Mathematical analysis

Programming language

Neumann boundary condition

Geometry

Turing

Physics

Nonlinear system

Population

Instability

Bifurcation

Mathematics

Mechanics

Diffusion

Biology

Quantum mechanics

Demography

Pattern formation

Computer science

Differential equation

Reaction–diffusion system

Statistical physics

Equilibrium point

Allee effect

Sociology

Boundary (topology)

Parameter space

Applied mathematics

Genetics

Thermodynamics

International Journal of Biomathematics

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.

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