Abstract. A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis quadrature for Fredholm integral equations of the second kindwhose kernel k(t, s) is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when k(t, s) is infinitely differentiable away from the diagonal t = s. Relation to the singular value decomposition is indicated. Application to integro-differential Schrödinger equations with nonlocal potentials is given.

In the Hartree-Fock approximation the Pauli exclusion principle leads to a Schrödinger Eq. of an integro-differential form. We describe a new spectral noniterative method (S-IEM), previously developed for solving the Lippman-Schwinger integral equation with local potentials, which has now been extended so as to include the exchange nonlocality. We apply it to the restricted case of electron-Hydrogen scattering in which the bound electron remains in the ground state and the incident electron has zero angular momentum, and we compare the acuracy and economy of the new method to three other methods. One is a non-iterative solution (NIEM) of the integral equation as described by Sams and Kouri in 1969. Another is an iterative method introduced by Kim and Udagawa in 1990 for nuclear physics applications, which makes an expansion of the solution into an especially favorable basis obtained by a method of moments. The third one is based on the Singular Value Decomposition of the exchange term followed by iterations over the remainder. The S-IEM method turns out to be more accurate by many orders of magnitude than any of the other three methods described above for the same number of mesh points.2

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Previous studies have explored the relationship between termite branch tunnel geometry and foraging efficiency in a model simulation in which foraging efficiency, γ, for two termite species, Coptotermes formosanus Shiraki and Reticulitermes flavipes (Kollar) (Isoptera: Rhinotermitidae), was investigated in response to two variables, the probability of tunnel branching (Pbranch) and the probability of tunnel branch termination (Pterm). It was found that simulated tunnel patterns based on empirical data did not have maximum foraging efficiency. We hypothesized that termites could increase their foraging efficiency in response to landscape heterogeneity. The present study investigated how termites could control the two variables, Pbranch and Pterm, in response to the external environment in terms of tunnel network connectivity. It was found that the best simulated strategy for C. formosanus and R. flavipes termites would occur if both Pbranch and Pterm were increased together. This study provides possible mechanisms for foraging strategies in subterranean termites and a baseline for future empirical work.

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