 So, today we will be looking at expanding on the simple structures we saw we have been seeing both positive and negative feedback systems. So, start with let us consider this scenario as shown in the diagram. So, we have population is affected only by the birth rate or net birth rate that as you can see is a positive feedback system. So, we are going to introduce a new variable called as a carrying capacity which is going to define how much the population can grow, nothing can grow forever there has to be some limiting constraint some limiting resource which finally slows down the growth. So, in this simple example let us assume it is called as carrying capacity. Now as and as you can see carrying capacity is external to the system now in the first order system that we saw the simple positive feedback system we have seen that in population affects net birth rate, net birth rate affects population and that is net birth rate is defined by something called as fractional birth rate. Now by introducing this carrying capacity what you are saying is will as the population grows towards the carrying capacity it is going to affect the effective fractional birth rate which in turn will affect the axial birth rate. So, we have positive feedback system here followed with a and a negative feedback system right here. So, as the population comes closer to the carrying capacity we would like the effective birth rate to fall down when population is far less than the carrying capacity then there is no real restriction on the birth rate it can be very high as it as population approaches carrying capacity we expect the birth rate to fall or when population is exceeds the carrying capacity we can even expect the birth rate to be negative. So, this is a very simple scenario before we start simulating these things let us try to see what will the expected behavioural system be we will try to plot the expected rate level graph level is another name for stock and how we can expect the fractional birth rate to change with population divided by capacity ratio. So, let us look at those three things right now convenience have reproduced this the same causal loop diagram here. So, here we have a positive feedback system and here is a negative feedback system. Let us take this scenario the questions from let us say bottom up kind of approach let us mark on the y axis effective birth rate and on x axis here let us define it as p divided by c. So, let us denote carrying capacity as c so, the population is nothing but the ratios nothing but p by c ok. So, on the x axis we have ratio p by c and y axis we want to plot what is going to be the effective birth rate. So, that we call here so, we can place it in context we have population we have births and we have. So, this is a simple positive feedback system that we had seen and we had modeled it like a stock and flow like this and here we put a growth factor g. So, this is what we had done when we studied positive feedback systems where we defined as a stock and then we told it is going to increase where rate is equal to g into p and it became a positive feedback system. So, this is a model that has been expanded on the left side where this fractional birth rate now g does not directly affect g affect the rate that is right here. So, I am here using this is birth rate we are talking about I put a small b it of g equivalent. So, we are saying that that is will change based on the carrying capacity. So, that is the additional loop that has come into play right here. So, now, how would we like this fractional birth rate to change? So, for convenience you can think you know as this p by c ratio changes what will happen to this value of g or this value of b what we expected to happen. So, for convenience let us just have some three values here let us just put 1 and let us put some large number see. So, now, if population is very low initially carrying capacity is very large then there is really no restriction on the birth rate right we can allow the birth rate to be as high as it can be and. So, let us assume that birth rate starts here somewhere and for some time we can even expect in the birth rate will be kind of it is not at all affected by the carrying capacity right. Even suppose we have an excess of 10,000 suppose is the carrying capacity and your current population is just 10, we can expect the birth rate to be constant, but again the current population is say 100 still 10,000 is quite far off or population is 500 or whatever number. So, up to some extent we can expect that the effective birth rate can continue to remain constant nothing really affects it to change its direction right. So, this can continue up to some point. Another way to think about it is how much you are going to consume for example whatever you can take let us say the food you eat suppose there is let us say you are hungry and there is say 10 pizzas available. So, as per normal rate whatever you can your hunger is satisfied when there are when you eat say one full pizza then hunger is satisfied. So, there are 10 pizzas still you eat only hunger is satisfied. But even if there is 5 pizzas still you eat one and hunger is still satisfied. So, when there is excess capacity your consumption continues to remain unaffected right up to a point. So, as the population comes closer to the carrying capacity or then I will start to say ration the amount of intake. So, we can expect that this slope you know can slowly reduce over time and as we approach 1 we continue to reduce and as P and C starts to go very very large at some point the growth get get halted. The population is way way high as I told there is ok. So, give an example like as you told as you started with there is a 100 pizza and 10 guys see each of you have one it is fine then there is 50 pizzas 10 guys no problem or 50 guys 100 pizzas still no problem then suddenly it is 100 guys 100 pizzas still may not be any problem imagine there are 10,000 guys 100 pizzas probably you can just get a small piece of it 1 million guys and there is 10 100 pizzas probably nobody will eat will all be busy fighting you may not get chance to eat. So, that means your consumption effectively has dropped to 0 as P by C close to large right. Your pizzas you are carrying capacity and population is a number of people who want to have a slice of the pizza or slice of the resource here we are denoting simply as the carrying capacity. This is the effective birth rate that we can expect. So, given this fractional rate value for G if G is constant what will be the behavior? So, when G is constant we can expect a exponential growth now here in this graph it shows that this G is falling down then what is the behavior we can expect as G goes down we can expect an asymptotic convergence to the goal or a goal seeking behavior. This now if we model the population P over time then we can expect a kind of a behavior like this where initially there is a exponential growth and here it became let us say a goal seeking or asymptotic growth and this point we call it as the inflection point. The point where the loop dominance changes at inflection point what happens at inflection point loop dominance changes from positive loop to negative goal seeking. So, after the inflection point this loop is active after the inflection point I mean this loop is dominant rather after inflection point this loop can be expected to be dominant given this it has to eventually saturate in this kind of system. So, let us expand into one more type of graph there is a rate level graph how will this graph be how can we expect this graph to be. So, initially remember I need to get this growth behavior. So, if you recall in a pure positive feedback system when there is a population of the stock value those exponential growth then rate level graph your rate level graph had a line like this and we saw that whatever the positive slope it will show this exponential growth behavior and if it is goal seeking then we told that the line has to be like this. So, if I am going to get a figure like this then I need to get a triangular or a hump shape graph like this if you get something like this this is your rate or the net rate there this net rate is what is to occur here. So, just to you confirm to look at this graph we have shown three different graphs with three different x and y axis. The first one we told was first effective birth rate or how we can expect this to change with respect to this ratio get just hypothesized on that and based on that it will ok here it is constant. So, we can expect exponential growth where we did value of population over time graph and as this continues to fall and reaches 0 then it has to the addition to the stock will incrementally reduce till successfully reduce until it reaches the maximum value. The third graph we plotted over net rate versus level as we told level large stock in our case it is the population p and this is the net birth rate that will be kind of hump shape. Now, let us try to come up with a very simple model which can simulate this behavior for us to do that I am going to make an assumption. See this curve here as you can see is non-linear right. Now, trying to fit an equation for it whereas, p by c changes in value I need to come up with a curve like this it is kind of little more difficult. Now, I need to come up with some if there are conditions or I need to draw it as a graph and then say if the value of p by c is say between 0 to 0.5 take this value if it is between this take this value etcetera that is one way or if I know the equation for this curve then I write how this birth rate can change as a function of p by c.