By Professor Tönu Puu (auth.)
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We then replace the partial differential equation by two interlocked ordinary ones: This case can also represent a one-dimensional space of unlimited extension (infinite, or closed like the perimeter of a circle) if population is assumed to alternate between two values. The case can also represent a two-dimensional chessboard pattern of alternating population density, again either infinite or closed. 30 The discrete model can be easily simulated. The low diffusivity, chosen in order to make the computational instabilities introduced by discretization more harmless, moves the four equilibrium points (with zero or unitary population densities in the two regions) innoticeably.
The low diffusivity, chosen in order to make the computational instabilities introduced by discretization more harmless, moves the four equilibrium points (with zero or unitary population densities in the two regions) innoticeably. In Figure 3 we see the results of simulation. There is in fact only one stable solution (1, I). The solution (0,0) is, as expected, unstable, but so are the spatially inhomogeneous solutions (1,0) and (0,1) as well. We have one stable and one unstable node and two saddle points.
The basic method still is one suggested by Poincare and perfected by Lindstedt. In order to apply the method we pretend that the right hand side, containing the nonlinearity, is small. To this end we multiply it by a small parameter E, obtaining: 55 (16 ) We try to find a solution in the form of a power series in E: (17) where (18) is a new time scale. The attempted solution is now substituted into the differential equation, which itself becomes a power series in E. , we equate each of the coefficients to zero.
Nonlinear Economic Dynamics by Professor Tönu Puu (auth.)