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.

Impact of predation in the spread of an infectious disease with time fractional derivative and social behavior

The main purpose of this paper is to explore the influence of predation on the spread of a disease developed in the prey population where we assume that the prey has a social behavior.The memory of the prey and the predator measured by the time fractional derivative plays a crucial role in modeling the dynamical response in a predator–prey interaction.This memory can be modeled to articulate the involvement of interacting species by the presence of the time fractional derivative in the considered models.For the purpose of studying the complex dynamics generated by the presence of infection and the time-fractional-derivative we split our study into two cases. The first one is devotedto study the effect of a non-selective hunting on the spread of the disease, where the localstability of the equilibria is investigated. Further the backward bifurcation is obtainedconcerning basic reproduction rate of the infection. The second case is for explaining theimpact of selecting the weakest infected prey on the edge of the herd by a predator onthe prevalence of the infection, where the local behavior is scrutinized. Moreover, for thegraphical representation part, a numerical simulation scheme has been achieved usingthe Caputo fractional derivative operator.

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.