Efficient nonanthropocentric nature protection

Abstract

This paper analyzes nature protection by a social planner under different ‘utilitarian’ social welfare functions. For that purpose we construct an integrated model of the economy and the ecosystem with explicit consideration of nonhuman species and with competition between human and nonhuman species for land and prey biomass. We characterize and compare the efficient allocations when social welfare is anthropocentric (only consumers have positive welfare weights), when social welfare is biocentric (only nonhuman species have positive welfare weights) and when social welfare is nonanthropocentric (all species have positive welfare weights). Not surprisingly, biocentric social welfare calls for suspending all economic activities. It is more important, however, that both anthropocentrism and nonanthropocentrism make the case for nature protection through different channels, though. Our analysis suggests that one may dispense with the concept of nonanthropocentric social welfare provided that in the anthropocentric framework the consumers' intrinsic valuation of nature is properly accounted for.

Appendices

Appendix A

Proof of Proposition 1

The associated first-order conditions to the Lagrangean (11) are

$${\user1{\mathcal{L}}}_{y} = \lambda _{y} - \lambda _{Y} \leqslant 0,\quad y{\user1{\mathcal{L}}}_{y} = 0,$$

(50a)

$${\user1{\mathcal{L}}}_{{y_{h} }} = \delta _{h} a_{h} n_{h} U_{y} - \lambda _{y} n_{h} \leqslant 0,\quad y_{h} {\user1{\mathcal{L}}}_{{y_{h} }} = 0,$$

(50b)

$${\user1{\mathcal{L}}}_{{r_{y} }} = - \delta _{h} a_{h} n_{h} U_{r} + \lambda _{Y} Y_{r} - \lambda _{r} \leqslant 0,\quad r_{y} {\user1{\mathcal{L}}}_{{r_{y} }} = 0,$$

(50c)

$${\user1{\mathcal{L}}}_{{z_{{yj}} }} = \lambda _{Y} Y_{{z_{{yj}} }} - \lambda _{{z_{j} }} \leqslant 0,\quad z_{{yj}} {\user1{\mathcal{L}}}_{{z_{{yj}} }} = 0,$$

(50d)

$${\user1{\mathcal{L}}}_{{r_{i} }} = n_{i} \lambda _{{b_{i} }} B^{i}_{r} - \lambda _{r} n_{i} \leqslant 0,\quad r_{i} {\user1{\mathcal{L}}}_{{r_{i} }} = 0,$$

(50e)

$${\user1{\mathcal{L}}}_{{z_{{ij}} }} = n_{i} \lambda _{{b_{i} }} B^{i}_{{z_{{ij}} }} - n_{i} \lambda _{{z_{j} }} \leqslant 0,\quad z_{{ij}} {\user1{\mathcal{L}}}_{{z_{{ij}} }} = 0,$$

(50f)

$${\user1{\mathcal{L}}}_{{b_{i} }} = \delta _{s} a_{i} n_{i} V^{i}_{b} + \mu _{{n_{i} }} n_{i} - n_{i} \lambda _{{b_{i} }} = 0,$$

(50g)

$$\begin{array}{*{20}c} {{\mathop \mu \limits^. }_{{n_{i} }} = \rho \mu _{{n_{i} }} - {\user1{\mathcal{L}}}_{{n_{i} }} = \rho \mu _{{n_{i} }} - \mu _{{n_{i} }} b_{i} + \lambda _{r} r_{i} + {\sum\limits_j {\lambda _{{z_{j} }} z_{{ij}} - \delta _{h} a_{h} n_{h} U_{{n_{i} }} } }} \\ { - \delta _{s} a_{i} {\left( {V^{i} + n_{i} V^{i}_{n} } \right)},} \\ \end{array} $$

(50h)

$$ \begin{array}{*{20}l} {{{\user1{\mathcal{L}}}_{{\lambda _{{b_{i} }} }} \geqslant 0,\lambda _{{b_{i} }} {\user1{\mathcal{L}}}_{{\lambda _{{b_{i} }} }} = 0,} \hfill} & {{{\user1{\mathcal{L}}}_{{\lambda _{y} }} \geqslant 0,\lambda _{y} {\user1{\mathcal{L}}}_{{\lambda _{y} }} = 0,} \hfill} & {{{\user1{\mathcal{L}}}_{{\lambda _{Y} }} \geqslant 0,\lambda _{Y} {\user1{\mathcal{L}}}_{{\lambda _{Y} }} = 0,} \hfill} \\ {{{\user1{\mathcal{L}}}_{{\lambda _{r} }} \geqslant 0,\lambda _{r} {\user1{\mathcal{L}}}_{{\lambda _{r} }} = 0,} \hfill} & {{{\user1{\mathcal{L}}}_{{\lambda _{{zj}} }} \geqslant 0,\lambda _{{z_{j} }} {\user1{\mathcal{L}}}_{{\lambda _{{zj}} }} = 0.} \hfill} & {{} \hfill} \\ \end{array} $$

(50i)

