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Unformatted text preview: 8.7. NONLINEAR DIFFERENTIAL EQUATIONS AND PHASE PLANE METHODS 763 Substituting this in the first equation yields . 354(- 4701)- . 354 k 2 =- 900 ⇒ k 2 = 631 . 3 Thus the particular solution to the problem for the given initial conditions is x ( t ) =- 500 + 0 . 485(1868) e . 028 t- . 485(631 . 3) e- . 178 t =- 500 + 906 e . 028 t- 306 e- . 178 t and y ( t ) =- 1000 + 0 . 379(1868) e . 028 t + 0 . 621(631 . 3) e- . 178 t =- 1000 + 708 e- . 0085 t + 392 e- . 291 t d. The solution points x ( t ) and y ( t ) versus t are plotted in Fig. 1.17A. and the values of x ( t ) versus y ( t ) for corresponding implicit values of t are plotted in the phase plane in Fig. 1.17B. Figure 8.17: The two solutions x ( t ) (black) and y ( t ) (red) are plotted as functions of time in panel A. and in the phase plane in panel B. Note that the points in the plane get further and further apart as t increases and the trajectories grow without bound. 2 8.7 Nonlinear Differential Equations and Phase Plane Methods At the end of the previous section we saw that if we are not careful how we formulate our models, we could write down a model in which populations grow without bound. One way to deal with this is to always make sure that population growth is nonlinear, of which logistic is the simplest example. Assume that the growth rate of frogs in two lakes 1 and 2 are described, as in Fig. 1.18, by the functions F ( x ) and G ( y ) respectively. In this case, if we have the same steady stream of individuals migrating between lakes 1 and 2 at per-capita rates g and h respectively, then our model becomes dx dt = F ( x )- gx + hy dy dt = G ( x )- hy + gx Since the above coupled equations are no longer linear, no general method is available to integrate the differential equations: we cannot integrate dx dt without knowing y ( t ) and vice-versa, and the method of transformations no longer uncouples the equations when F ( x ) and G ( y ) are not linear. Phase Plane Method In two dimensions, however, a general approach is available to obtain a qualitative understanding of the solutions to couple systems of nonlinear differential equations with constant parameters. This method is called the phase plane method and relies on plotting all equations the arise from setting the right-hand-side of the differential equation © 2008 Schreiber, Smith & Getz 764 8.7. NONLINEAR DIFFERENTIAL EQUATIONS AND PHASE PLANE METHODS Figure 8.18: A well mixed two compartment model in which populations at densities x and y growth nonlinearly at rates F ( x ) and G ( y ) in lakes 1 and 2 respectively and that individuals in each lake move between lakes at per-capita rates g and h respectively. models to 0 and plotting the resulting implied algebraic equations, called the null-clines in the phase plane. First we lay out this general method and then demonstrate the method in a number of classical models in population biology....
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This note was uploaded on 02/09/2011 for the course EEP 115 taught by Professor Waynem.getz during the Fall '10 term at Berkeley.
- Fall '10