Dissertation Part One:

In between all of the drama-space of my endless self-inquiry and the challenging yo-yo dynamics of the relationship and the cancer diagnosis and all of that, I have been nose to the grindstone on the PhD dissertation. It is the only peace and quiet I get, a lot of the time. It's a balm to just retreat into data, lit review, formal dry and spare writing. Dissertation hermit. I have been frustrated, sometimes angry and unhappy lately, so it's great to have this extremely focused arena to ice out. There's a distinct difference between my current unhappiness and depression, by the way, but that's for another post. 

Mammillaria halei, the study species for a lot of my research

The first chapter of the dissertation, which I am currently working on while also working on the last two chapters, is an attempt to understand the population biology of a rare endemic cactus of Baja California, Mammillaria halei. The plant is an island isolated taxon, and its range is only about 20 km by 50 km (12 miles by 30 miles). It is relatively abundant over this restricted range, although even within its island habitat, it displays distinctly patchy, fragmented distribution. 

A common site location for Mammillaria halei, well established on serpentine rock

The "Holy Grail" of conservation biology of single species is to get relatively strong inference for estimates of risk of extinction and time to extinction. This has proven to be far more challenging than one might imagine. Appearances can be quite deceiving in a lot of ways. Consider a vast hillside that appears covered with thriving aspen trees, Populus tremuloides. Large stands of this species that look like a dense population of many individuals are usually actually only a very few individuals that have sent out new above ground trees via root suckers. Imagine the sudden onset of some fatal pathogen that relies on a certain genetic vulnerability in the aspen host. In a few short years, what seemed like a robust population could be completely wiped out, since they are all the same clone. On the other extreme, imagine a difficult to find, tiny cactus plant, that seems sparsely distributed over a very small geographic range and limited to a particular kind of soil. It would be easy to conclude that this plant is at risk of extinction. However, it may well be very tenaciously persistent, genetically robust and much more common than has been detected. 

These scenarios represent issues with determining rarity, per se, not necessarily risk of extinction or time to extinction. But a determination of rarity is often the first step in recognizing a plant species as at risk. 

Rabinowitz outlined seven types of rarity (1981), and Mammillaria halei fits the following category:

"Locally abundant in a specific habitat but restricted geographically; endemic." 

The key problem regarding estimates of the extinction risk and time to extinction of Mammillaria halei is, at least in part, this local abundance. The experience of visiting its habitat is surreal, since there are none in any known large or established populations on the peninsula, not even adjacent to the islands, except for a small patch of about four individuals near San Carlos. But as soon as one gets to, for example, Isla Magdalena, the plant is everywhere. It is one of the dominant features of the flora. This is a common feature of geographically restricted plant species, especially those that appear to have evolved as a result of geographical separation into a highly isolated geographic region from an ancestral population. There is perhaps one location on Earth to see them, but in that single locale, they are relatively abundant, especially in what appears to be preferred sites. 

The questions are: 

Is the high population density in parts of the habitat sustainable?

Is the overall population expanding, contracting, or staying relatively stable?

What stage is the population in? Is it mostly young plants, mostly senescent plants, or a balance of demographic stages? 

Are different patches and fragments of the population more at risk than others, and, if so, what impact might that have on the whole population?

Is the geographical locale itself subject to natural or androgenic disturbance that might put the species at risk? How stable is the only known habitat?

Is there competition in the only known habitat from other plant species? Is there a risk of predation via herbivory from animals, from historical or introduced species?

Is the habitat reliably populated by the plant's pollinators? Or has the range of the pollinators changed so that the isolated habitat has fewer pollinators? 

There are other questions that are important to answer, especially regarding the fitness of various demographic stages and populations (fitness is the biological measure of the likelihood of an individual passing down its genes), the genetic structure of the overall population and subpopulations (in particular genetic diversity and allele frequency vs. possibly deleterious inbreeding or outbreeding depression), etc. I am not going to be doing population level genetics for the dissertation, for funding reasons, but will be analyzing the DNA I collected in the future, hopefully. The great thing about DNA is you can keep it in a -20C freezer forever. 

I did get subjective estimates on an ordinal scale of the condition of each individual I recorded for another chapter on species distribution modeling, but that's not the most reliable quantitative data. Condition seems to correlate with various types of habitat where Mammillaria halei occurs, and it may be useful data. In particular, simply comparing apparently "barely surviving" individuals against the general habitat types of rain shadow with low vegetative cover versus ridgetop and exposed to fog with high vegetative cover, the "worst looking" individuals definitely correlate strongly to the dry, exposed, rain shadow habitat. 

I think any vulnerability of the island habitat to severe enough change that would wipe out the species is relatively low, even with climate change. The vegetation does seem to respond to regular fogs that come in from the Pacific that only go so far up and over the ridgetops, or travel more through deep erosion channels that cut through the islands. The interspersal of heavier vegetative cover with low cover corresponds quite exactly with landscape features that would amplify the effects of fog. It is possible that climate change would reduce the frequency or moisture content of these fog washes. It is also possible that climate change would increase the frequency and severity of higher category hurricanes, but there's little evidence that hurricanes are harmful for the cactus in any way. 

