1) Questions/ Comments on the paper by Chad Brassil: Stability Analysis: What do we mean by dynamical system? My Response: For us a dynamical system will be any model that includes time as the independent variable, or one of the independent variables. So in general these will be Differential Equations (Ordinary or Partial), Discrete models such as difference equations, Delay differential equations, and integro-differential equations and integro-difference equations. ----------------------------------------------------------------- What is a time series solution? My Response: This means a solution that is expressed as a sequence (maybe finite) - that is it gives a solution to some dynamical system at specific times, as a list of values of the solution at these times. ----------------------------------------------------------------- I was confused on the Discrete Time notation. Does the N sub t refer to the population size in respect to time? Also what is the reasoning for the eigenvalue to be the metric to describe the growth rates? My Response: Yes you are correct that N sub t in these models refers to the population size at time t, but the times t are viewed as discrete so we get n sub 1, N sub 2, ... as a sequence of population sizes - this is called a time series ----------------------------------------------------------------- On page 680, he mentions that "neutrally stable is generally considered to be an unrealistic outcome in ecological models, but a form of it does occur in classic predator-prey models." Why does something generally unrealistic still occur in that particular model? I'm sure we will get there eventually, but it intrigued me. My Response: We will see this in Chapter 7 as an example of a math model that has some aspects of reality (in this case the fact that there are cycles of population size observed in predator-prey situations) but it is not all that realistic because the math model with slight perturbations in species density leads to cycles which are just shifted. Some the classic Lotka-Volterra model has behavior that is not "robust" in the manner this is discussed at the end of Chapter 2 - a slight change in the model leads to a large shift in patterns of behavior. The lack of robustness means that this model is not that realistic though it has served as a base to build more realistic models. ----------------------------------------------------------------- On page 680: I'm sure we will eventually talk about it when we get to discrete-time models, but I don't fully understand why |g'(N*)|<1. My Response: Yes, we'll see this in Chapter 4. The idea is similar to the theorem we have on stability of an equilibrium on page 10 in the text, but because the model is discrete, the conditions for stability of an equilibrium change. ----------------------------------------------------------------- I am a still a little confused on what a nullcline is? What would it look like in terms of an actual population and why is it important mathematically? My Response: The nullcline is the graph of points (typically in the case of a model with more than one species so there are at least two coupled differential equations) on which the rate of change of one of the species is zero. To find it, you set the derivative = 0 and then solve for one of the species denisties in terms of the other(s) - this gives you a function in the phase space for the species. So if N1(t) is one of the species densities and N2 is the other species density, then you'd set dN1/dt = 0 and solve this for N1 as a function of N2 and then graph this in N1 vs. N2 space and that is the nullcline for N1. ----------------------------------------------------------------- How can you determine if an equilibrium is globally stable? My Response: In general this is not easy to do - there are some theorems that apply to particular systems (ones called competitive or cooperative - there's a whole series of papers on this by Morris Hirsch that gives results on global stability). Aside from these general results, you need to look at each system on its own to analyze global stability. ----------------------------------------------------------------- ----------------------------------------------------------------- 2) Questions/ Comments on the paper by Sebastian Schrieber: ODE Why do nonlinear equations model density-dependent populations? My Response: The idea of density dependence is that the per capita growth rate of a population is a function of the population density, rather than a constant as it is in the exponential growth model. This means that the differential equation giving the population growth rate is a non-linrear function of population density. ----------------------------------------------------------------- I do not understand heteroclinic cycles. Can you provide another example besides the rock-paper-scissor example that was given? My Response: We will see another example of this when we discuss multi-species models in Chapter 9. In general it arises when there are two solutions that "connect" two different equilibria with the connecting paths (one going from equilibrium #1 to #2 and another one going from #2 to #1) forming a cycle. ----------------------------------------------------------------- Brassil talks about how analytical analyses of stability become much tougher with more complex models, as does Schreiber in reference to nonlinear equations. Will we get a chance to go over procedures for numerical analyses in complex models? My Response: There are many different programs now that are specifically designed for analysis of ODE models that are more complicated than the ones we can investigate with simple analysis. Some are generalpurpose ones (e.g. Mathematica, Maple, Matlab) and some are specific to differential equations and difference equations (XPP, Berkeley Madonna). These are "solvers" - they do not specifically focus on the formal "numerical analysis" such as stability of the numerical scheme. You may wish to use some of these programs in a project to learn more about them, but to learn more about numerical analysis of ODEs there are several Math courses you might consider. ----------------------------------------------------------------- ----------------------------------------------------------------- 3) Questions/ Comments from Dercole and Rinaldi: Bifurcations While reading the Bifurcations article, I wondered what would be an example of a catastrophic transition? My Response: In general this means a transition that follows one of the types of "catastrophes" I mentiuoned in class. It is called a catastrophic transition if, as you slowly change one of the underlying model parameters, the equilibrium solution suddenly shifts, in a non-continuous manner. We illustrated that in class with the cusp catastrophe. ----------------------------------------------------------------- I was unable to find an