Spatiotemporal patterns induced by cross-diffusion in predator prey model with prey herd shape effect

In this paper,we investigate a predator-prey model with herd behavior and cross-diffusion subject to the zero flux boundary conditions.First,the temporal behavior of the model has been investigated,where Hopf bifurcation has been obtained.Then,by analyzing the characteristic equation it has been proved that the cross-diffusion generate a complex dynamics such as Hopf bifurcation,Turing instability,even Turing-Hopf bifurcation.Further,the impact of the prey herd shape on the spatiotemporal patterns has been discussed.Furthermore,by computing and analyzing the normal form associated with the Turing-Hopf bifurcation point,the spatiotemporal dynamics near the Turing-Hopf bifurcation point has been discussed and allso justified by some numerical simulations.

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.

Global threshold dynamics of an age structured alcoholism model

We consider in this research an age-structured alcoholism model.The global behavior of the model is investigated.It is proved that the system has a threshold dynamics in terms of the basic reproduction number(BRN),where we obtained that alcohol-free equilibrium(AFE)is globally asymptotically stable(GAS)in the case R_(0)≤1,but for R_(0)>1 we found that the system persists and the nontrivial equilibrium(EE)is GAS.Furthermore,the effects of the susceptible drinkers rate and the repulse rate of the recovers to alcoholics are investigated,which allow us to provide a proper strategy for reducing the spread of alcohol use in the studied populations.The obtained mathematical results are tested numerically next to its biological relevance.

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.

Numerical inversion method for the Laplace transform based on Boubaker polynomials operational matrix

We investigate through this research the numerical inversion technique for the Laplace transforms cooperated by the integration Boubaker polynomials operational matrix.The efficiency of the presented approach is demonstrated by solving some differential equations.Also,this technique is combined with the standard Laplace Homotopy Per-turbation Method.The numerical results highlight that there is a very good agreement between the estimated solutions with exact solutions.

Backward,Hopf bifurcation in a heroin epidemic model with treat age

Heroin is considered as damage on the health of the individuals.In this paper,we consider a heroin epidemic model with treat age,where the inner rivalry in between the drug users for a small dose of drugs is investigated.The influence of treatment period for a drug consumer before quitting treatment is investigated,where it is obtained that the considered model can undergo backward bifurcation,which shows the possibility of having two endemic equilibriums,this type of bifurcation is discussed in terms of the basic reproduction number R0.The bifurcation diagram is drown in the case of an age structured model.Also,we show the existence of Hopf bifurcation is shown under suitable conditions on the model parameters.The obtained mathematical results are confirmed numerically.