After invoking pplane8 a window appears where you type in your differential

After invoking pplane8 a window appears where you

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After invoking pplane8 a window appears where you type in your differential equation, including the limits on your state variables . This will generate a new window showing the direction field for the phase portrait. With the mouse you can click at any point and have a solution trajectory be drawn. Default for this is both directions in time. (To get a forward trajectory only one select Solution direction under the Options menu.) One of the most powerful features is the ability of this program to find equilibria and determine the eigenvalues near that equilibrium. This feature is found under the Solutions menu saying Find an equilibrium point . Select this feature, then click on the graph where you think an equilibrium point may be. A large red dot
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appears at the equilibrium, and a new window opens telling the coordinates of the equilibrium, the nature of the equilibrium, and the eigenvalues and eigenvectors of the linearized system near the equilibrium. Another useful feature under the Solutions menu is to Show nullclines . The nullclines tell where trajectories are either horizontal or vertical. The intersection of two different color nullclines shows the location of an equilibrium. Below is a figure showing the direction field for our example. Four solutions were added along with the equilibrium and the nullclines. Nonlinear Example The pplane8 program is particularly useful for systems of two nonlinear differential equa- tions . A competition model for two species y 1 and y 2 competing for the same resource satisfies the following system of differential equations: ˙ y 1 = 0 . 1 y 1 (1 - 0 . 022 y 1 - 0 . 03 y 2 ) , ˙ y 2 = 0 . 12 y 2 (1 - 0 . 037 y 1 - 0 . 024 y 2 ) . Though it is not the case for this system of differential equations, it can be quite challenging finding the equilibria for a nonlinear system of differential equations. MatLab does have a powerful tool for solving nonlinear systems of equations to find where they are zero, and it is called fsolve . The MatLab function fsolve requires entering a function f ( x ), which can be a vector function, and an initial guess, x 0 , then it tries to find the closest x that solves f ( x ) = 0 . If we define the right hand side of the DE above as the following MatLab function: 1 function z = compet ( y ) 2 % competition model : f s o l v e f o r e q u i l i b r i a 3 zt1 = 0.1 * y (1) * (1 - 0.022 * y (1) - 0.03 * y (2) ) ; 4 zt2 = 0.12 * y (2) * (1 - 0.037 * y (1) - 0.024 * y (2) ) ; 5 z = [ zt1 , zt2 ] ; 6 end
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If we enter the following in MatLab: ye = fsolve(@compet,[10,20]); then MatLab gives the unstable equilibrium y e = [10 . 3093 , 25 . 7732] T .
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