I remember — two years ago — being in my Differential Equations class and learning about the Lotka-Volterra predator-prey model.
I found it (and many other mathematical models like it) to be fascinating!
The predator-prey model is a differential equation that describes (oftentimes) two sinusoidal wave functions that impact each other in this specific way:
y’ = ax – αxy
x’ = -cy + γxy
where a, c, α, and γ are all positive constants.
Now, this might look like some random gibberish, but the resulting y(t) from these differentials would be the population of predators, and the x(t) would be the population of prey.
Without going way too far into the details of assessing differential equations — that would take too much time — these differential equations lead to two sinusoidal functions that are based on what the different constants are set to. These constants determine the rate at which the prey reproduce and are eaten and the rate at which the predators compete and reproduce due to an abundance of preys to eat. A video of a sample use of the model can be viewed here and a view of the two sinusoidal functions over time can be viewed here. In the video, imagine that time is just moving forward, and the dots would be denoting both the positions of the x(t) and y(t) functions of different predator-prey systems with varying constants that effect the initial and changing population sizes of each species.
What is neat about this model is not only how it can be used to predict population change between two simple populations in a small food-chain, but also what it can say about our world and the impacts that a change in these systems can have…with a great example being what can happen if someone uses pesticides.
An important phenomena that this model shows(and what actually does often happen as a result of using pesticides in farming) is that you could throw an entire system out of balance and effectively cause more problems than you initially planned to solve.
Here’s how: imagine that the predator-prey system currently acts like the blue circle in the video above. So, there are moderate numbers of predators and preys, your everyday pests. These two populations are sinusoidal; they generally move up and down over time with a small amplitude about an average population. Now, imagine that, in an effort to reduce both populations, you use some sort of pesticide and then bam!, both populations move to a point near the bottom left of the graph in the video — perhaps near where the red line is closest to the x=0 and y=0 point. This would configure the constants in the above functions to result in two sinusoidal functions that together function like the red line in the video — and as the video shows, the prey population would explode!
With a greatly reduced number of predators around, the prey’s population can grow rapidly. Consequently, with a large number of prey around to eat, the predator’s population begins to grow rapidly, although lagging behind the prey’s population. Eventually the preys are overeaten, and with less food, the predators die off rapidly, bringing both species back to low numbers…only to set the conditions for the cycle to happen all over again.
While the pest-controlling was meant to reduce both populations, the results of the act of using pesticides actually forces both populations to grow and shrink in a less stable way, resulting in moments in time where both populations are many factors larger than before pesticides were even used!
Now, while this is an important model for ecology, I have noticed that many, if not, most systems are fairly sinusoidal. There seems to be rhythm to life.
And many noticeable examples of this are in human systems. We tend, in many ways, to be sinusoidal functions — a different kind of sinusoidal than a predator prey model — but a sinusoidal function in some way nonetheless.
I mean, with lots of variables, and with different kinds of behavior than a simple sine curve, but still a sinusoidal function nonetheless.
The important detail here is that we all tend to be rhythmic creatures, and that an important part of our rhythms as beings is achieving balance and maintaining this balance.
Sometimes, an important question that is asked by our Methodist friends here at Marsh Chapel is “Is it well with your soul?” and, honestly, that may be by far one of the most important questions that we should ask each other and ourselves often.
Is it well with your soul?
With the mystical-to-me-and-yet-honestly-I-get-what-you-mean-by-soul-here language and everything.
Is it well with your soul?
With the quiet and slow voice that, for a moment perhaps forces you to pause and wonder: wait, is it well with my soul? and then when you pause for a second, you realized you haven’t sat down for more than 5 minutes at a time in the last week and you’ve had a peanut butter and jelly sandwich for lunch the last 17 days and now you know why you’ve been dreading lunch and your ankles hurt. So maybe you should join your friends for tacos during lunch and your work will totally be waiting right there on that desk over there when you get back and those 30 minutes of a lunch break is definitely more valuable than that small amount of extra work you could complete each day by not stopping.
Is it well with your soul?
Or when I asked myself this question earlier this summer and noticed that the number of worship services I could attend weekly declined a significant amount and I missed the moments of peace and angelic music and man I really wish I could be reading that Rob Bell book sitting on my night stand but I always get home too exhausted to read and too tired to even just watch Netflix.
Is it well with your soul?
A question that often forces us to peer into our selves and see the constants that are driving the differential equations behind the many variables that govern our psyche and our health.
And then, hopefully, we can either prevent a change in the parameters that push us out of balance, or we can work to bring ourselves to more stability. Perhaps with meditation or contemplation. A good one is to put on some meditative music and to yawn a few times and then breath in slowly and when breathing out, lay a sacred word or phrase from your identities and traditions onto that breath you let out. And then, for like ten minutes, just do that…free from all distractions and the outside world.
A thought crosses your mind? It is okay, allow it to cross your mind and let it pass.
In all honesty, ten minutes of that can help with restoring a little balance to these functions by changing the constants to a more balanced place — or if you want to get all technical about differentials, it would cause some forcing on this sinusoidal system, or something like that.