Diffusion driven instability and Hopf bifurcation in spatial predator-prey model on a circular domain

https://doi.org/10.1016/j.amc.2015.03.070Get rights and content

Abstract

In this paper, we investigate theoretically and numerically a 2-D spatio-temporal dynamics of a predator-prey mathematical model which incorporates the Holling type II and a modified Leslie–Gower functional response and logistic growth of the prey. This system is modeled by a reaction diffusion equations defined on a disc domain {(x,y)R2/x2+y2<R2} with Dirichlet initial conditions and Neumann boundary conditions. We study the local and global stability of the positive equilibrium point. We show that the diffusion can induce instability of the uniform equilibrium point which is stable with respect to a constant perturbation as shown by Turing in 1950s and derive the conditions for Hopf and Turing bifurcation in the spatial domain. Numerical results are given in order to illustrate how biological processes affect spatiotemporal pattern formation in a spatial domain. We perform the computations and generalize, on a circular domain, the results presented in Camara and Aziz-Alaoui [6].

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 t=a1T,u(t)=b1a1x(T),v(t)=r2b1a1a2y(T),a=a2r1a1r2,b=a2a1,e1=b1k1a1,e2=b1k2a1,system (1.3) becomes {dudt=(1uavu+e1)u,dvdt=(bbvu+e2)v,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: {U(T,x,y)T=δ1ΔU(T,x,y)+(a1b1U(T,x,y)c1V(T,x,y)U(T,x,y)+K1)U(T,x,y),forT[0,+[,(x,y)Ω,V(T,x,y)T=δ2ΔV(T,x,y)+(a1c2V(T,x,y)U(T,x,y)+K2)V(T,x,y),forT[0,+[,(x,y)Ω,U(·,x,y)·η=V(·,x,y)·η=0;onΩNeumannboundaryconditions,U(0,·,

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

  • (i)

    0 < a < 1 ≤ e1e2 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 D={(r,θ):0<r<R,0<θ<2π}, we denote Df(ρ)dρ=0R02πf(r,θ)dθdr and V(

Diffusion driven instability

By setting Z=(uu*vv*)φ(r,θ)eλt+ikr,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.: {Δrθφ=k2φ,φr(R,θ)=0.Then by linearizing around (u*, v*), we have the following equation: dZdt=AZ+DΔZwhere A=(fufvgugv)=(12u*e1(1u*)e1+u*au*e1+u*bb),D=(100δ)and

fu=uf(u*,v*),fv=vf(u*,v*),gu=ug(u*,v*),gv=vg(u*,v*).

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 n=1,,N,withN=TΔt,i=1,,P+1,andj=1,,M+1 find {ui,jn,vi,jn} such that {nui,jn=Δriθjui,jn+

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 {(x,y)R2/x2+y2<R2}. Initially, we presented the model on the circular domain (i.e. polar coordinate D={(r,θ):0<r<R,0θ<2π} 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.

References (27)

Cited by (0)

View full text