Pattern dynamics of a reaction-diffusion predator-prey system with both refuge and harvesting

We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge(proportion of both the species)and harvesting of prey species in this contribution.Criteria for asymptotic stability(local and global)and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point.For partial differential equation(PDE),the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated.The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species.A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space,that is,spots,stripe-spot mixtures,labyrinthine,holes,stripe-hole mixtures and stripes replication.Finally,we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations.

Dynamical behavior of a mathematical model of early atherosclerosis

Atherosclerosis,a continual inflammatory disease occurring due to plaque cumulation in the arterial intima,is one of the main reasons behind deaths from diverse cardiovascular diseases.The basic interactions between oxidized low density lipoprotein(LDL)and macrophages in the formation of atherosclerotic plaque are modeled here in terms of a reaction-diffusion system in one-dimensional(1D)space under Neumann boundary conditions.Two simple mathematical models are considered which differ by the influx term only in the case of the interaction of oxidized LDL.Both the spatial and nonspatial systems are simply analyzed theoretically and numerically.Numerical bifurcation analysis confirms the existence of Hopf bifurcation concerning four significant model parameters.Examining the gravity of the model offered in this investigation,an obvious insight into this inflammatory response can be achieved both qualitatively and quantitatively.