 Today, we are going to look at modeling of positive feedback systems in detail and the underlying behavior. So, the positive feedback system or a reinforcing type of system, the variable will continual feedback on to itself and will reinforce its own growth or collapse. This will be the arms race, you can see the threat perceived by nation B will drive the weapons of nation B which then affects the threat perceived by nation A, which then drives up the weapons of nation A. So, this entire loop is a positive feedback system which continuously reinforce itself and which in popular literature is also known as arms race. Or a more simpler example could be the population growth where if the net birth rate is positive then we expect the population to grow. So, here the population will not grow linearly because whenever there is more births it increases the population and after some time the that increased population will contribute to further increase to the higher birth rates. So, this is going to continuously feed itself and due to the result in a exponential growth of the variable population. So, the characteristic behavior I think we saw it last time also it is an exponential growth but to get a more realistic feel of what is it that we are looking at because it is not perceivable in the short run we need to give it some time. So, consider a paper thickness 0.1 mm, you fold it in half then you again fold it how thick will it be? It should be 0.4 mm. When you fold it in half it becomes 0.2, then you fold it again then it becomes 0.4. So, with every fold it doubles right there. How thick will it be if you fold it 42 times or 100 times imagine the paper is big enough. Any ordinary piece of paper probably not more than 7 times you can actually fold it but imagine if you can actually end up doing it just 42 times, there is just 2 times 3 times it is not really infinite number of times they are folding it is countably finite number of 32 times. To do the math that is more than 400,000 kilometers. Kilometers you just change the units from millimeters to kilometers 400,000 which is more than distance from here to the moon. So, that is how big your exponential can grow without if it is left unchecked. So, this exponential growth that we just saw characterizes most of the positive feedback system. The other one is characterized by accelerated decay or exponential decay. So, with time initially when it seems like you know systems is going fine, but after some time you can find that there is an exponential growth that has occurred. So, it has a big feedback or reinforcing systems what we are looking at is net flow rate. So, this is a positive feedback loop or a reinforcing loop as a plus or r can denote it and the characterizing characteristic behavior could be exponential growth whatever the variable of interest let us assume it is a stock. So, this is a exponential growth, other part is accelerated decay exponential xx is always time. How will the shape of this be accelerated decay or exponential collapse? We have how many options do we want to it is going to be this way ok. Of course, this is growth whatever options we have it is going to be this way. This is the accelerated collapse. Since very less quantity seems to get down this is does not increase it is actually flat. It is part of the diagram, but after some time it just after crossing point it just starts collapsing faster and faster before it hits 0. For example, I have illustrated was panic selling in stock markets or panic buying or panic buying panic selling nobody rise in panic why people sell in panic. Panic selling or withdrawing cash from the banks just a message that the bank is running dry of cash is enough for people to start queuing up initially they get less and less amount then suddenly your phase is right scenarios that we characterize as a exponential collapse in the system. So, there is a what characterizes these two is while system is here or even system is here it seems that things are all fine, but then the same amount of time you wait suddenly the system becomes an unmanageable size. So, up to here things are ok and then you just waited a little longer and system becomes unmanageable or here it was fine till here then you waited little longer and now system becomes unmanageable size which is here which you did not want. So, that is a characterizing behavior or exponential system. So, that thing seems to be ok, but the growth is being unchecked was in the system. Now, to model this as a stock flow diagram let us see how we are going to go about doing that. Let us see look at the SFD representation model there is a stock. So, it is a rectangle and there is a net flow rate, net flow or net inflow and we see that the stock again feeds back into a net flow rate. So, I have to represent that the positive sign. So, the top causal link is represented by this thick arrow with a valve the bottom causal link is represented by an explicit causal link in your stock flow diagram and to now capture the relation between the net inflow versus stock let us introduce a new variable called as fractional growth rate. Let us have the this is G that has let us define the stock as just a variable S for simplicity. So, the underlying equations here since stock would be given as change in stock or DT is nothing but net inflow that is DS by DT my net inflow let me just simply define