Table 3. Qualitative Hessian optimization outcomes for the eco-social model 4 and scenarios analysed in the current study.

D = the determinant of Hessian matrix, and i = number of variables. For more details see Supporting Information (S3 Appendix).

Model 4
Determinant of	MEABRs	OAAs	Sustainable scenario
Hessian matrixes
A. Qualitative analysis
Di = 1	(-) P, < 0	(+) L, > 0	(-) P, < 0
Di = 2	(+) PT, < 0	(+) LS, > 0	(+) PT, > 0
Di = 3	(-) PTU, < 0	(+) LSV, < 0	(-) PTU, < 0
Di = 4	(+) 2PTUϕ—PTm2—PUg2—TUa2, > 0	(+) 2LSVϕ—LSm2—LVg2—SVa2, > 0	(+) PTUϕ—PTm2—PUg2—TUa2, > 0
Di = 5	(+) 4PTUϕe - 2PTm2e - 2PUg2e - 2TUa2e, > 0	(+) 4LSVϕe - 2LSm2e - 2LVg2e - 2SVa2e, > 0	(-) 4PTUϕe + 2PTm2e + 2PUg2e + 2TUa2e, < 0
Hessian results for GFF	Saddle-point	Rel. Max.	Rel. Max.