 We will continue to look at structures causing a shaped growth today, basic structure of population affected by births as well as carrying capacity and we defined a ratio P by C, this is P and this is your C and we had defined a fractional birth rate B and we defined a effective fractional birth rate as the model. So, we have taken up this basic model where we are looking at how population growth gets a constraint at a later stage by the carrying capacity initially when it is not constrained good exponential growth when the and as a population approaches carrying capacity we can see an asymptotic growth. So, that is the dynamics we saw, so let us try to draw it right here, let us say use your let us draw a rate level chart, let us make it level or stock on the x axis and rates on the y axis. So, for this system as it was a population increases the rate rather here in this case it is only the birth rate continues to increase and after the inflection point the birth rate falls down and reaches an equilibrium somewhere later, so the expected graph here we can see will be like this, this point here is your let me just write it here unstable equilibrium and this point here becomes your stable equilibrium and this line here that we have drawn will be both the, so this is the births birth rate as well as the net rate. This case birth rate net rate is the same graph we just have only one particular curve right here then we had gone and drawn the deaths as a another variable within the system, we had defined deaths and initially we defined deaths as just fractional death rate D, we kept it as an external factor here. So, what we did for this particular scenario let us call it say scenario let us just make this part as a scenario A, so this is the curve I get for scenario A, the stable equilibrium for scenario A and when I include the scenario B, let us see what happens, so this is scenario B, since scenario B I have defined a death rate which increase in proportion to the level of stock, it is just a D multiplied by your total population, so it is going to be a linear curve. So, this is your death on death rate as in scenario B, as a result we are going to have a net, a net rate which could be which is nothing but the birth rate minus the death rate. So, death rate continues to get affected independent of this, so we will assume it is the same curve there will be some small changes in the population, but for practical purpose we will assume it to be a or rather analytical purpose we will continue, so assume this is the birth rate and this is the death rate. So, we will compute a new net rate for scenario B which could be something like this which will intersect this curve somewhere here, so this is your net rate for scenario B, this is your now new equilibrium point, this is a new equilibrium right here, as you can see by adding a constant or a proportionate outflow, your equilibrium moved from this point to this point here that means system is going to saturate at a much lower value of stop, so it may not reach the carrying capacity. Once again change the model, we again introduce something called as an effective fractional death rate and then we computed that, so this part written here is only for scenario C. So, scenario C this link is not there, so as we make death also change with respect to the carrying capacity as a population approach carrying capacity death rate is going to increase further or increase non-linearly, so this can be captured graphically as the level increases my death rate also changes, so let us assume that death rate kind of say change like this, let us assume this is death rate for scenario C, so as per this my effective my net rate for scenario C is further defined by this birth rate blue color and this dotted line as a death rate, so the equilibrium point further shifts downwards and this becomes a new equilibrium point right here. So, as more and more constraints keep happening, so we can learn couple of things here as more and as the constraints start acting on the actual state of the system and as it start affecting the flows, the point of equilibrium will be lower as more constraints keep coming, initially the constraints only on the births, so I stop here even if I had a constant exogenous outflow or proportionate exogenous outflow, I found that the equilibrium shifted downwards because more inflows and outflows are acting on the more outflows are acting on the stock in this case suppose both birth rate and death rate is getting affected by the carrying capacity then my new equilibrium point is much lower, so equilibrium point refers to the value of the stock, so if you assume this is the carrying capacity we are able to reach it that means here we cannot reach the carrying capacity, we can visualize it very easily that I am going to saturate at a point much lower than the carrying capacity of the system because of the non-linear dynamics on the birth rate as well as the death rate that you see right here. See if death rate and birth rate then they are independent of each other, then it makes sense to actually do the comparison among them but the equilibrium point will be the point of intersection between death rate and birth rate that is the net, so here this point, so this is the equilibrium point compared to this, this is the equilibrium point against which this is happening, it is the point at which net rate, this is the point at which birth rate equal death rate that is inflow is equal to outflow, so that becomes the equilibrium point right. Anything here that means birth rate is lower, death rate is higher that means level of stock has to go down, so that means it