necessarily have to equal the number of populations we sample from. Some populations may not be distinct from one another. There are a couple of ways to estimate $K$. The most straightforward is to run the analysis for a range of plausible values, repeat it 10-20 times for each value, calculate the mean ``log probability of the data'' for each value of $K$, and pick the value of $K$ that is the biggest, i.e., the least negative (Table 1). For the barberry data, $K=3$ is the obvious choice.

Table 1: Mean log probability of the data for $K=2,3,4,5$ in the Berberis thunbergii data (adapted from [1]).

K	Mean L(K)
2	-2553.2
3	-2331.9
4	-2402.9
5	-2476.3

Having determined that the data support $K=3$, the results of the analysis are displayed in Figure 1. Each vertical bar corresponds to an individual in the sample, and the proportion of each bar that is of a particular color tells us the posterior probability that the individual belongs to the cluster with that color.

Figure 1: Analysis of AFLP data from Berberis thunbergii [1].

Figure 1 may not look terribly informative, but actually it is. Look at the labels beneath the figure. You'll see that with te exception of individual 17 from Beaver Brook Park, all the of the individuals that are solid blue are members of the cultivated Berberis thunbergii var. atropurpurea. The solid red bar corresponds to Berberis thunbergii 'Atropurpurea', another modern cultivar. You'll notice that individuals 1, 2, 18, and 19 from Beaver Brook Park and individual 1 from Bluff Point State Park fall into the same genotypic cluster as this cultivar. Berberis $\times$ottawensis is a hybrid cultivar whose parents are Berberis thunbergii and Berberis vulgaris, so it makes sense that individuals of this cultivar would be half blue and half red. The solid green bars are feral individuals from long-established populations. Notice that the cultivars are distinct from all but a few of the individuals in the long-established feral populations, suggesting that contemporary cultivars are doing relatively little to maintain the invasion in areas where it is already established.