The secondary tumor is stated to be more dangerous among cancer patients and this tumor is created through metastasis. Metastasis is the process of spreading a tumor from the primary location to the second part of the human body. This is an unpleasant problem among cancer patients because metastasis can contribute to high mortality cases among them. The presence of finger-like protrusions on the plasma membrane of cancer cells is known as the invadopodia. This structure can contribute to cancer cell invasion through the metastasis process. The formation of invadopodia involves several molecular interactions between extracellular matrix (ECM), ligand, actin, and matrix metalloproteinases (MMPs). The degradation of the ECM by the MMPs is mentioned as the starting point for the occurrence of cancer cell invasion. In this paper, the concentration of MMPs is taken in several functions of g to observe the formation of invadopodia on the plasma membrane. Two-dimensional mathematical model of ligand and signal is solved numerically using the method of level set, ghost fluid with linear extrapolation, and finite-difference. Credit is given to the level set method which successfully detected the movement of the free boundary interface (plasma membrane) by setting the interface as a zero-level set function. Also, the neighboring meshes can be identified using this method. Results showed that the above-mentioned integrated methods effectively describe the movement of the free boundary interface and this directly points out the formation of protrusions on the plasma membrane.

Invadopodia are finger-like protrusions located at subcellular membrane which can lead to cancer cell invasion. The formation of invadopodia involves several steps such as actin polymerizations, degradation of extracellular matrix which produce ligand and signal stimulation that is occurred from the binding of ligand with epidermal growth factor receptor. In this paper, a mathematical model of signal transduction is investigated. Both signal and ligand are represented by Laplace equation with Dirichlet boundary condition for each region. The cell membrane is treated as free boundary surface to separate any activity that occurred in intracellular and extracellular regions. The motion of the interface is taken as gradient of  nterior signal and the cell membrane is set as zero level set function. The problem is solved numerically using finite difference scheme of upwind, interpolation and extrapolation methods. The results showed that the formation of invadopodia is formed when protrusions exist on the cell membrane.

Invadopodia are protrusions that are commonly spotted at the plasma membrane of the invasive cancer cells. In forming invadopodia, several molecular interactions are involved such as the ligand, extracellular matrix (ECM), matrix metalloproteinases (MMPs), actin, and signal which are interrelated. In this paper, the mathematical model of ligand and signal transduction is taken in the heat equation with the MMPs is set as function . Besides, the actin regulation moved the interface and thus computed as the signal gradient. The mathematical model is solved using the combination of methods finite difference, ghost fluid with linear extrapolation, and level set. Apart from that, the convergence results are also given to determine the effectiveness of the above-mentioned methods. Results showed that the stimulation of signal transduction from the ligand and membrane-associated receptor binding consequently moved the plasma membrane. Also, the methods used gave a good agreement in the convergence results.

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Predator-prey model is the study to show the interaction between prey and preda- tor. A stage-structured model is introduced where the mature and immature of the species is considered. Also, the effect of harvesting is taken into account for sustainable development. The model in this paper is explained by using the ordinary differential equations to show the dynamic behavior of the predator and prey. Hence, the predator-prey model with stage- structured in prey with the effect of harvesting in predator is considered in this paper. In analyzing the model, the stability of the equilibrium point is obtained and described by using the properties of eigenvalues and Routh-Hurwitz criteria. Finally, numerical simulations are given to verify the analytical results with the help of graphical illustrations.