This paper suggests and analyses a model consisting of two commensal populations with Michaelis-Menten type of harvesting for the first population. The first harvested commensal species draws strength from the second hosted species. The overall dynamics are provided to achieve the coexistence, stability and persistence of the equilibrium points for the proposed system. The local bifurcation near the positive equilibrium point is attained. Moreover, numerical simulation using MATLAB is investigated to the impact of the commensalism interaction on the behavior of the planned model. The analysis shows that the role of commensalismpr events the first population from extinction, which could be helpful for the survival of both species.
In this study, we set up and analyze a cancer growth model that integrates a chemotherapy drug with the impact of vitamins in boosting and strengthening the immune system. The aim of this study is to determine the minimal amount of treatment required to eliminate cancer, which will help to reduce harm to patients. It is assumed that vitamins come from organic foods and beverages. The chemotherapy drug is added to delay and eliminate tumor cell growth and division. To that end, we suggest the tumor-immune model, composed of the interaction of tumor and immune cells, which is composed of two ordinary differential equations. The model’s fundamental mathematical properties, such as positivity, boundedness, and equilibrium existence, are examined. The equilibrium points’ asymptotic stability is analyzed using linear stability. Then, global stability and persistence are investigated using the Lyapunov strategy. The occurrence of bifurcations of the model, such as of trans-critical or Hopf type, is also explored. Numerical simulations are used to verify the theoretical analysis. The Runge–Kutta method of fourth order is used in the simulation of the model. The analytical study and simulation findings show that the immune system is boosted by regular vitamin consumption, inhibiting the growth of tumor cells. Further, the chemotherapy drug contributes to the control of tumor cell progression. Vitamin intake and chemotherapy are treated both individually and in combination, and in all situations, the minimal level required to eliminate the cancer is determined.
This paper treats the interactions among four population species. The system includes one mutuality prey, one harvested prey and two predators. The four species interaction can be described as a food chain, where the first prey helps the second harvested prey. The first and the second predator attack the first and the second prey, respectively, according to Lotka-Volterra type functional responses. The model is formulated using differential equations. One equilibrium point of the model is found and analysed to reveal a threshold that will allow the coexistence of all species. All other equilibrium points of the system are located, with their local and global stability being assessed. To back up the conclusions of the mathematical analysis, a numerical simulation examination of the model is carried out. The system's coexistence can be achieved as long as the harvesting rate of the prey population is lower than its intrinsic growth rate.
This paper aims to study the prey refuge impact on the dynamic behaviour of a stage structure predator-prey model. The model consists of four ecological species: prey in the protected and unprotected area and immature and mature predators. It assumes the grown predator can feeds only on the prey in an unreserved area. The conditions that guarantee the existence of the possible fixed points are found. Further, the local stability around all of the equilibria is considered. Then, using the Lyapunov direct method, the essential conditions for the global stability of the equilibria are adopted. Numerical simulations are illustrated to confirm our results. It concluded that the protected area positively affects the system's coexistence.
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