Since 2010, there have been at least 12 tropical storms or hurricanes to hit Isla Magdalena, including four category 2 or higher hurricanes and one of the most powerful hurricanes ever recorded for the southern peninsula, Hurricane Odile. My survey period from 2014 to 2018 shows no measurable impact of all of this storm activity, either in contributing to mortality or to increased recruitment of new plants due to increased moisture. The stems of Mammillaria halei are usually quite low to the ground and are capable of bending all the way back toward the proximal end and then returning to their original position, without any apparent damage. These features probably prevent increased mortality from hurricanes. Also, the preferred habitat for many plants is pure rock or very heavy rock gravel, a substrate relatively more stables that loose soil. However, possible benefits of heavy rainfall seem negatively affected by the short time when the water is available to plants and the high runoff from the exact rocky habitat that is preferred. I anticipate that the amount of winter precipitation providing available water for longer periods of time will prove more important to the species than dramatic summer rain events. 

Sea level rise doesn't seem to present a direct threat to habitat, as even with a global average sea level rise twice the most likely value of one meter, the islands are projected to remain largely unaffected. (which you can check out using this otherwise depressing and alarming web tool). This is a result of the islands being true oceanic islands that resulted from tectonic plate action. They rise very quickly out of the ocean and have generally steep sides. There is a central, low lying trough on Isla Margarita that looks like it could fill in, but this entire region currently hosts practically no Mammillaria halei, so that is unlikely to affect distribution. 

The intuitive sense is that Mammillaria halei is not going anywhere any time soon and is not at risk. It is listed under Mexican federal government regulations as a species of special protection, and in the International Union for the Conservation of Nature assessment as threatened. But, as is probably the case with many species that seem at risk, these perspectives are not based on very much data. 

I am building what are called stage class demographic matrix models to explore the risk of extinction as well as the population dynamics. These models have been in use among conservation biologists for about 40 years as the primary tool for what's called population viability analysis (PVA). The models are built using a "transition matrix," a square array of the probabilities from one time step to another (usually one year) that a particular stage class will either stay in that class or transition out of it (via either mortality or graduation to the next highest class-- from a stage class point of view, it doesn't matter what the reason is). This matrix is repeatedly multiplied by another column matrix that includes the starting counts in the various stage classes at time zero, and each multiplication represents one time step, again, usually a year. Because the mathematical tools are matrices, they are subject to eigenanalysis, a set of tools for uncovering a variety of discrete population parameters such as the discrete population growth rate, the individual stage class growth rates, the contribution of each stage class to the persistence of the population, and the "stable stage distribution," a theoretical population structure for a persisting, non at risk population where the stage counts would be the same forever. 

t--------à t+1	Seedlings	Stage I	Stage II	Stage III	Stage IV	Stage V
Seedlings	0	0	0	.07	,08	.09
Stage I	.02	97.35	0	0	0	0
Stage II	0	.02	96.2	.001	0	0
Stage III	0	0	.03	95.4	.001	0
Stage IV	0	0	0	.015	92	.001
Stage V	0	0	0	0	.013	97.3

The above is a matrix constructed from best estimates of the probabilities of transitions based on four years of time series survey data. Multiplied out over 100 years:

Bad news for Mammillaria halei, with a discrete growth rate below one, which inevitably leads to extinction. The key stage class impact on this scenario is low survivorship of seedlings to reproductive age. 

However, when I run a model only on the ridgetop populations that are more regularly exposed to fog:

The scenario is more promising. It looks like the increased robustness of the species over time is a direct result of young plants surviving long enough to produce flowers and set seed. Of course, this makes neat biological sense. 

The problems with these models are legion, however. Four years of time series data on a taxon that probably diverged a million years ago and has been persisting on the islands ever since is kind of hilarious. Slow growing plants have a way of going through extreme cycles but persisting perfectly well over very long time cycles. The seed bank in the soil causes resiliency for a lot of plant populations that is very difficult to model. The models I have run so far are deterministic, assuming that transition probabilities remain the same over 100 years. I'll be running some stochastic models that introduce random changes to various transition rates, probably modeled after what could happen to survival with a rare three year precipitation event (these cycles happen every 50 years or so in the Sonoran) or with a prolonged reduction of precipitation. The stochastic models supposedly provide stronger inference on the long term prospects for the species, but we'll see what they look like. They can easily be run out to 1000 or 10000 or even one million years, also, and sometimes it's interesting to derive transitions in 10 year time steps and then do much longer time periods in the models, especially with slow growing perennials. 

A powerful way to improve small data sets is by using Bayesian statistical inference, so I am working with a fellow grad student who is a statistician on developing Bayesian transition estimates over high confidence intervals using Bayesian likelihoods. This may well significantly improve the inferential power of these demographic models. One of the enduring problems of conservation biology is that PVA is often based on only a few years of data and a limited sample size. The temporal autocorrelation of the data sets in itself is a serious problem-- since the time series data is usually collected over consecutive years, and climate cycles, for example, usually span longer time periods. Perhaps Bayesian estimates of the stage class transitions will offer a clearer picture and be a useful solution for other scientists who are trying to understand the population dynamics of potentially endangered plants using limited data. 

As is often the case, a seemingly innocent and straightforward question such as "Is this species endangered?" ends up being a complex challenge to answer. So far it looks like a few things are important to consider, namely, better estimates of the demographic parameters, incorporation of likely stochastic events, and spatially explicit analysis of the viability of subpopulations in widely different habitats. 

And that, dear readers, is how I am spending most of my time, when I am not agonizing over love or wondering what to do about cancer. 

If you made it this far, I applaud you. Here is a nice picture of a pretty flower of Mammillaria halei, as a reward.