 Now, let us improve the model. Let us add one more thing. Suppose, there is a constant birth rate within the system, not constant, what I meant is there is a birth rate to a system, but it is not influenced by the carrying capacity. Birth rate is affected by the fractional death rate and death rate is not influenced by the carrying capacity at all. What will we expect to behave as a system, plot the expected rate level graph and how can we expect fractional rates to change with the population slash capacity ratio. So, let us keep this graph in mind, keep this diagram. So, again the left half is exactly the same, only thing now is population is also decreasing at a proportional rate as per the fractional death rate in addition to the net birth rate. So, model including the death rate is nothing, but the fractional death rate d into the population p. So, that is changing in proportion. What can we expect our behavioural system to be? It will continue to be a shape. In the first case, we reached the carrying capacity c, this point became equal to the carrying capacity, let us recall. So, looking at one graph like this, one graph like this and one for the rate, drawing that small rates, pollution divided by p, not over time. So, the population will continue to exhibit a shape growth, but because there is a thing, when you study system compensation, what happens? When there is a constant outflow, what happens? We stop below the, below or above the goal. When there is a constant exogenous factor, then we are not able to reach the goal. We either saturate lower or higher depending on direction whether it is exogenous in flow or outflow. Similarly, when we have a proportional outflow, outflow is proportional to the total stock, but the proportionality is constant. Hence, we will continue to exhibit a shape, but we will be stopping below the goal. We will not be able to achieve the same carrying capacity. So, here probably the carrying capacity is here much higher than where we are, we might be able to stop. We will see why that is in a minute. This is the level. So, as the level is large, what happens to the death rate? As level increases, death rate keeps increasing. It is a constant proportion. So, this is the death rate, but the birth rate was exhibiting a hum shaped behavior. So, let us model birth rate. Birth rate was exhibiting a exponential, I mean not hum shaped graph previously. So, we will continue to have the same graph, but as I draw the hum shape, it will intersect the death rate at some point. So, that means, at that point it has to be 0 because when system attains equilibrium, birth rate has to be equal to the death rate. Earlier, when it attained equilibrium when the birth rate, net birth rate became 0, but here it will attain equilibrium when birth rate is equal to the death rate. That happens at this point. So, we can actually plot that is birth rate minus death rate as a kind of net rate graph. So, at this point now becomes a stable equilibrium point. This is an unstable equilibrium point that is point 0 and a point of inflection is now not net, net graph it is called net rate, sorry. And this point now is the peak of the net rate where it achieves that represents the point at which there will be the point of inflection. That need not be same as the point at which birth rate peaks, net rate where it can peak at a different point that peak of birth rate. This is the expected behavior. We can continue to visualize it here. Fractional rate, this is p by c, this is large, this is 1. So, your death rate will be 0. So, this is constant d. So, nothing changes in a similar fashion if you are going to draw the curve. Probably we are having a looking at a birth rate which is constant and then we wanted the birth rate to kind of reduce and then come to this point. So, if you assume that as the d, then the net rate is minus this probably net rate is somewhere here and at this point it starts to reduce and then press at somewhere. So, this becomes your net rate, net, that is not called as rate, net fraction. So, let us incorporate this in a simulation model and see what how does that look like. So, let us simulate the deaths. Incorporate deaths in your SFD model. Fractional death rate d is 0.07, death rate is d into p. Once we do that we will simulate the SFD model. We will look at what patterns of behavior we observe, what is the stable population size, when does population reach stability and when is the inflection point, we will look at all that. And the model from we can also include a variable called net rate and then we look at then Wensim will also update this rate level graph. So, I am going to go back go to Wensim now, we will go to file, save this and we change it. Then it include the deaths, I read a flow death rate and I have another variable called as fractional death arrow connecting population to death rate, fractional death rate to death rate, click equation. Fractional death rate we wanted 0.07 and over month the fractional death rate is 0.07, death rate is d j per month nothing but population times fractional death rate. So, death rate is product of these two fractional death rate is 0.07 a constant in population p. So, once you click it you will find that death rate is also included here to leave it and initial value we had 2000 just change it to 2, change the population back to 2, let us click ok, let us click the play button, click population, click the process strip. Now, you get interesting graph, the population of stock continues to exhibit a S shape pattern to see in here. So, the S shape growth is always with reference to stock value ok, death rate is just a in proportion to the population. So, it also has to exhibit the same shape just a constant constant times death rate. But if you look at net birth rate in the previous time when you observed it increased to a peak and then fell down and hit 0. But here now net birth rate went up then as it is coming down it intersected with the death rate and then both became equal at around 46 and then both are now constant value. To see the actual net rate falling to 0 we need to actually model the net rate. So, to model the net rate we can just introduce a new variable called as net rate, connect your birth rate and death rate to this net rate, click equation, click net rate, click birth minus death, birth minus death. So, it is the death rate that is again simulate. Now, if you plot net rate you will find that the net rate will fall to 0 as you see here and the value of net rate. So, to visualize all the values select all 4, click the tables, go to the table of values, keep scrolling the net rate here peaks at about around time 50, where is net rate here? Around the same value of 49.99, it is around time 56 quite close to when the population value piece near 500. So, the death rate and net birth rate are still different, but as that this is the net rate. The net rate peaks at actually 20 at time 47, if you observe the net rate is the third column for me, it is a 21.1134 value, very small to read, but you can see it from your computer, it is around 47. Just observe that the net birth rate actually peaks much later at time 52 or 53, rather 56 is when birth rate peaks, but net rate at peak much earlier at around time 47 itself, so it can be different. And then death rate and birth rate both will converge to in this case around 45.5. The population peaks at around 650 or something, when it peaks is about around time 80, the inflection point is somewhere around 44, 45, it coincides with the peak 47 which coincides with peak of the net rate, not the birth rate or the death rate at the peak of the net rate is inflection point. You can try this, it includes the next rate, two rates also in the graph and see whether we are getting the graph very similar to what we have plotted straight line and the curve intersect and everything can be visualized. Let us try to find on this. Suppose other limiting factors such as starvation affects death rate non-linearly, birth rate be affected, let us assume it also affects death rate. What will be expected behavior of system, plot the expected rate level graph, how can we expect fractional rate to change with population capacity ratio? For that, suppose death rate is also changing non-linearly, what will happen? That is the question. Let us understand it here. So, death rate also changes non-linearly. Just let us look at this graph, Urim changes non-linearly, this shape is going to remain the same. Perhaps it may, the point of saturation may be lower. What does it mean by changing death rate non-linearly? That means death rate is going to increase as I come closer to, as the level becomes larger and larger, as level goes closer to carrying capacity, my death rate is going to increase. Suppose the death rate increases like this, death rate is changing non-linearly now. So now, birth rate we already the same graph. Now you can see the point of intersection is here and my new net rate graph comes like this. So, this becomes a new net rate graph. So, what we are observing is, as death also changes non-linearly, my equilibrium point shifts left towards or equilibrium point becomes lower or the level at which I expect to saturate becomes lower. So, if it did not have the death rate at all as a constraint, the first case the birth rate increase then it fell down and hit level that is the maximum I can reach. If I have death rate linearly increasing, then my birth rate graph is going to intersect at much early point. That means equilibrium point is shifts lower and if death is also increasing non-linearly, then the equilibrium point will be even lower. This will be the new equilibrium point. It will be somewhere here. So, you can expect the similar behavior and equilibrium point is lower. We can also model this. We will continue it in next class.