Diffusion driven instability and Hopf bifurcation in spatial predator-prey model on a circular domain
Section snippets
Introduction and mathematical model
Most natural phenomena are modeled by a reaction diffusion systems which are special cases of systems of parabolic partial differential equations [13]. These systems of reaction–diffusion equations describe how the concentration or density distributed in space varies under the influence of two processes: local interactions of species, and diffusion that causes the spread of species in the space. These systems of equations can describe the dynamic processes in biology, physics and ecology.
Asymptotic behavior of ODE system
In this subsection we recall some results on the asymptotic behavior of the system without diffusion (1.3).
To simplify system (1.3) we introduce some transformations of variables. After applying the following rescaling system (1.3) becomes We also require that ae2 < e1 which ensures that, system (1.6) has a positive equilibrium point corresponding to constant coexistence of the two
Equilibrium points and stability
Let us now consider the reaction diffusion system defined on a circular domain with Neumann boundary conditions (which means there are no flux of species of both the predator and prey on the boundary of the circular domain) and Dirichlet initial conditions as follow:
Global stability of the non-trivial steady state
In this section, we study the global stability of the homogeneous non-trivial steady state E* = (u*, v*).
Theorem 4.1 Let
0 < a < 1 ≤ e1 ≤ e2 and ae2 < e1.
If (i) is satisfied, then the steady state E* is globally asymptotically stable for system (3.5).
Proof
The proof is based on a positive definite Lyapunov function.
The hypothesis 0 < a < 1 ensures that ae2 < e2 and ae2 < e1 ensures the existence of the non-trivial positive steady state E*.
Since we denote and V(
Diffusion driven instability
By setting where k is the wave number and φ(r, θ) is a eigenfunction of the Laplacian operator on a disc domain defined in Section 2.2 with zero flux boundary, i.e.: Then by linearizing around (u*, v*), we have the following equation: where and
.
Theorem 5.1 Suppose that δ < δc b + e1 > 1 or 0 < u* < α1 orα2 < u*, so that E* = (u*, v*) is
Numerical method
Eq. (3.5) is not defined at the origin r = 0, to avoid this singularity and to have the desired regularity the finite difference scheme [25] uses a uniform grid with some steady at the origin. This poles condition play the role as a boundary condition which is a necessary condition in the finite difference scheme. By discretization, the associated approximation of problem (3.5) takes the following form:
For find such that
Conclusion
In this paper, we present the analysis of two developed predator-prey systems in the last decades in ecology. For this, we have presented a spatio-temporal prey-predator system given by a reaction–diffusion equations which is based on a Holling-type II modified functional responses. We assumed that the two populations diffuse in a disc domain . Initially, we presented the model on the circular domain (i.e. polar coordinate and analyzing the nature of
Acknowledgement
We are very grateful to the Editor and to the anonymous Referees for their valuables remarks and suggestions which help us to improve the quality of the present paper.
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