Optimal harvesting and spatial patterns in a semi arid vegetation system

TL;DR

This paper develops a mathematical framework to determine optimal harvesting strategies in a semi-arid vegetation system, revealing that spatially patterned controls outperform uniform ones and that social optimization benefits vegetation survival and yield.

Contribution

It introduces a novel numerical approach to analyze spatially distributed optimal harvesting with bifurcation analysis and control of pattern formation in vegetation models.

Findings

Patterned controls outperform uniform harvesting in profit.

Social control yields higher harvest and vegetation survival.

Optimal tax can be derived for private optimization.

Abstract

We consider an infinite time horizon spatially distributed optimal harvesting problem for a vegetation and soil water reaction diffusion system, with rainfall as the main external parameter. By Pontryagin's maximum principle we derive the associated four component canonical system, and numerically analyze this and hence the optimal control problem in two steps. First we numerically compute a rather rich bifurcation structure of flat (spatially homogeneous) and patterned canonical steady states (FCSS and PCSS, respectively), in 1D and 2D. Then we compute time dependent solutions of the canonical system that connect to some FCSS or PCSS. The method is efficient in dealing with non-unique canonical steady states, and thus also with multiple local maxima of the objective function. It turns out that over wide parameter regimes the FCSS, i.e., spatially uniform harvesting, are not optimal. Instead, controlling the system to a PCSS yields a higher profit. Moreover, compared to (a simple model of) private optimization, the social control gives a higher yield, and vegetation survives for much lower rainfall. In addition, the computation of the optimal (social) control gives an optimal tax to incorporate into the private optimization.