A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Autonomous Equations and Stability Sections 2.4-2.5 Dr. John Ehrke Department of Mathematics Fall 2012
Autonomous Differential Equations In this lecture we will consider a special type of differential equation called an autonomous differential equation. Autonomous differential equations are characterized by their lack of dependence on the independent variable. The general form of a first order autonomous equation is given by = f (y). (1) dt Keep in mind that y is still a function of an independent variable t, but it does not appear explicitly in the forcing term, f. These equations are sometimes called time independent. These equations are separable and so normally are easily solved depending on the nature of f (y). Rather than dwell on solution techniques we will be concerned with how to get useful information out of the ODE without actually solving it. In this section we will investigate: Qualitative techniques for describing solutions of autonomous ODEs. Models with harvesting, parameter studies. Bifurcation analysis. Slide 2/14 Dr. John Ehrke Lecture 5 Fall 2012
Characteristics of Autonomous Slope Fields The nature of autonomous equations makes spotting constant solutions and interpreting the general behavior of solutions fairly straightforward. A typical slope field for an autonomous differential equation is given below. What can we observe? Observations: 1. Every horizontal line, y = y 0 in the direction field is an isocline with slope f (y 0). 2. The integral curves (solutions) are invariant under translation. (i.e. A horizontal shift of a solution is another solution.) 3. The critical points of the ODE are the solutions to the equation f (y) = 0 where f (y) is the right hand side of the autonomous ODE. Saying that y 0 is a critical point is akin to saying f (y 0) = 0, but notice this means y = y 0 is a solution and hence a horizontal solution (barrier) through which other solutions cannot pass. Slide 3/14 Dr. John Ehrke Lecture 5 Fall 2012
Population Dynamic Equations In the next few examples, we will be layering a few basic principles of population growth onto the uninhibited growth model, dt = ry (2) to obtain a more realistic model. In the case of equation, (2), r is called the growth rate if r > 0 and the decay rate if r < 0, and has solution y(t) = y 0e rt where y 0 is the initial size of the population. While we understand the namics of this model well from pre-calculus and calculus, it paints a very limited picture of population behavior since it predicts uninhibited exponential growth. We will make this a little more interesting in the coming examples. Example Let s assume a constant amount of fish, ω, are pulled from a fresh fish farm with uninhibited growth rate r, and are modeled by the equation dt = ry ω (3) Find the critical points and analyze the behavior of solutions as a function of ω. Slide 4/14 Dr. John Ehrke Lecture 5 Fall 2012
Phase Lines and Equilibria Observations: The figures to the left show the phase portrait (above) and slope field (below) associated with the model /dt = ry ω. The value y 0 = ω/r corresponds to the zero of the right hand side of the equation. This point is a constant solution to the differential equation, often called an equilibrium solution. Looking at the phase portrait above the green arrow together with critical point ω/r denote the increasing/decreasing nature of the solutions, y(t). The arrow pointing left of ω/r indicates solutions which begin below ω/r decrease away from ω/r while solutions that begin above ω/r grow without bound. The y-axis of the phase portrait is often called the phase line and is usually oriented vertically with the key points labeled. Slide 5/14 Dr. John Ehrke Lecture 5 Fall 2012
Phase Analysis Example Example Given the first order autonomous equation /dt = y 3 y perform the following quantitative analysis: 1 Find the critical points (if any exist) and sketch the phase line. 2 Label each critical point according to its stability type. 3 Sketch the direction field and solutions corresponding to each of the following initial points: y(0) = 0, y(0) = 1/2, y(0) = 1/2, y(0) = 2, and y(0) = 2. Solution: The critical points are the solutions of the equation y 3 y = 0 which are y = 0, ±1. The critical point at y = 0 is a stable sink, while the critical points at y = ±1 are unstable sources. Slide 6/14 Dr. John Ehrke Lecture 5 Fall 2012
Logistic Equation While the previous example is interesting it does not reflect the true conditions under which populations grow/decay. Limitations including spatial requirements, food/water, and other resources can effect the rate at which populations grow. Factoring this into our analysis requires we replace the constant r by a function h(y) and examine the autonomous equation = h(y)y (4) dt choosing h(y) in such a way that h(y) r when y is small (i.e. grows exponentially for a while, but h(y) decreases as y grows larger, and eventually h(y) < 0 when the population is too large to sustain. Remarks: The simplest function that has all the necessary properties is h(y) = a by where a, b 0. Under these conditions, equation (4) becomes the logistic equation = (r ay)y (5) dt Note, the logistic equation has two critical points (and hence two equilibrium solutions) y = r/a and y = 0. Slide 7/14 Dr. John Ehrke Lecture 5 Fall 2012