will go up here which is defined by this net rate value. For this curve when you do not have this part of the model that is only birth rate and population then there is nothing to stop it going all the way down to 0 which becomes a new equilibrium point. So, this is what we did last class, the summary of it. This pattern is exhibited in various scenarios as population trends of many animals and plants, learning curves, diffusion of news, riots, epidemics, rumors, for all exhibits is a shade pattern if you look at it the aggregate level. Initially just take the third point on diffusion of news and rumors. So, think about it initially when the news spreads it actually we use the term it spreads like wildfire that means it is actually going an exponential growth in the news. But after sometime as the entire population gets to know the information or gets to know the news or rumor or whatever it is then there is no new person hearing it. So, then it will saturate and hit the capacity. So, there the diffusion again becomes a shade. When growth of new products and various socioeconomic activities can also be attributed to this exponential growth. Initially everybody is full of enthusiasm and then we all exhibit exponential growth in various aspects. Then as time goes on new information we find it more and more difficult to absorb or new news too difficult to spread and so on. Now, even the new products as and when the market gets saturated. So, that is what we mean initially the growth is very nice lot of people are buying it, but then we use the term then the market becomes saturated. What we mean is it is achieving a kind of hitting the carrying capacity or number of people who would like to buy the product is already reached no new products are sold. So, that is the time when we want to introduce another new product. These are some example curves exhibiting as shaped here is a plot of growth of sunflowers their height and days. So, if you plot your own again what can you say your own growth part you will probably get a shaped curve all your growth are saturated you may grow wider but not taller. So, that is kind of saturated right here. Again cable TV growth again it is such nice shaped pattern here and saturate as everybody starts you own cable TV or satellite dish. So, only new customers are going to come or the people are going to own second TVs or going to buy new TVs and so on. Adoption of cardiac pacemakers of physicians again exhibits a nice shaped pattern. This is from various different papers that I found these curves this amount of yeast and the time it takes for it to for it to grow again a shaped growth. This is the growth of Tasmanian sheep which has been shown in various other literatures where again classical shaped pattern is shown after which it has reached a kind of a steady state and revolving or fluctuating around its main. So, this growth is also called as logistic growth or a sigmoid growth. So, these are the terms that you may come across in literature they all mean this shape pattern. So, time path includes two distinct behavior as we saw. So, general behavior is initially you have a exponential growth until inflection point and beyond the inflection point we have a saturation or asymptotic growth. So, you show the general behavior, the general pattern, your time, is your stock, this is a shape pattern that you are looking at. So, initially we have exponential growth and later we will have asymptotic growth or goal seeking. So, this point here is referred to as inflection point. So, inflection point is a point at which after which the negative feedback loop starts to dominate the positive feedback loop. So, this is exact behavior the general structure for that the general structure of a shape growth is we have state of system or stock then we have the net rate. This is your typical positive feedback system you can model a resource adequacy you can define a carrying capacity fraction of increase. This is your positive feedback loop and this becomes your negative feedback loop. So, this is the general structure which is going to cause a shape growth. So, we are going to pretty much be having a positive feedback as well as a negative feedback which is constrained by your carrying capacity or which affects a resource adequacy. Finally, note that it is slightly different from the birth model that we saw. In birth model we had a negative link here because we divided by carrying capacity and we have positive link here and we made a negative link here. It does not matter eventual loop is a negative feedback loop that is what we want as long as that is satisfied we are going to get a shape growth where the net rate and state of system getting affected positive feedback and negative feedback. At point of inflection the negative feedback will start to dominate after the point of inflection until then the positive feedback loop dominates causing exponential growth. So, when you read it or we write about it it is always initially the system state is driven to an exponential growth in the system when the resources are adequate as the resource adequacy approaches the carrying capacity the net rate slows down causing an asymptotic growth and finally, growth ceases. Having shown this generic structure actually this is the first structure that we have seen which can model the growth with a limiting factor whatever we have discussed till now is with based on this limiting factor or carrying capacity. We can have a second structure the rate from systems involving epidemics new product diffusion rumors where these two different loops may not be very apparent to you, but it still causes a shape growth.