This paper examines a bionomic model of competing organisms in the presence of allelochemicals with differing time-dependent densities. A competition mathematical model is proposed in this paper. It is shown that when one plant produces an allelochemical, it gives stimulatory effect to itself and inhibitory effect to the other plant. Equilibrium points are calculated and stability analysis is performed about non-zero equilibrium point. Hopf-bifuraction is observed with the help of delay parameter introducing in term of allelochemicals. Model is verified with already existing data for the effect of allelochemical on plant growth by Zahid et.al (2016). The numerical results are substantiated using MATLAB.
In this paper, a mathematical model is applied to investigate the effect of density of allelochemicals on plant population involving delay. The model is constructed using a system of nonlinear delay differential equations. In the model, there are three state variables: plant populations (P1 & P2) and density of allelochemical (T). It is assumed that in the presence of density of allelochemicals, plant populations are affected. It further delays the conversion of resource into the density of allelochemicals and hence affects the plant population adversely. This effect is observed by using delay in the competition model. Equilibrium points are calculated. Using the Routh–Hurwitz theorem, certain explicit formulations defining the stability and path of the Hopf‐bifurcation periodic solutions at the positive equilibrium are obtained. Further, the direction of these bifurcating periodic solutions is also studied. Numerical simulation and graphical support are provided for analytical findings using MATLAB.
Wood plays a very important role for daily household needs. In this paper, we have taken three parameters, forest biomass density
FB, wood‐based industries density
WI, and synthetic‐based industries density
SI in the proposed mathematical modeling. The forest biomass is assumed to grow logistically, which requires a fixed time to achieve full growth. The excessive cutting of wood for the growth of wood‐based industries not only depletes these forest resources, but there comes a delay in the growth of forest biomass as well. On the contrary, the growth of synthetic‐based industry is independent of forest biomass, yet there is a competition between wood‐based products and synthetic‐based products as per market demands. With the help of differential equation, we studied the stability of the system. Delay plays a significant role in the system to understand the complex behavior of the system about non‐zero equilibrium point. Hopf‐bifurcation observed and directional analysis is studied using normal form theory and manifold reduction. MATLAB software is used to graphical representation of the system.
Pituitary adenomas are tumours that occur in the Pituitary gland. Depending on the size it can be divided into Microadenoma (<10mm) and Macroadenoma(>10mm). It can present with endocrine manifestations secondary to hypo or hyper function of pituitary gland and ophthalmological manifestations due to mass effect. Case Report: A 36-year-old female was admitted to our institution with complaints of loss of vision of right eye, headache and amenorrhea for 3 months, which were progressive in nature. Her MRI Scan revealed a well-defined sellar mass, measuring 27 x 23.18 mm, with suprasellar extension, causing expansion of pituitary fossa and superiorly extending into Hypothalamus with mild compression over optic chiasma. Prolactin level was 72.16 ng/ml. Surgery was done under general anesthesias, by Endoscopic assisted transsphenoidal approach. The adenoma was completely removed. Sella opening was closed by synthetic graft. On awakening from anesthesias visual acuity in right eye was immediately and completely restored. Conclusion: Pituitary adenoma is a complex set of benign tumours that present with hypersecretory syndrome and mass effect. An appropriate imaging of pituitary region using MRI and endocrinological consultation is the standard for detection of pituitary adenoma. Although, medical management and radiotherapy offer effective treatment for these tumours in specific situations, Endoscopic Trans-sphenoidal approach continues to provide optimal outcomes with low incidence of morbidity.
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