[MUSIC] Welcome again to the third week of the course on simulation modeling of natural processes. In this module, I will talk to you about balance equations. This module aims at generalizing a bit what we saw in the previous module about growth of a population. And this generalization is called balance equation. So, what is the idea here? We want in a dynamical system, describe the variation of a given quantity and basically this variation will be changed by either a creation of a quantity or destruction of a quantity. So, in our previous example the variation of the population was given in terms of the births minus the deaths. So we can simply, in a formal way, write it with our notations for dynamical systems. So we have s dot, which is equal to f of s, and f of s is simply the creation rate minus the destruction rate. Now if we want to also write it for a discrete system, what we will write is that basically the evolution, the variation of s. So f of s plus delta t minus s of t, so this is the variation iver one delta t step is equal to delta t times the creation rate minus the destruction rate. So something that is nice to notice here is that delta t goes to 0, we simply have exactly the same equation as for the continuous case. Why is that? If we divide our first equation by delta t, we'll see on the left hand side a term which contains s of t plus delta t minus s of t, which is divided by delta t. In the limit delta t goes to zero this is exactly the definition of the timederivative of s. As you will see later, this discrete way of describing a dynamical system is what you will usually, or something close to what you will implement on your computer to solve your model. So to illustrate these balance equations, we can consider a system that is a bit more complex. We consider a system which contains two populations, one population of antelopes, which is a prey, and one population, which are the cheetahs, which are predators. So, basically, a cheetah will eat antelopes and antelopes will reproduce. And the only other process that we will consider is that cheetahs will die. So let me describe a bit the variation of each population. So on the first equation on the left of the slide, we have the variation of the population of antelopes, which is noted a. Which is equal to their reproductions or they reproduce fast, and the birth rate of the antelopes is noted ka. Then the second term here is the rate at which the antelopes will get eaten by cheetahs. So this process is proportional to the probability that they meet, so basically, this term is, of course, negative because it will decrease the population of the antelopes. And will be given by a proportionality constant or rate of being eaten. K of ca times c times a, where c is the population of the cheetahs. What we can notice here if there are no cheetahs, the antelope population will simply increase exponentially fast. Then, on the second equation, we described the evolution of the cheetah population, which is denoted by a c. And so the first term in this equation, which is numbered three on the slide, is an exponential decrease of the population of cheetahs. So here we assume that the cheetahs do not reproduce on their own if they are not fed. Which means that in absence of any antelope, the cheetahs will simply starve and die. And this sort of the death rate is minus a constant, kc, times the number of cheetahs in the population. And the fourth term, so the second term of this equation, is the growth of the population, which is proportional to the the probabilities that cheetah eat antelopes. So we have another proportionality constant, which is kac times a times c, which describes the growth of the cheetah population. So this system can be solved analytically, but it's not the point here. I just give you the time evolutions of the two populations. So here in blue you have the cheetah population, and in red you have the antelope population. So, we start at some initial condition, here it's three for the antelope population, and four for the cheetah population. And we see that first the cheetah population is increasing because they eat antelopes. And then you reach a point where you have almost no more antelopes, and the cheetahs start dying of starvation. So their population decreases. Then since there are no more cheetahs left or almost no cheetahs left, the antelopes get not any more eaten and their population starts to increase again, exponentially fast. Then at some point you reach some critical value where the cheetah can see the antelope population grow again and eat them all, try to eat them all, at least. And you have your cheetah population that increases while the antelope population decreases. And you continue these periodic time evolution. This function can also be described as a parametric function, where we do not have the time evolution anymore. But we can plot on the y axis the population of cheetah, and on the x axis the population of antelopes. So what we see here is that basically for any initial condition, we'll be on a loop on this parametric plot. So we see that when we have a low number of cheetahs, we'll usually have relatively large amount of antelopes and vice versa. With this I end my first example on balance equations and this generalization of dynamical system modeling, and in the next module I will extend a bit the notions presented here. Thank you for your attention. [MUSIC]