adequate definition for manifolds. My Response: For us, just think of a manifold as a rubber sheet, or a function representing a surface, though in general this "surface" could be in multiple dimensions. In topology, a manifold "looks" locally like standard Euclidean space. ----------------------------------------------------------------- I found that the Hopf bifurcation is supercritical if the first Lyapunov coeffecient is negative; otherwise, it is unstable and the bifurcation is subcritical. Is that how I should be understanding super- and subcritical? My Response: We'll be going over Hopf bifurcations when we discuss predator-prey models in Chapter 8. In general these arise when, as a model parameter changes, an equilibrium becomes unstable and a periodic solution arises. If the periodic solution is stable as the parameter changes (e.g. solutions are attracted to it) then it is supercritical, otherwsie it is subcritical. ----------------------------------------------------------------- ----------------------------------------------------------------- 4) Questions/ Comments on the paper by Jim Cushing: Difference Equations On page 170, he explains that difference equations are used for a variety of reasons, one example being if there are "significant gradients in spatial habitat." What exactly does that mean/ a specific example? My response: This refers to the possibility that the underlying habitat can viewed as having discrete types across space, rather than a continuum of types. This is a "patch" view in which the movement of populations jumps from one patch to another and since these are discrete patches, the underlying model is not a differential equation but a difference equation. --------------------------------------------------------------------------- On page 170, how do difference equations have "a more transparent parameter identification and estimation" compared to continuous-time models? My response: This is in part because data from any field observations or experiments don't arise continuously but at discrete time (and possibly space) points. So it is natural to use time series methods to estimate parameters, for which there is considerable statistical theory to guide the parameter estimation. For continuous models estimated from field data, we never have anything other than discrete time samples, and the statistical theory is rather more complex (because we are convolving model fitting with time-point sampling). ---------------------------------------------------------------------------- On page 172, I was confused by the description of the Naimark-Sacker Theorem. Could you re-explain that? My response: This is a discrete-time analog of what we will see in other contexts as a Hopf bifurcation. It arises as a parameter in the model is varied, causing an equilibrium to lose stability and creating a cycle of points which repeat, and which are a dynamic equilibrium in the sense that if a solution starts on this cycle, it remains on it. ---------------------------------------------------------------------------- On page 173, he says "Mathematically, the equilibria that occur when r=1 constitute a continuum that bifurcates from the extinction state x=0." Could you explain the bifurcation piece of that? My response: In this simplest geometric growth or decay model, when r = 1 any initial condition is an equilibrium because if the solution starts there it stays there. So there is a vertical line of equilibria as illustrated in Fig. 1. When r > 1 however, there are no positive equilibria, since all solutions except thoise starting at 0 grow without bound. ---------------------------------------------------------------------------- On page 175, I was confused about the fundamental theorem of demography and how it was used? Was it used to help find bifurcations? My response: The theorem he refers to (which will be discussed in Math582) states that under appropriate assumptions on birth and survival rates in an age-structured population (e.g. one in which we consider the population with a set of discrete age classes), the population structure approaches a certain distribution with an overall growth rate. This doesn't relate to the bifurcation diagrams directly, but if you take a structured population modela and vary a parameter, you can use the theorem to state what the long-term growth rate of teh population is and view this as a model outcome that could "bifurcate" under appropriate conditions. ---------------------------------------------------------------------------- How is the theta-logistic used? My response: This is one of many extensions of the logistic growth model (in either continuous or discrete time) in which another parameter is added that modifies the growth characteristics. It adds another non-linearilty and this can lead to different dynamical behaviors as the parameter theta is varied. SO it is simply another model that some have applied to model population growth. ---------------------------------------------------------------------------- What characterizes a spatial class and a spatial habitat? My response: These are characteristics that depend upon the focal species (or group of species) under consideration. These can be determined by underlying habitat conditions, such as soil type, or by vegetation type (e.g. grassland, shrubland, forest), or by other conditions. ---------------------------------------------------------------------------- In the Jacobian matrix (for the linearization principle), what happens if an eigenvalue equals 1? My response: In this case, in the long-term the population neither grows nor declines. It arises in Markov chain models since the dominant eigenvalue for the transition matrix for a Markov chain is 1. This means there is a long-term stable structure for the states in the Markov chain. ---------------------------------------------------------------------------- When is the Leslie Model typically used? My response: This is the basic model for populations in which age-structure is considered and in which there are fixed birth rates that depend upon age and fixed survival rates that depend upon age. It is covered in detail in Math582. ---------------------------------------------------------------------------- How do we read Figure 4 on p. 175? My response: This is an example of a bifurcation diagram for which a parameter is varied (the parameter r in the Ricker model) and the value is shown on the hozontal axis. the vertical axis gives all equilibria points for any particular fixed r value. When there are several points on the vertical axis for a given r value, this typically means that there is a cycle (a dynamic equilibrium). When the vertical axis looks "filled in" this is chaos - solutions sweep through in time an entire continuum of values as time goes on. ----------------------------------------------------------------------------