Questions raised from class participants Chapter 1 of the text: I was reading through my 581 notes from yesterday, and had a question. When you were going through Lyapunov Asymptotic Stability, you posed a question to the class that I wrote down quickly but didn't fully understand. I believe your question was something to the effect of "Can a function be Lyapunov Asymptotically Stable but not Lyapunov Stable"? Maybe I even wrote the question down wrong. Either way, I missed what the argument was for answering that question and was wondering if you could outline that again for me so I can fill in my notes. My response: When we define asymptotic Lyapunov stability we include the condition that the equilibrium is Lyapunov stable. What I was trying to ask is if an equilibrium could meet the limit conditions for asymptotic stability but not meet the conditions for Lyapunov stability. I then made an argument that there could be irregular "jumps" away from an equilibrium as time gets large that imply the conditions for Lyapunov stability would not hold except for very large times even though in the limit the solution approaches the equilibrium. ----------------------------------------------------------------- Chapter 3 of the text: Can we use them both deterministic model and stochastic model at the same time. My Response: Yes, in general we use use both and are interested in questions such as under what circumstances the mean of the stochastic model is the same as the solution of the deterministic model. We saw for example that this was true for the pure birth process. ----------------------------------------------------------------- will we investigate any nonlinear stochastic models this semester? My Response: In general, non-linear stochastic models are much more complicated than linear ones - so we will only see a few examples in this course. The end of chapter 3 has one example. ----------------------------------------------------------------- Questions from Chapter 5. Regarding the stability of the equilibrium at carrying capacity for the differential delay equation dN/dt=rN(t)[1-N(t-T)/K]. If mu has negative real parts => lambda has negative real parts => the carrying capacity is stable. The equation mu=-rTe^-(mu) has real solutions up to rT=1/e. From that point, the next piece we did in class was determining when the complex solutions had real parts that shifted > 0. How do we know they had negative real parts for 1/e<= rT <= pi/2? My response: One way to see this is to write down the equations for solutions that are complex. So you assume a solution mu = a + bi and plug this into the equation to get two equations - one for the real part and one for the imaginary part. Then think of what happens as rT varies - the real part is a continuous function of rT. Since the real part is negative for rT=1/e (the graph in the book Figure 5.17 shows that the root is negative just as the double root arises when rT = 1/e ), complex solutions have real part that varies continuously as rT varies so it can't suddenly "jump" - the real part must vary continuously up until the real part disappears (we get a purely imaginary solution) and that gives the equations 5.49 and 5.50 (a), (b) in the text from which we saw this shift occurs at rT=pi/2 ------------------------------------------------------------ A second question relates to the above. The carrying capacity was stable up to rT=pi/2. How do we know when the carrying capacity is stable vs. when it is stable with oscillations? Is it because the oscillations occurred from the complex solutions with negative real parts? Is this always true? My response: Yes indeed this is correct - in general, when we linearize around an equilibrium, the linear system has solutions which are exponential. The exponential solutions with exponents that have negative real part either die out monotonically to zero (if they are real and not complex) or die out with oscillations (if they are complex). The shift from declining monotonically to declining with oscillations around the equilibrium occurs when the solutions to the characteristic equation obtained by plugging in e^(lambda t) for solutions shift from having lambda having negative real solutions to having complex solutions with negative real part. ----------------------------------------------------------- For problem 5.2, to categorize when certain behaviors happen, are the following categorizations correct? grow monotonically: happens when all lambdas are real and at least one has |lambda|>1. decay monotonically: happens when all lambdas are real and 0< lambda <1. decay in oscillatory manner: happens when all lambdas are real and -1< lamdba <0 OR when some lambdas are complex but still all |lambda|<1. grow in an oscillatory manner: happens when all lambdas are real and all lambdas are such that |lamda|>1 OR some lambdas are complex but there exists |lambda|>1. My response: We need to be careful here because I think you are mixing up the cases of a discrete model and a continuous model. Problem 5.2 deals with a continuous model, in which case we linearize near an equilibrium and look for exponential solutions e^(lambda t). These solutions die out if the real part of lambda is negative. In the earlier part of Chapter 5, we were looking at discrete-time models in which we linearize around an equilibrium and look for solutions lambda^t . In the discrete case therefore, solutions die out if |lambda| <1. What is stated above is correct for Problem 5.1 which is a discrete-time model. For the Nicholson blowfly case though of Problem 5.2, it is a continuous model so grow monotonically: happens when all lambdas are real and at least one is positive decay monotonically: happens when all lambdas are real and negative decay in oscillatory manner: happens when all lambdas are complex and have negative real part. grow in an oscillatory manner: happens when all lambdas are complex and have positive real part -----------------------------------------------------