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5.3 Geomatric Analysis and 290 The model equations and analysis. The predator-prey assumptions yield the system of equations =as-bry Factoring gives =y(cx-d) for = 0 are I = 0 and y = a/b. The nuliclines for oullclines.) A solution. The (d/c,a/b) solution has trajectories cirding abrein - (d/c,a/b). (The hand-drawn phase plane is shown in Figure 5.39. Naturally, we y = 0 and x = d/c. points (d/c,0) and (0,a/b) are not points of intersection of itd The nuliclines Graphing these in the phase plane reveals two fixed points (0.0 interested it is in not the clear (0,0) motion. Jacobian solutions analysis determines spiraling this. in from this analysis if the are or out or if the though in pure oscillatory are Here circling f(z,y) = ex - by and g(z,y) = axy - dy, and the Jacobian matrix is = cy CZ -ba d While we are at it, we compute ((10.0)-(:-a). y a/b x d/c FIGURE 5.39 Phase Piane and Nulicline Diagram for Predator-Prey Model. Scanned with CamScanner 5.6 308 6. Consider the system of equations fixed points of this system (there are two). of b. a. Find Use Jacobian the analysis to determine the nature the fixed points (i.e, source, saddie, spiral, etc.) the nullclines on a phase plane. d. e. Use Graph phase plane analysis to mark the flow arrows. Do the directions of flow the differential equation solver capable of drawing phase plane plots to resalts of b? these several trajectories compare with what you expected from the previous amalysis by haxd Use trajectories a (at least one from each region formed by the nullclines Hos do 7. Consider the system of equations :=sin(x)-y dy y 2 a. Find the fixed points and determine their nature using Jacobian analysis. b. Draw the nullclines and flow arrows on a phase plane and compare with the behavios suggested by the Jacobian analysis. & For a final project, a student of ours was attempting to build a model for an with a number of interacting variables changing over time. He set the equations 25 a system of first-order homogeneous linear equations with constant coefficients. Without consulting us, he was arbitrarily assigning values to the coefficients trying to find a si of coefficients that would result in a stable ecosystem. Try to help him Set up a software model that solves (numerically or symbolically a system of three first-order linear homogeneous differential equations with constate coefficients. Arbitrarily (and without mathematical analysis) assign coefficients and initial conditions to try to get a system with solution curves that level out and are not all zere Keep trying It can be done. it can be done, but probably not by arbitrarily picking numbers. Try again. this tim carefully choosing numbers and, perhaps, referring to some of the discussions on thest types of systems in the text. If this student wanted a stable model, what advice would you give him? Also feel free in an ecosystem. to advise about randomly choosing parameters which are supposed to have some Assume that the following two logistic differential equations for time t between 0 and al. 10 aumerically 9. Use Euler's method (either directly or with a solver that uses Euler's method) to wht Compare z(0) = 100. Solve both using the h values: 1. 0.5. 0.25. and the solutions of the first equation for following the different h values. Explain whal if Scanned with CamScanner

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