Autonomous Differential Equations and Population Growth–Differential Equations

During this week’s lecture, the professor talks about the idea of autonomous differential equations, which is essentially the differential equations without independent variables, namely x. The general look of such equation should be dy/dx=f(y). I was confused at first: how can equations regarding x be without it? Then I realized it is  a derivative rather than a equation, so its integral would naturally be with x, just like the integral dy/dx=2 would be y=2x+c. (Click here for more information about autonomous differential equation)

After some research, I found this kind of particular equation has great social and economic use. The reason for it to be without x is that the independent variable in the integral function is going in a steady rate; its velocity on x-axis never changes. There are only a few of things in this world is moving at a completely unchanged rate, and since velocity is measured in time, so time itself must be moving at an unchangeable rate. For that reason, autonomous differential equations can describe the change of anything’s speed, as long as it has one and the speed it is constantly changing. Everything has a speed, every moving object, planets are moving around stars, and stars are moving around the centers of galaxies, even the universe itself is expanding at certain rate; every creature, dead or alive, has a speed: the living one is aging, the dead one is decaying. Autonomous equations, however, are more commonly used to measure the expansion rates of things, from population growth to the savings in bank accounts to how fast is a rumor spreading, all of which have great social or economic values.

Autonomous differential equations are typically used to describe a exponentially or logistically changing object, such as population. Many think that population grows at a constant rate; they are right, in theory, because in a micro perspective, population growth is simply birth rate minus death rate. In real life, however, it is much more complicated. Any living creature lives on resources, human especially does. There are so many things human need to have a decent life: food, clothes, a shelter, etc. There are only so much food the earth can produce, and once reach that point, the nature starts to balance itself: people start to die of hunger, more realistically, the population would slow down when it is approaching the maximum. The maximum of population can be measured by using the direction field of autonomous differential equations; population growth is generally expressed in the form of dy/dx=by-ay2 , which is an eventually unsolvable equation. At that moment, things I learned from former classes start to come back—although unsolvable, by using the direction fields of the equations, we can know the pattern of how the integral (aka every possible original function) of the derivative behaves. (click here for the first blog about direction fields) For the simple natural growth of population, once the starting population is lower than the maximum, it would infinitely approaching the maximum without making contact (reaching the maximum), and if the population starts higher than the maximum, it would again approach the maximum, but by decreasing. It is such an interesting idea because it is related to history and sociology. In fact, many crucial events in early human history are related to such a growth. Humans started to be nomadic, hunters. People lived in small packs, hunted animals as food, lived in caves or forest. Once such living pattern was settled, they gave birth to babies, and population started to grow. At first, all was well, there were more people in the hunting group, hence more chance of getting food. As time passed, however, they found it was harder to sustain the population growth, because there were too many people to feed, and the efficiency of the hunting group was decreasing. Soon it became impossible to keep the population at an increasing rate. Normally in such situation, mathematically, people would just gave birth to fewer babies and finally reach a balance, and there would never be any civilization (civilization is based upon massive population), nor the world we have today. A miracle happened. Instead of reaching the balance and keep the population level, we breached the border, and become a race with 7 billion population nowadays. At some point, a woman, it was believed; found it would be more efficient to grow grains in order to feed people. Then people started farming, domestication of animals, a pattern was capable of sustaining much more population. Then there were cities, nations, empires; there came religions, wars and trades.

population_growthpicture from

Our civilization is entirely based on that moment of breaching of the mathematical border, the theoretical maximum. The instinct of achieving impossible is in our DNA, and there are so much we can do, as long as we want them bad enough.

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