## How to take candy and favors from your family and coworkers

Ask your favorite family member who is bad at math if they will take the following bet. (As a word to the wise, I would play with pennies, or M&Ms, or my personal favorite: chocolate covered espresso beans. Just make it something small just in case they look up Jensen’s Inequality and get mad. Or in case you lose, because there is some chance you will. The odds are in your favor in the long term, though, so if you do, find yourself another sucker. )

Here is what you say:

“I will pay you in chocolate covered espresso beans to roll a die, as long as you pay me based on your winnings. Since I know you like candy, let’s make it the square of the number. The average number rolled should be 3.5, and that squared is 12.25. Since you are my _____ (boss, little sister, etc.) and I want to be generous, and besides, we don’t want to quarter candies, I’ll pay you 13 beans for each roll. You just have to give me the square of what you roll. So if you roll a 2, you pay me 4 candies out of the 13. Do we have a deal?”

Try to get them to roll at least 10 times if you can. The more the better, because the long term odds are in your favor. Below I’ve graphed the outcome of one person playing another. Each starts out with 50 chocolate covered espresso beans. Red is the House (you) and blue is your favorite coworker who is bad at math. See how many candies you could win if you let your friend have 100 rolls of the dice?

The trick here is that you’re paying the square of the average roll (13ish), but the average of the squared rolls is higher! It is just over 15. Basically, you get so many more beans when you roll high that it more than makes up for all the lower rolls where you are playing your friend. This game illustrates, as I allude to above, a form of Jensen’s Inequality. What is unequal? The square of the averages is not equal to the average of the squares.

What if we are measuring something other than rolling dice?  The insects I wrote about in the last post essentially play this game against the environment. Imagine little bugs picking one of the ephemeral bedrock pools Galen studies, and laying its eggs there. Many of them are small, and dry up and the eggs bake. Sorry, bugs. This round you pay me more than I pay you.

But…. When they do survive, and they hit a pool with few competitors and few predators and all of their hundreds or thousands or hundreds of thousands of offspring grow up and move out into their own pool and lay their own eggs, they win big. Kind of like rolling a six. This was well described in Peter Chesson and Robert Warner’s 1981 paper in the journal American Naturalist. This works for plants, too. Little seedlings wait in the baking Arizona earth as part of a seed bank, waiting for a winter thunderhead to explode above them, moisten the soil, and sound the starting gun for them to bust out of their seed covers and grow and reproduce like crazy. I’m not saying this happens for every species. Just that it can, and has been demonstrated for some, like in Anna Sears and Peter Chesson’s 2007 paper in Ecology.

The environmental variation is not the whole story. Bugs (and plants) play this kind of game against one another, too, essentially each one rolling their own die, paying the other. That gets more complicated.

Now go win yourself some chocolate covered espresso beans. With the resulting caffeine buzz, write a comment about how many people fell for it.

[Note: huge thanks to Simon for feedback on the game design!!]

## That’s it

What if your life and your children depended on living in water, but every so often the pool you were living in dried up? Maybe it was no more than a puddle, and, though part of a flowing stream after every good rain, between storms just disappeared – sometimes? Let’s say you’re an insect, and you have wings to fly off in search of other water, but your offspring won’t for a month or two. Why would you ever lay your eggs somewhere so risky?

In my last post, I wrote about how the landscape we live in, even a city block, is full of good patches and bad. What makes a spot “good” or “bad” to a dandelion or a dragonfly depends on their individual needs: a dragonfly needs water and plants to land on and other insects to eat, while a dandelion needs soil and light and of course some water, too. But the number of other conspecifics (other dragonflies or dandelions) they have to compete with can also make a patch better or worse.

Besides wanting to understand life, the universe, and everything, I want to be a biologist because I like being outside. So I was excited to help another student in my lab with field work last week. Galen studies stream ecology on Mount Lemmon, the tallest mountain overlooking Tucson in the Santa Catalina range to the north. Specifically, he studies the response to environment and competition in several kinds of aquatic insects. That sounds like a mouthful, so I’ll just summarize that he counts backswimmers and water striders up and down these mountain creeks.

I am planning to study these communities of aquatic insects, too, because they are crazy cool. First of all, they have short life spans compared to a PhD timescale (a few weeks to a few years), and are really small, and live in small little pools, so it’s easy-ish to study their population dynamics. Well, easier than studying lions and hyenas on the African savanna anyway, since I can just box these guys up in a kiddie pool or aquarium and watch to see what happens.

But if you get down on your hands and knees, these guys are no less spectacular than crocodiles and elephants. You ever have backswimmers invade your pool? Did they bite you? They haven’t gotten me yet (I scoop them up in a net and let Galen do the handling), but I hear the notonectids (science-y name for backswimmers) pack a punch that’s not as teeny tiny as they look. Then there are the “toebiters,” also known as giant water bugs, like Abedus herberti (that last year’s Biosphere 2 fellow Chris Goforth studies and blogs about here). Brilliantly colored sunburst beetles (Thermonectus marmoratus) bustle around the rocky beds of these glorified puddles. What is their business? What are they eating? Above them, water striders (Gerridae) skim the surface.

Beyond the alien-like life forms fighting their daily battles in the pools, liquefying and sucking the guts out of one another, these streams perfectly illustrate the variability I wrote about last time. While after a good rain, water cascades over the Gneiss bedrock like designer waterfalls, most of the time it seems these “streams” dry up into a series of puddles. Some are the size of a hot tub, while others are hardly larger than a pot you would boil spaghetti in. Some are shaded by trees, others bake in direct sunlight, surrounded by bare rock radiating heat back. (Those are absolutely unbearable to count bugs in on a hot summer day, even at the higher elevation.) Some are full of fallen trees, grass, or algae, and others are nearly empty.

But these pools also change from week to week, from year to year. Last year, Galen picked a bunch to monitor, counting the insect inhabitants regularly. He tried to pick some that he thought were permanent, and some that were ephemeral, or dried up entirely rather frequently. (Note for geeks: he did this by dividing all the pools into these categories of permanence and elevation, then randomly selecting a percentage of each of them using a computer random number generator I think. That’s called a block design. It’s very important to prevent researcher bias in experimental design by doing things like flipping coins or using random numbers. Human beings are inherently nonrandom, however random, say, your parents may seem sometimes.)

This year, however, the pools are behaving totally differently. Pools that were always dry last year now boast water after weeks without rain. Ecologists call this spatio-temporal variability, and it can help promote coexistence, just like the spatial variability I wrote about in the last post, although as you can imagine, it gets more complicated when you consider that fourth dimension (time). Spatio-temporal variability and its effects on community structure is what my adviser specializes in, and it’s the title of the class I am taking from him this fall.

So why risk the variability when you could opt for a more consistently safe environment? Maybe it’s a case of high risk bringing high reward. A small pool may dry up, but if it lasts, your offspring may have it all to themselves, low competition, low predation. For species with buffered population growth (a relatively long adult lifespan, or a seed bank, or something else that keeps them around), a good year can help more than a bad year hurts. That’s one condition for the storage effect my lab studies. Hopefully Galen can enlighten us in the next couple years about what’s really going on with these pools.