“Predators and Prey” Parashat Ekev 5774
My Rav and my teacher, Rabbi Silberman, always used to quote the Mishnah in Pirkei Avot [5:22] “Turn [the Torah] and turn it again, for everything is in it.” There is nothing that is not alluded to in the Torah. I remember my Grade 7 teacher, Rabbi Tatel, telling us that the Torah even spoke about ecology, a hot topic at the time. Rabbi Tatel, who had a degree in Biology, showed us the verse in Parashat Ekev in which Am Yisrael are promised that if we keep Hashem’s mitzvot we will be rewarded with success in battle. We will utterly vanquish their enemies. However, we don’t want to vanquish them too quickly [Devarim 7:22]: “Hashem will drive out those nations from before you gradually. You will not be able to destroy them quickly, lest the beasts of the field outnumber you.”The inhabitants who are being vanquished by Am Yisrael have been keeping the population of wild animals in check. If these inhabitants were to suddenly disappear, the population of the wild animals might skyrocket, and Am Yisrael could find themselves with a plague of scorpions, snakes, and other unpleasant Middle Eastern pests. In order to prevent such an occurrence, our enemies will be defeated gradually. Am Yisrael will slowly become responsible for keeping the wildlife population under control, and a plague will be averted. And there you have it: Ecology par excellence.
What Rabbi Tatel didn’t tell us is that Rashi’s comment on the above verse is mind-boggling: “But is it not so that if [Am Yisrael] obey the will of Hashem, they need not fear the beasts? As it is said [Job 5:23] ‘And the beasts of the field made peace with you’. It was, however, revealed before [Moshe] that [Am Yisrael] would sin in the future.” In other words, the mere fact that the wild animals are not making peace with Am Yisrael means that we are being punished for our sins. Rashi’s comment is problematic in at least three ways:
- The first problem is that the verse that talks about driving out the animals is located after a lengthy description of what Am Yisrael will merit if we keep the Torah and mitzvot: We will defeat our enemies, our enemies will flee, and Hashem will make sure that the animals don’t cause us too much trouble. Everything here is positive. What would make Rashi interpret this verse in a negative way?
- The second problem is that Moshe couldn’t have known for certain that Am Yisrael would sin in the future because man has freedom of choice. While Moshe later tells them that [Devarim 31:27] “I know your rebellious spirit and your stubbornness. Even while I am alive with you today you are rebelling against Hashem, and surely after my death!” he is really only saying that based on their past history, it is highly probable that Am Yisrael will continue to sin. Here, however, it was “revealed” before Moshe that they would, one would assume, certainly, sin.
- The third problem is with Rashi’s source: Hashem promises Am Yisrael in Parashat Bechukotai [Vayikra 26:6] that if we keep Hashem’s mitzvot “I will remove wild beasts from your land”. Conversely, if we don’t keep His mitzvot, then He won’t remove the wild beasts, so if there are any wild beasts at all, it means that we are sinning. Why does Rashi prefer to quote a verse from Job? Things become even stickier when we note that the person who speaks the verse in Job is Job’s friend, Elifphaz, who is neither a prophet nor a wise man, just a good friend giving advice.
I would like to propose a (slightly mathematical) way ahead. Our way ahead is based on the population dynamics of biological systems in which two species interact and one of the species – the “predator” – feeds on the other – the “prey”. How can we calculate the numbers of both species as a function of time? Well, let’s assume that one year has a nice amount of rainfall, and so there is lots of food and water available to the prey. As a result, their birthrate increases and their numbers rise. The predators are only too pleased with the large numbers of fresh prey that they have to munch on, and their population begins to rise as they devour the prey. Eventually food and water begins to run out, and the prey find themselves fighting for limited resources, causing their numbers to dwindle even further. So if we plot on a graph the populations of the predators and the prey as a function of time, we would expect to see up-and-down oscillations in both populations. These dynamics have been modeled in what is called the Lotka-Volterra equations and their behavior is well-understood.
Now let’s see what happens when we take the predator out of the equation, leaving only the prey. The Lotka-Volterra equations are replaced by what is known as a “Logistic Map” that looks like this:
X(n+1) = X(n)*k*(1-X(n))
The Logistic Map contains two terms that pull in different directions. The first term means that the more animals you have today, the more you’ll have tomorrow. The second term means that the more animals you have, the less food there is to go around, and the population drops. The two terms work against each other, and like the Lotka-Volterra equations, one would expect the population of the prey to cyclically rise and fall.
Except that that it doesn’t always work this way. Notice the letter “k” in the equation. “k” is a number between 0 and 4, and its value has a surprising effect on the population. For k values less than 3, the population quickly stabilizes into one value. When k is set to about 3.01, the population bounces back and forth between two numbers. Raise k a little more and the population bounces back and forth between four values, and then eight and then sixteen and so on. This is called “period doubling”. For a k value of 4, the behavior is completely unpredictable, and is, for all intents and purposes, random. It is important to understand that the value of k is not determined a priori by anyone or anything. The value of k is determined after the fact (a posteriori) in order to model an already-existing population. But here’s the thing: While the population that lies before you looks completely orderly and predictable, all you have to do is turn the “k-dial” just a little bit to the right and all sense of order and predictability is lost.
For the third week in a row, it bears mentioning that the Book of Devarim is Moshe’s last will and testament, a message for Am Yisrael as they prepare to cross the Jordan River. Imagine it’s ten years down the line and Am Yisrael have just convincingly defeated the Canaanites or the Amalekites or the Hamas. Hashem has been good to us. He has shielded us from our enemies. We are even large enough to give Him credit for His share in our victory. The wild animals are a metaphor. Moshe is warning Am Yisrael, not that they will certainly sin, but to remember that it is always a possibility. We must never become drunk with our success, because the situation can change with the slightest turn of a dial. We will always be precariously close to complete chaos. We can never let our guard down. This explanation addresses the three problems we had above:  Moshe is not chastising Am Yisrael, and  he is not predicting that we will sin.  Like Job’s Eliphaz, he is just giving us good advice: never let your guard down, because the Evil Inclination is waiting in the wings. Rabbi Silberman was so right: The Torah really does contain everything. It makes it that much more critical that we contain the Torah.
Ari Sacher, Moreshet, 5774
X(n+1) is proportional to X(n)
X(n+1) is proportional to (1-X(n)). It is assumed that X lies between 0 and 1.
In a subsequent verse Moshe warns Am Yisrael lest they take credit for winning, saying “My strength and my hand have done all this for me”
Did Moshe know about Lotka-Volterra? This is no simple question. The answer divides Jews along the rationalist – mystic schism, and I am aware that some of you will not like my answer. That said, I tend to believe that Moshe did not know about the theory of chaotic population dynamics. However, it must have been well known that two nearly identical populations can have dynamics that are completely different. Moshe knew the “what”. Lotka and Volterra came along and explained “why”.