We continue our study on the racetrack model. In the previous paper, we have shown that the global solution has an -limit which is a stationary solution. In this paper, we introduce a simplified racetrack model and study stability and instability of stationary solutions by using the linearization principle.

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Recently the geometric operator mean is extended to the multi-variable one; the Karcher mean. Including these multivarible means, we discuss a construction method by the Moore-Penrose inverse. The key concept is the orthogonality of operator means.

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This paper is devoted to studying a complete two-dimensional Daisyworld model on a sphere. The Daisyworld model which has been originally introduced by Andrew Watson and James Lovelock (1983) describes the process of planetary self-regulating homeostasis by a biota and its environment. After formulating our two-dimensional model, we construct global solutions, dynamical systems and exponential attractors. We also show some numerical results suggesting pattern formation of stripe segregation.

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We study the bifurcation problem for a chemotaxis-growth system with logistic growth in a two-dimensional rectangular domain. We apply the local bifurcation theorem by Ambrosetti and Prodi that does not require one-dimensional degeneration of the linearized operator around trivial solutions. We then obtain bifurcation solutions with two- and three-dimensional degeneration indicating spatially regular nesting patterns.

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In [1], the authors introduced and studied the notion of almost contra-bcontinuity in topological spaces. In this paper, we investigate some more properties of this type of continuity.

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We introduce the concept of interior ideal and the concept of fuzzy interior ideal in hypersemigroups and we prove, among others, that in regular also in intra-regular hypersemigroups the interior ideals and the fuzzy interior ideals coincide. We also prove that an hypergroupoid H is simple if and only if every fuzzy ideal of H is a constant function; and that an hypersemigroup H is simple if and only if every fuzzy interior ideal of H is a constant function, equivalently if, for every element a of H, we have H = H * {a} * H.

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