Stability Analysis, Dispersion Relation and Pattern Formation of Damaged Cells on Biological Tissues in the Body of Living Organisms

This study is geared towards investigating stability analysis, dispersion relation and pattern formation potential of damage in the body of living organisms using 3-coupled nonlinear system of partial differential equations (PDEs). Obviously, this study proved that there are possibilities of asymptomatic stage of the damage with the first equilibrium point and the second is symptomatic stage of damaged cells at equilibrium which is usually and physically experienced in diabetic patients. The second equilibrium point predicts initiation of damaged cells, progression or inhibitions for healing. The stability analysis revealed steady state solutions with the exudation surface activated by the stress, temperature and viscous terms causing increased growth rate of damaged cells in the body of the organism. The solutions of dispersion relations is an archetype of spatial heterogeneous and homogeneous solutions of the linear biological systems with unbalanced and exponential growth term linearly (linear growth rate of damaged cells and its wave vector k) with respect to time. The study identified that the quadratic nature of Ψ(<i>k</i><sup>2</sup>) would lead the system to bounded finite amplitude. The study explored into complex dispersion relations and ascertained some special cases where one or more parameters affected damaged cells mechanically at equilibrium with bifurcation surface between spatial homogeneity and heterogeneity. Here, the bifurcation parameter τ increases monotonically beyond the critical value and renders <i>B</i><i>(k</i><i><sup>2</sup></i><i>)</i> zero and the uniform steady state bifurcates to a spatial unstable state. This study showed pattern formation potentials are quadratic and bounded domain of the injured cells, spatial pattern of random and viscous initial damaged cell densities and patterning process of no uniform damaged cell density due cell traction is bigger than the critical value of cell traction which is the resistance of the extracellular matrix and spatial non-homogeneities activates which grows out of bounded region for some k. The study used partial Differential equation and determinant methods to proffer the solutions. Among others, this study recommends that the affected individuals should explore the use of antioxidants as to combat with the out growing inbuilt stress called oxidative stress which is activator of damaged cells densities in the body of the organisms.