 But not all systems we need to explicitly give the goals, there are some systems where the goal is zero value, many systems have zero value goals, example is say you can assume radioactive decay that is also a exponential system, exponential decay, the general structure is that we do not really explicitly specify the goal because the goal is anyway 0, so you do not need to explicitly define it. So, the system becomes much more compact, so your state of system S net outflow rate is simply S times the fractional decay rate D, so your change in stock or net outflow rate is just stock times D, so you do not need to subtract with the 0. So, let us see what happens and typically in these systems we are interested in half life, for all we can just go with the example also radioactive decay we are interested in half life, nothing but the time interval required cut the discrepancy in half. Let us quickly check that out, zero value goal, what we want to do is we have stock S and they have rate and they have fractional adjustment or decay rate D, so let us decay fraction D. So, the equation for this is equal to minus rate which is minus D into S because I am just reducing it, you can solve it similar to the way we solved for exponential growth or the previous scenarios by moving S to the other side and integrating. So, upon solving we can get a nice expression for S at time t which is nothing but initial value t power minus t into D or S naught into e power minus t by same adjustment time. So, A t again stands for adjustment time which is just one over the fraction D, this is an A. This curve as, so we need to get a expression e power minus x, so that we can get the curve similar to this. So, for all these systems this is your S naught and this is your goal, right goal is 0. So, this is your S naught and this is your goal. Let us assume we have a adjustment time twice the adjustment time, three times adjustment time and so on. So, this is your time axis A t, 1 A t, 3 A t, etcetera. So, this is a behavior expected because e power minus x function I am just plotting that. So, one thing we are interested in this is we want to know when is the half life. So, we want to know at half point where I will reach this. So, let us denote this as the half life t subscript h, the time at which the stock value becomes 50 percent. We can just quite straight forward to compute it, half life. Half life is when S of t is 50 percent of your initial value. So, from the above equation we get this is equal to e power minus t h, t h by A t. So, once you solve it we will get it is minus adjustment time is 0.5. So, that means, your adjustment time is nothing but 0.6931 and 0.6 adjustment time whereas, your half life is equal to 0.6 times your adjustment time. This will be very similar to your exponential doubling time. In fact, it will be exactly the same. So, exponential doubling time and your half life here you get the same multiplier of 0.69 times your adjustment time. So, there is a required to cut your discrepancy half. So, whether you are going to double or half you just multiply 0.69 or 0.7 you can just imagine it like a 70 percent rule. So, after 70 percent of time has passed the stock value is cuts into half or stock value doubles in case of exponential growth systems. So, you will know we have looked at positive feedback systems. Then we looked at negative feedback system. Then we looked at negative feedback system with constant exogenous rate either in inflow or an outflow and then we had a zero value equal to special case but again negative feedback system. Now, it will be interesting to take it to the next step of natural question is what happens when both positive and negative feedback occurs in the same system. So, if you only look at one part of it that is if you just focus your attention on population and birth rate that we expected to be exponential growth system. On the other side if you focus your attention on only population and death rate we expected to be a negative feedback system or a zero value goal system where system is the population is eventually going to die off. If there is no additional birth rate to a system at some point system is going to die off. So, with death rate we are going to have an exponential decay with a zero value goal or with birth rate we are going to have a positive feedback system which is a result in exponential growth correct. So, now a birth rate is governed by fractional birth rate and death rate is governed by fractional death rate D. So, what are the possible patterns of behavior is there anything else when both these values both these take values how do we expect the system to behave. Individually we know when both are there together both B and D are there then how will the system behave and buying positive and negative feedbacks we are looking at population model deaths with a fraction D births a fraction B ok. So, the system we are looking at there are only three possible ways the system cannot oscillate the system cannot have produce an S shaped growth because there is nothing causing any other dynamics within the system. So, we just have two constants B and D. So, we are going to get only three modes of behavior for your population if in steady state when births exactly equal deaths then population will remain a constant when the births will be equal to deaths. See here deaths the equation for deaths is nothing but deaths is equal to P into D births is equal to P into B the equation of births is just P into B the equation of deaths is P into D. So, in the system if P is equal to P is equal to B I am sorry D is equal to D then system is going to not oscillate nothing system will always be in steady state as this. Now, if you have more births and deaths what will happen it will be an exponentially increasing system. So, this when B is greater than D B is less than D in less births than deaths then what will happen it will decay with a zero value goal right it would not be a mirror image of this instead you will get a behavior like this this is when D is less than D right. So, this is your goal seeking behavior zero value goal this is the shape we drew that is what you will get when deaths are more than births the population is going to keep declining until it hit zero. If births are greater than deaths it is going to have exponential growth births are equal to deaths then you are going to get a constant rate. So, it comes out because of this the net rate as opposed to the level if we plot it the net rate B equal to D case is constant sorry this is constant line B equal to D it is constant when B is greater than D say again F I am going to get a B is greater than D or B is less than D. So, if the slope is positive I am going to get exponential growth slope is negative then I have I have to get exponential kind of a decay and this it is constant I mean it is equal then I am going to get a linear growth this is only three possible modes of behavior within the system right. And again the system continues to be linear and at any point we just figure out which is dominating and appropriately we can compute the net rate. And if suppose I know the value of B and value of D I take the difference and depending on that I can figure out whether system is going to increase or decrease and I can compute what is going to be for example, as doubling time or the half time or half life. Level is a stock P this is a stock or level. When we saw this like let me just know this for positive feedback systems we had made a rate level chart I have shown you the slide. So, there the rate level chart was like this and we told that this slope here defined your growth rate G. For negative feedback systems we also had a rate level chart where we drew it something like this this direction it went on the negative side also if you remember. So, we just drawing the same graph except that we are already taking the difference and drawn it here the same graph that we drew from that same order point of order and that is why the system continues to be linear.