关键词: renewable resource, optimal harvesting, stability, bifurcation, limit cycle

Abstract: We study the vector boundary value problem with boundary perturbations: ε²y⁽⁴⁾=f(x,u,y",ε,μ) (μx-μ=A₁(ε,μ),y(x,ε,μ)|_x-μ=B₁(ε,μ) y"(x,ε,μ)|_x-μ=A₂(ε,μ),y"(x,ε,μ)|_x-μ=B₂(ε,μ) where yf, A_j and B_j (j=1,2) are n-dimensional vector functions and ε, μ are two small positive parameters. This vector boundary value problem does not appear to have been studied, although the scalar boundary value problem has been treated. Under appropriate assumptions, using the method of differential inequalities we find a solution of the vector boundary value problem and obtain the uniformly valid asymptotic expansions.

Key words: renewable resource, optimal harvesting, stability, bifurcation, limit cycle