Table 1. Qualitative Hessian optimization outcomes for the eco-social models 1 & 2 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 1			Model 2
Determinant of	MEABRs	OAAs	MEABRs	OAAs	Sustainable
Hessian matrixes					scenario
A. Qualitative analysis
Di = 1	(-) 2b, < 0	(+) 2b, > 0	(-) P, < 0	(+) L, > 0	(-) P, < 0
Di = 2	(+) 4bϕ—a2, > 0	(+) 4bϕ—a2, > 0	(+) 2Pϕ—a2, > 0	(+) 2Lϕ—a2, > 0	(+) 2Lϕ—a2, > 0
Di = 3	-	-	(+) 4Pϕe - 2a2e, > 0	(+) 4Lϕe - 2a2e, > 0	(-) 4Lϕe + 2a2e, < 0
Di = 4	-	-	-	-	-
Di = 5	-	-	-	-	-
Hessian results for GFF	Rel. Max.	Rel. Min.	Saddle-point	Rel. Min.	Rel. Max.