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Zbl 1055.35046
Dancer, E. N.; Du, Yihong
Effects of certain degeneracies in the predator-prey model.
(English)
[J] SIAM J. Math. Anal. 34, No. 2, 292-314 (2002). ISSN 0036-1410; ISSN 1095-7154/e

Summary: To demonstrate the influence of spatial heterogeneity on the predator-prey model, we study the effects of the partial vanishing of the nonnegative coefficient functions $b(x)$ and $e(x)$, respectively, in the steady-state predator-prey model $$\matrix -d_1(x)\Delta u=\lambda a_1(x)u-b(x)u^2-c(x)uv,\\ -d_2(x)\Delta v=\mu a_2(x)c-e(x) v^2+d(x)uv, \endmatrix \quad u\vert _{\partial \Omega}=v\vert _{\partial \Omega}=0,$$ where all other coefficient functions are strictly positive over the bounded domain $\Omega$ in $\Bbb R^{N}$. Critical values of the parameter $\lambda$ are obtained to show that, in each case, the vanishing has little effect on the behavior of the model when $\lambda$ is below the critical value, while essential changes occur once $\lambda$ is beyond the critical value.
MSC 2000:
*35J60 Nonlinear elliptic equations
35J20 Second order elliptic equations, variational methods
35J55 Systems of elliptic equations, boundary value problems
35B45 A priori estimates
35B32 Bifurcation (PDE)
47J10 Nonlinear eigenvalue problems
92D25 Population dynamics

Keywords: global bifurcation; a priori estimates; predator-prey model

Cited in: Zbl 1022.92025

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