as G multiplied by stock S. As I told G is a fractional growth rate units can be fraction per time the stock units are just simply units net inflow rate becomes this units per time. So, this is a simple equation that is underlying this diagram that we just draw here drawn here the stock which is DS by DT and fractional growth or net inflow rate is nothing but G into S. So, at every point in time I am adding G into S to the stock value. So, if I want to simulate what I want to do is stock at time t plus DT is minus say stock at time t is nothing but DT into net inflow at time t or stock at t plus time DT is stock at is a minus sign stock at time t plus DT into G into S which is again at time t. So, DT is your DT is the time step. If you are still using Euler's method to say time step of 1 then DT is simply 1. So, S at time t plus 1 is S at time t plus 1 into G into S of t and to simulate this model all we are going to do is keep solving this equation again and again. So, if you want to do it manually how will you go about doing it? First is say we have to initialize some value of stock. So, actually the behavior depends on initial value of stock right behavior depends on initial value of stock at t equal to 0 right. Why does it behave? Why does it depend on that? The simple reason is stock value is 0 what will be the behavior? Will be nothing it will be just the stock value will continue to be 0 forever because the initial value of stock is 0. So, if it is 0 then we can expect no growth if it is any value greater than 0 then we can expect exponential growth. Two possible behaviors are there if it is assume G is greater than 0. As long as G is strictly positive we are going to get a exponential growth. Of course, stock has to be non-zero for this to kickstart L system will not will exhibit no growth. So, let us see the diagram we just saw underline equations. So, we want to contextualize it with this very focused example as you may have put money in a bank and it is going to accumulate interest compounded interest every year for the next say 15 years. So, this simple diagram here represents how the interest accumulates in a bank and though you may feel that bank is not giving you adequate interest and the money does not seem to grow the speed which you may want it but actually the growth is exhibits a exponential growth. To simulate it so, let us we will let us do a quick hand simulation. We typically initialize the time at 0 and initialize stock at the initial value then see first the net flow is calculated based on the current value of stock then we add the increment that is delta t into the net inflow increment time and then we add this whatever increment we calculated to the stock. So, this time is not easy and time we keep looping. This may look unnecessarily confusing but what we want to do is shown here. So, let us assume for that interest rate example the initial value of stock is 100. So, first at time 0 the level is 100 then at time 0 so, up to this is what is given? Time 0 at initial value of stock or level is given at the 100. So, we calculate the rate which is 15 the value of G is taken as 15 the interest rate is taken as 15 percent. So, we take 0.15 into 100 which is 15 and let us assume a time step of 1 the dt is equal to 1. So, 15 into 1 is 15. So, the next time period the increment time period by 1 then level will be 100 and 15 100 plus 15 then we repeat the same step around 15 into 0.15 17.25. So, the next period we add 17.25 we add 17.25 to the stock. So, from this I calculate this and I calculate this then I add this to here and from this I calculate this value 17.25 of course, calculate the increment and add the increment oops sorry to that. So, I can see when you start with 100 with every time unit passing I am adding an increment value equal to interest rate multiplied by that current value of stock. We are assuming we are not taking the money out. So, let us see what happens when we plot it. So, we started with 115 the value is now 813.75. So, it is easier to visualize to see this graph. So, this now it starts to exhibit a exponential growth. So, this is the principal value that is the stock value and this is net interest income which is your net inflow rate which will also again exhibit this is nothing net income is nothing but a constant multiplier of the stock. So, stock exhibits a exponential growth this is just a fraction of that it will also exhibit exponential growth offset by the value of G which in case is 0.15. Let us see initially I started with 100 and then by the time it reached 200 it was 1, 2, 3, 4 around time period 5 it reached 200. Let us see let us see how long 200 takes to double from 5 start 1 again around time period it reaches 400 that is double and from. So, as you can see after 5 time period passed it doubled from 100 to 100 after another equal interval of same 5 time period passed the current value of stock is 200 double to 400 and to the same equivalent amount of say another 5 time units passed the stock value doubles to 800. So, with a constant. So, interesting thing about this exponential system is there is a constant doubling time that is after constant time unit passes my stock value doubles in value which is the characteristic of such a exponential system. So, in this particular example it happens to be 5, 5 time period approximately but we can actually compute this.