Solution of the logistic equation Example The logistic equation is often written in the form (1 dt = r y ) y (6) k where k = r/a. This form is used to make the analytic solution more intuitive. Using this form, find the analytic solution of the logistic equation, and verify the solution behavior qualitatively. Solution: The analytic solution to the logistic equation can be found in detail on pg. 82 of the text. The logistic equation has two critical points y = 0 and y = k. The y = 0 equilibrium corresponds to population extinction. The y = k equilibrium denotes the population saturation level, often called the carrying capacity of the population s environment. Above the saturation level, /dt < 0 and so populations decrease because their environment cannot sustain them. Similarly, below the saturation level, /dt > 0 and so populations have the required resources to thrive. Slide 8/14 Dr. John Ehrke Lecture 5 Fall 2012
Critical Points and Stability As we saw in the previous example, there are different types of solution behaviors depending on the nature of the critical points. There are three types of critical point behavior possible. Stable Critical Point: The solution y = k in the previous example was a solution that all other solutions wanted to be near. Such critical point solutions are called stable solutions. Unstable Critical Point: On the other hand, the solution y = 0 is the solution which in some sense repels the other solutions. Such critical point solutions are called unstable. Semistable Critical Point: Functions, f (y) which have critical points with multiplicity often exhibit critical point behavior where solutions on one side are attracted and solutions on the other side are repelled. These critical points are called semistable. In the examples which follow, we will look at how a change in parameters can effect critical point behavior. Slide 9/14 Dr. John Ehrke Lecture 5 Fall 2012
Logistic model with harvesting In this example, we will consider the logistic model that includes harvesting. Think about a fresh fish farm in which a constant amount of fish are pulled out for processing. The ODE under this case looks like dt = ay by2 h (7) where h is the amount of fish harvested. If h = 0, this is the model explored in the previous example. Analyze the behavior of solutions depending on the value of the parameter. Assume h > 0. Solution: h m = a/2b is the maximum harvesting rate. Harvesting rates higher than this quantity lead to solutions which are everywhere decreasing and in the context of a population would lead to extinction. Harvesting rates above h m exhibit two critical points, one stable and one unstable, while harvesting rates less than h m have no critical points. The point h m is the point at which this change occurs, and is a special type of point called a bifurcation. Slide 10/14 Dr. John Ehrke Lecture 5 Fall 2012
Autonomous Equations and Parameters Definition (Bifurcation) A phase line is a single line containing the information about the equilibrium points and the nature of solutions on each side of them. When a series of phase lines are plotted on a graph of the parameter vs. the dependent variable (i.e. µ vs. y in this example) we obtain a bifurcation plot. A bifurcation occurs for a value of the parameter which changes the basic nature of solutions. Example Consider the first order autonomous ODE, dt = y(1 y)2 + µ. (8) Analyze the equilibria and behavior of solutions. Include a description of the bifurcation points and bifurcation diagram. Slide 11/14 Dr. John Ehrke Lecture 5 Fall 2012
Analysis of Parameters We consider the first order autonomous ODE /dt = y(1 y) 2 + µ. To find the points of bifurcation, we look for the critical points of Taking derivatives, yields f (y) = y(1 y) 2 + µ. (9) (y(1 y) 2 + µ) = 0 = 3y 2 4y + 1 = 0. Solving for y gives y = 1 and y = 1/3. Plugging these critical values of y back into (9) and solving for µ we obtain µ = 0 and µ = 4/27. This creates three regions of interest: 1 For µ < 4/27, y has a single equilibrium point. 2 For 4/27 < µ < 0, y has three equilibrium points. 3 For µ > 0, y has a single equilibrium point. 4 For µ = 0 and µ = 4/27, there are two critical points, y = 0 and y = 1/3. Slide 12/14 Dr. John Ehrke Lecture 5 Fall 2012
Bifurcation Plot Slide 13/14 Dr. John Ehrke Lecture 5 Fall 2012
Stability Theorem Stability Theorem Let f (y) and f (y) be continuous. The equation dt = f (y) (10) is stable at y = y 0 provided f (y 0 ) = 0 and f (y 0 ) < 0. The equation (10) is unstable at the point y = y 1 provided that f (y 1 ) = 0 and f (y 1 ) > 0. If both f (y 2 ) = 0 and f (y 2 ) = 0, then the point y 2 is a point of bifurcation for (10). Slide 14/14 Dr. John Ehrke Lecture 5 Fall 2012