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Existence, Multiplicity, and Stability of Positive Solutions of a Predator-Prey Model with Dinosaur Functional Response

We investigate the property of positive solutions of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions. Firstly, using the comparison principle and fixed point index theory, the sufficient conditions and necessary conditions on coexistence of positive solutions of a predator-prey model with Dinosaur functional response are established. Secondly, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of the model and obtain the stability and multiplicity of the positive solution under certain conditions. Furthermore, the local uniqueness result is studied when and are small enough. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions when is sufficiently large.

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Heterotopic Purkinje Cells: a Comparative Postmortem Study of Essential Tremor and Spinocerebellar Ataxias 1, 2, 3, and 6

AbstractEssential tremor (ET) is among the most common neurological diseases. Postmortem studies have noted a series of pathological changes in the ET cerebellum. Heterotopic Purkinje cells (PCs) are those whose cell body is mis-localized in the molecular layer. In neurodegenerative settings, these are viewed as a marker of the progression of neuronal degeneration. We (1) quantify heterotopias in ET cases vs. controls, (2) compare ET cases to other cerebellar degenerative conditions (spinocerebellar ataxias (SCAs) 1, 2, 3, and 6), (3) compare these SCAs to one another, and (4) assess heterotopia within the context of associated PC loss in each disease. Heterotopic PCs were quantified using a standard LH&E-stained section of the neocerebellum. Counts were normalized to PC layer length (n-heterotopia count). It is also valuable to consider PC counts when assessing heterotopia, as loss of PCs extends both to normally located as well as heterotopic PCs. Therefore, we divided n-heterot…

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