On positive solutions of some system of reaction-diffusion equations with nonlocal initial conditions. (English) Zbl 1242.35141

Walker, Christoph

J. Reine Angew. Math. 660, 149-179 (2011).

The author focuses on the positive solutions \(u=u\left( a,x\right) \) and \(v=v\left( a,x\right) \) of the system of parabolic equations \[ \partial _au-\Delta _Du=-\left( \alpha _1u+\alpha _2v\right) u,a\in \left( 0,a_m\right) ,x\in \Omega , \]
\[ \partial _av-\Delta _Dv=-\left( \beta _1v-\beta _2u\right) v,a\in \left( 0,a_m\right) ,x\in \Omega , \] with the nonlocal initial conditions \[ u\left( x,0\right) =\eta \int\limits_0^{a_m}b_1\left( a\right) u\left( a,x\right) da,x\in \Omega , \]
\[ v\left( x,0\right) =\xi \int\limits_0^{a_m}b_2\left( a\right) v\left( a,x\right) da,x\in \Omega . \] The subscript \(D\) indicates that Dirichlet conditions are imposed on the boundary \(\partial \Omega \). Such system arises when studying stationary solutions to a particular predator-prey system with age structure.
Using global bifurcation techniques, the author describes the structure of the set of positive solutions with respect to two parameters measuring the intensities of the fertility of the species. Of particular interest are coexistence solutions, solutions \(\left( u,v\right) \) with both components nonnegative and nonzero.

Reviewer: Cristian Chifu (Cluj-Napoca)

MSC:

35K51	Initial-boundary value problems for second-order parabolic systems
35K57	Reaction-diffusion equations
35B09	Positive solutions to PDEs
92D25	Population dynamics (general)

Keywords:

predator-prey model; coexistence solutions; Dirichlet conditions; age structure; global bifurcation techniques

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