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In this paper, we study the equilibrium points, local asymptotic stability of an equilibrium point, instability of equilibrium points, periodicity behavior of positive solutions, and global character of an equilibrium point of a fourth-order system of rational difference equations of the form x n+1 = αx n-3 β + γ y n y n-1 y n-2 y n-3

Surghar mountain belt has comparatively less natural resources of floral diversity because it is composed of minerals of different kinds making it less favorable for the growth of different vegetation. The pollen morphology of some selected plants from Surghar belt Mainwali has been evaluated. The pollen grains were measured and demonstrated using scanning electron microscopy (SEM). The examined plant specimens have a difference in size, shape, polarity, and their exine ornamentation. The pollen taxa show a huge variation in size and sculpture. Pollen fertility has also been estimated, shows that the selected plants are well-known in the Surghar belt. The need of the hour is to conserve these plants having a higher fertility rate to cope with the deforestation in an area. The conclusion does not favor theories in which deforestation results in fast growth in population. It shows that the irrational management and unlawful cutting down of woods neglected by the forest department are the main causes of deforestation in the mountain belt of Mianwali. The findings show the importance of morphological characteristics in the identification of natural resource species in the area. K E Y W O R D S deforestation, irrational management, pollen fertility, pollen morphology, Surghar belt

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In this paper, we study the qualitative behavior of a competitive system of second-order rational difference equations. More precisely, we investigate the boundedness character, existence and uniqueness of positive equilibrium point, local asymptotic stability and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the system. Some numerical examples are given to verify our theoretical results.

In the present work, we study the qualitative behavior of two systems of higher-order rational difference equations. More precisely, we study the local asymptotic stability, instability, global asymptotic stability of equilibrium points and rate of convergence of positive solutions of these systems. Our results considerably extend and improve some recent results in the literature. Some numerical examples are given to verify our theoretical results. MSC: 39A10; 40A05

We study the qualitative behavior of the following exponential system of rational difference equations:xn+1 = αe-yn+βe-yn-1/γ+αxn+βxn-1,  yn+1 = α1e-xn+β1e-xn-1/γ1+α1yn+β1yn-1,  n = 0,1,…,whereα,β,γ,α1,β1, andγ1and initial conditionsx0,  x-1,  yo, and  y-1are positive real numbers. More precisely, we investigate the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions that converges to unique positive equilibrium point of the system. Some numerical examples are given to verify our theoretical results.

Fifty-seven patients were studied over a period of three years to analyse the efficacy of surgical pleurectomy for spontaneous pneumothorax. Thirty-one and 26 patients underwent open and video-assisted thoracoscopic surgery (VATS) pleurectomy, respectively. VATS was the main modality used for primary spontaneous pneumothorax (PSP) (21 vs. 8). However, secondary spontaneous pneumothorax (SSP) was mainly managed with open pleurectomy (23 vs. 5). The median operating time was significantly longer in open group (72.4 vs. 55 min; P=0.005). The amount of analgesia required in the first five days was significantly more in open group (108 mg vs. 46.9 mg; P=0.02). Chest drainage was significantly more in open group (1027.1 ml vs. 652.8 ml; P=0.04). However, chest drain duration and hospital stay had no significant difference. VATS emerged as a cost-effective modality (1770 pounds vs. 3226 pounds). The ability to return to work was significantly earlier in VATS group in PSP patients (6 weeks vs. 10 weeks; P=0.007). There were 3 (5.27%) recurrences in VATS group for patients with SSP. This experience suggests that VATS pleurectomy is an appropriate modality for PSP. However, open pleurectomy is a viable alternative to treat SSP.