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Two novel iterative methods for a class of population models of competitive type are considered. This numerical solution is related to the positive solution as the competitive rate tends to infinity. Furthermore, the idea first is applied to an optimal partition problem. 3947As in [5], we consider the class of all possible segregated states S:where W + = max(W, 0) and W − = max(−W, 0).

Given a bounded open set ⊂ R 2 , we present numerical approximations for two problems related to minimal partitioning of the first eigenvalues of the Dirichlet Laplacian. The first problem is about minimizing the sum of first eigenvalues of the Dirichlet Laplacian. This partitioning problem arises as a steady state of a reaction-diffusion process.To do this, a new idea to approximate the second eigenfunction and second eigenvalue is presented. We use the qualitative properties of the minimization problem to construct a numerical algorithm to approximate optimal configurations. A rigorous analysis to show the convergence and the rate of convergence is given. Moreover, we discuss the numerical implementation of the resulting approach and present computational tests confirming the expected asymptotic behavior of optimal partitions with large numbers of partitions.