带扩散的Logistic单种群模型及其最优收获

在一些合理的假设条件下，就空间分布非均匀的Ｌｏｇｉｓｔｉｃ型收获模型 得到了与空间分布均匀的Ｌｏｇｉｓｔｉｃ型收获模型［１，２，３］完全平行的结论，其中包括种群持续生存和灭绝时收获努力量 Ｅ（ｘ）须满足的充要条件、种群持续生存时趋于正平衡状态的速度估计、种群灭绝时其密度趋于０的速度估计以及在种群持续生存条件下的最优收获努力量 Ｅ、最优平衡解 ｐ（ｘ）和最大收获量 ｈ＊

Logistic Model for Single-species with Spatial Diffusion and its Optimal Harvesting Policy

In this paper, the spatially nonhomogeneous Logistic model on which describes the growth of a single-species with spatially variable harvesting effort, is considered under the homogeneous Neumann boundary condition, where is a bounded domain in with smooth boundary, the Laplace operator in , positive constant D the diffusive coefficient, r(x) > 0 the intrinsic growth rate of the considered single-species, K(x) > 0 the environment capacity, and E(x) > 0 the harvesting effort. Results parallel to those of the spacially homogeneous Logistic equation discussed in 1], 2] and 3] are obtained. It is showed that if , where denotes the first eigenfunction of the linearized elliptic eigenvalue problem of (*) with the same boundary condition, then the population density u(t, x) tends to a strictly positive equilibrium solution p(x) as for any nonnegative and nonzero initial values of u, and that if then u(t,x) will tend to 0 as for any nonnegative initial values of u. The speeds for u to tend to p(x) when rE > 0 and for u to tend to 0 when rE 0 are estimated respectively. At last, the optimal harvesting policy is discussed by using the variational calculus and two application examples are given.