Dynamics of an autonomous food chain model and existence of global attractor of the associated non-autonomous system

In this paper, we have analyzed a tri-trophic food chain model consisting of phytoplankton, zooplankton and fish population in an aquatic environment. Here, the pelagic water column is divided into two layers namely, the upper layer and the lower layer. The zooplankton population makes a diel vertical migration (DVM) from lower portion to upper portion and vice-versa to trade-off between food source and fear from predator (Fish). Here, mathematical model has been developed and analyzed in a rigorous way. Apart from routine calculations like boundedness and positivity of the solution, local stability of the equilibrium points, we performed Hopf bifurcation analysis of the interior equilibrium point of our model system in a systematic way. It is observed that the migratory behavior of zooplankton plays a crucial role in the dynamics of the model system. Both the upward and downward migration rates of DVM leads the system into Hopf bifurcation. The upward migration rate of zooplankton deteriorates the stable coexistence of all the species in the system, whereas the downward migration rate enhance the stability of the system. Further, we analyze the non-autonomous version of the system to capture seasonal effect of environmental variations. We have shown that under certain parametric restrictions periodic coexistence of all the species of our system is possible. Finally, extensive numerical simulation has been performed to support our analytical findings.

Life-threatening Double-J stent migration with helicoidal appearance: Length matters

Background: The Double-J stent is one of the most commonly used devices in urologic practice. Due to its widespread use, numerous common complications have been reported, such as irritative symptoms, infection, and encrustation. More rare complications have also been described, such as up or downward migration and displacement outside the urinary tract. We present a rare case of downward migration of a Double-J stent in a 21-year-old Caucasian female. Case Presentation: A 21-year-old female with a solitary kidney presented to the emergency department with acute renal failure, left flank pain, and fever. She had undergone left Double-J stenting 1 week earlier in her homeland for left renal colic and anuria. A kidney–ureter–bladder X-ray revealed a 10-mm lumbar ureteral stone and the proximal coil of the Double-J stent making multiple loops along the ureter, resulting in a helical appearance. She underwent surgery to remove the previous stent and to place a new one. She was discharged 2 days later and her renal function had returned to normal values at her 1-week follow-up. Conclusions: Double-J Stent placement is a common procedure in the management of urinary tract diseases but is not devoid of life-threatening complications. Regular follow-up of stents and on-time evaluation of clinical complaints are mandatory for an aggressive treatment of complications.