Maximizing the total population with logistic growth in a patchy environment.

This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the population of a single species with logistic growth residing in a patchy environment and study the effects of dispersal and spatial heterogeneity of patches on the total population at equilibrium. Our objective is to maximize the total population by redistributing the resources among the patches under the constraint that the total amount of resources is limited. It is shown that the global maximizer can be characterized for any number of patches when the diffusion rate is either sufficiently small or large. To show this, we compute the first variation of the total population with respect to resources in the two patches case. In the case of three or more patches, we compute the asymptotic expansion of all patches by using the Taylor expansion with respect to the diffusion rate. To characterize the shape of the global maximizer, we use a recurrence relation to determine all coefficients of all patches.

Rho-associated protein kinase inhibitor induced morphological changes in type VI collagen in the human trabecular meshwork.

AIMS: To investigate morphological changes in type VI collagen in the human trabecular meshwork associated with the rho kinase inhibitor ripasudil. METHODS: This cross-sectional study evaluated the effects of ripasudil eye drop administration (RA) or no ripasudil eye drop administration (NRA) in patients with primary open-angle glaucoma (POAG; age range 60-80 years) who underwent conventional outflow reconstruction between December 2015 and September 2016 at Tokai University Hachioji Hospital. The juxtacanalicular tissue was removed and imaged using transmission electron microscopy. Type VI collagen comprises cross-banded aggregates with transverse bands 30 nm apart repeating every 105 nm. The transverse bands are called the outer rod-like region (ORR) and the intervals are called the inner rod-like region (IRR). The waveform intensity in the type VI collagen was analysed in electron micrographs using Fourier transformation to detect the IRR and ORR borders. RESULTS: Ten eyes of 10 patients were included (n=5/group). The baseline characteristics did not differ significantly between groups. ORR width was significantly smaller in the RA group (37.85±3.43 nm) than in the NRA group (50.62±5.23 nm, p<0.05), whereas IRR width was significantly greater in the RA group (70.68±10.84 nm) than in the NRA group (58.19±5.34 nm, p<0.05). Morphological changes in the type VI collagen total width tended to correlate with the duration of ripasudil administration (r=0.9, p=0.08). CONCLUSIONS: Ripasudil administration in patients with POAG induced morphological changes in type VI collagen. Patients with POAG administered RA had a significantly smaller ORR width and a significantly greater IRR width than patients with POAG not administered RA.