Now let's consider the another different type of problems, the so-called Population Dynamics. And let's now think about population dynamics. In many applications like medicine, or the economics, or the ecology, it is important to predict the change of population of certain species. So now, let small p(t) be the population of a species at time t. Population must be given by an integer, non negative integer. So even though this population p(t) is always an integer. We may assume that p(t) is a continuous or even differential function, when p(t) is large enough. Then the body experiments, people found the following. And it is the simplest case, some ideal case, we may assume that the population p(t), grows population. It might be true in the case of the bacteria growth on the enough space, and enough food supply. What does that mean? Is that the rate of change, that is p prime of (t), is proportional to the present population p(t). The proportional constant, that's denoted by r, which is a positive constant and the meaning of it, this is a growth rate. Very simple, first order linear differential equation. So that the solution will be p(t) = p naught times e to the rt. What is the p naught? That is equal to the p(0), the initial population. We call this a Simple First Order Differential Equation. p prime t = rp(t). The exponential model of a population growth, will sometimes be called as Malthusian model of a population growth. This is too simple or in a sense too idealistic. In a more realistic situation, in a more realistic situation, it's reasonable to assume that the individual growth rate by which I mean P prime / P. It mostly depends on the population and decreases as P increases. If there are more populations then, this growth rate slows down. That's a more reason of a hypothesis, so as a very simple search model, we will take P prime / P to be linear, and the decreasing in p. For example. We may take a p prime of p = (r- ap), where r is growth rate, which is positive constant. a is another positive constant. Multiply p on both sides. You will get the equation (4). p prime =( r = ap)p. So this is now first order but non linear. Because in here, you have ap square. So that this is a non-linear. We call this differential equation for p prime = (r-ap)p, the logistic equation with birth rate r and a death rate ap. Can you recognize this differential equation is non linear, but it's a probability fact? So we know how to handle it. However, we've also try to investigate the behavior of its integral curves, what is the integral curves? This is the graph of the solutions, another name of the solution. We can investigate the behavior of the integral curves, that which we call as logistic curves. Without solving the differential equation explicitly, simply by expecting the function right inside. Which I denote by capital F(p). Can you remind the logistic differential equation? Which is simple p prime = (r- ap)p. What I mean is, let's denote this variance side as a capital F/p. By inspecting the behavior of this function, capital F(p). We can say many things about the solution curve, integral curve in other words. First note that, this function F(p) = 0, when p = 0 or p = r / a, distribute, identical is 0 and p is identical r / a. They are constant solutions of the equation, of the given equation, logistic equation. Because, when p is a constant, any constant its derivative is 0. So, F(p) is equal to identical 0 then, right-hand side is 0, left-hand side is 0. Also when p = r / a, right-hand side is 0 and left-hand side is 0. So, constants. Where is it? p = 0 and p = r/a. They are solutions. Constant solutions. We call these specific two constant solutions as the equilibrium solutions. Also we can say, any integral curve of this logistic differential equation, is increasing or decreasing, depending on the sine of its force derivative. If a p prime is a positive than, it must increase. If a p-prime is negative than, the integral curve must decrease. Then we know quite well from the calculus. Also we can say something about the concavity of this integral curve, by looking at the sine of second derivative of p. The sine of second derivative, tells us the concavity of the curve p(t). More preciously, if the sine of the second derivative of p double prime of t is positive then, is a concave up. What is the concave up? The graph looks like this one. That's half on run P double prime is a positive. On the other hand, if p double prime has a negative sign is negative <0 then, the graph of p must to be equal to concave up. It should look like this one. Then we know quit to where from the calculus two, and you can compute to p double prime from this easily. Take the one more differentiation respect to t. And to simplify, you will get this expression. p double prime of (t) = p prime (r-2ap). If this is a positive, then concave up, if that is negative, then concave down. Are you ready to analyze the behavior of the integral curves now?