A strongly coupled diffusion effect on the stationary solution set of a prey-predator model. (English) Zbl 1167.35048

The author investigates the positive solution set of the following quasilinear elliptic system describing the predator-prey model with the strongly coupled diffusion \(\Delta(\frac{v}{1+\beta u})\):
\[ \begin{aligned} \Delta u+u(a-u-cv)=0 &\quad\text{in } \Omega,\\ \Delta \bigg[\bigg(\mu+\frac{1}{1+\beta u}\bigg) v\bigg] +v (b+du-v)=0 &\quad\text{in } \Omega,\\ u=v=0 &\quad\text{on } \Omega, \end{aligned} \]
where \(u\) (resp. \(v\)) is the population density of the prey (predator), \(\Omega\subset \mathbb{R}^N\) is a bounded domain, \(a,b,c,d\) and \(\mu\) are positive constants, \(\beta\geq 0\). In the previous article [T. Kadota and K. Kuto, J. Math. Anal. Appl., 323, 1387–1401 (2006; Zbl 1160.35441)] the existence of the bifurcation branch of positive solutions was proved, globally extending the relation to the bifurcation parameter \(a\). Here the nonlinear effects of large \(\beta\) on the positive solution continuum are studied.


35Q80 Applications of PDE in areas other than physics (MSC2000)
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J65 Nonlinear boundary value problems for linear elliptic equations
35B32 Bifurcations in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
92D25 Population dynamics (general)


Zbl 1160.35441