• (1) Set δ _h =0 and δ _s =1 in Eqs. 50–50i and consider a maximum of 10 in which r _i >0 for all i, z _ij >0 for all i, j, i≠j, and (hence) z _jj <0 for all i. In view of Eqs. 50e and 50f it is straightforward that either (a) λ _r >0 and therefore $\lambda _{{b_{i} }} > 0$ and $\lambda _{{z_{j} }} > 0$ or (b) λ _r =0 and therefore $\lambda _{{b_{i} }} = \lambda _{{z_{j} }} = 0$. Consider first the case (b). Raise $\overline{r} \;\,{\text{by}}\;\,{\text{d}}\,\overline{r} > 0$ and set $n_{i} {\text{d}}r_{i} = {\text{d}}\overline{r}$ and ${\text{d}}b_{i} = B^{i}_{r} {\text{d}}r_{i} > 0$. The impact of db _i >0 on social welfare is given by Eq. 50g where $a_{i} n_{i} V^{i}_{b} > 0$ is the direct impact on instantaneous welfare and where $\mu _{{n_{i} }}$ captures the shadow price of the population of species i or, more technically, the present value of the marginal future welfare induced by ${\text{d}}{\mathop n\limits^. }_{i} = n_{i} {\text{d}}b_{i} > 0$. Since the welfare index 9 is strictly increasing in n _i (for all i) $\mu _{{n_{i} }} > 0$ follows cogently. Yet $\lambda _{{b_{i} }} = 0$ and Eq. 50g imply $\mu _{{n_{i} }} = - a_{i} V^{i}_{b} < 0$ which is incompatible with a maximum of 10. This leads us to conclude that case (a) applies with $\mu _{{n_{i} }} = \lambda _{{b_{i} }} - a_{i} V^{i}_{b} > 0$.

  Suppose now case (a) holds but y>0, contrary to the assertion in Proposition 1i. Then λ _y =λ _Y ≥0 follows via Eq. 50a. If λ _y >0 then y _h =0 from Eq. 50b and therefore ${\user1{\mathcal{L}}}_{{\lambda _{y} }} > 0$. However, $\lambda _{y} {\user1{\mathcal{L}}}_{{\lambda _{y} }} = 0$ implies λ _y =0. This contradiction establishes λ _y =λ _Y =0. Moreover, Eq. 50c yields λ _Y Y _r −λ _r =−λ _r <0 and therefore r _y =0 via $r_{y} {\user1{\mathcal{L}}}_{{r_{y} }} = 0$. Since Y(0, z _y ) = 0 by assumption, y=0 follows from r _y =0. This contradiction proves that y=0 when 10 is maximized. y=0 and Eq. 6 yield y _h =0 for all h. Finally, r _y =0 and z _yj =0 for all j is the unique efficient way to produce y=0.

  In the remaining proof of (1) we establish Eqs. 12–15. Equation 12 follows from Eq. 50f. More precisely

  $$\frac{{n_{i} \lambda _{{b_{i} }} B^{i}_{{z_{{ij}} }} }} {{n_{i} \lambda _{{b_{i} }} B^{i}_{{z_{{ik}} }} }} = \frac{{\lambda _{{z_{j} }} }} {{\lambda _{{z_{k} }} }} = \frac{{n_{m} \lambda _{{b_{m} }} B^{m}_{{z_{{mj}} }} }} {{n_{m} \lambda _{{b_{m} }} B^{m}_{{z_{{mk}} }} }}\quad {\text{for}}\,i,j,k,m = 1, \ldots ,N.$$

  (51)

  Equation 13 is derived from Eqs. 50e and 50f:

  $$ \frac{{n_{i} \lambda _{{b_{i} }} B^{i}_{{z_{{ij}} }} }} {{n_{i} \lambda _{{b_{i} }} B^{i}_{{z_{{ik}} }} }} = \frac{{\lambda _{r} }} {{\lambda _{{z_{k} }} }} = \frac{{n_{m} \lambda _{{b_{m} }} B^{m}_{r} }} {{n_{m} \lambda _{{b_{m} }} B^{m}_{{z_{{mk}} }} }}\quad for\;i,j,j,m = 1, \ldots ,N. $$

  (52)

  Equation 14 follows from Eqs. 50e and 50g, and finally, using Eqs. 50c, 50e and 50g in Eq. 50h we obtain Eq. 15.

• (2) Set δ _s =0 and δ _h =1 in Eqs. 50a–50i. Now we restrict our attention to an interior solution such that all inequalities Eqs. 50a–50i hold as equalities. Equation 16 follows from Eqs. 50d and 50f. To establish Eq. 17 we combine Eqs. 50c–50f. To obtain Eq. 18 we combine Eqs. 50e and 50g, and using Eqs. 50c, 50e and 50g in Eq. 50h we get Eq. 19.

• (3) Here we set δ _h =δ _s =1 and make use of the same manipulations as in the derivation of Eqs. 16–19.

Appendix B: Comparative dynamics in the one-species model

Derivation of Eqs. 36a–36c: Total differentiation of Eqs. 35a–35c yields:

$$J \cdot {\left( {\begin{array}{*{20}l} {{{\text{d}}r} \hfill} \\ {{{\text{d}}z} \hfill} \\ {{{\text{d}}n} \hfill} \\ \end{array} } \right)} = {\left( {\begin{array}{*{20}l} {0 \hfill} \\ {{ - {\left( {{\text{d}}\beta W_{n} + {\text{d}}a{\left( {V + nV_{n} + \rho V_{b} } \right)}} \right)}B_{r} } \hfill} \\ {{ - {\left( {{\text{d}}\beta W_{n} + {\text{d}}a{\left( {V + nV_{n} + \rho V_{b} } \right)}} \right)}B_{z} } \hfill} \\ \end{array} } \right)}$$

(53)

where

$$ J: = {\left( {\begin{array}{*{20}c} {{B_{r} }} & {{B_{z} }} & {0} \\ {{nY_{{rr}} + \lambda B_{{rr}} + \phi _{1} Br}} & {{ - nY_{{rz}} + \lambda B_{{rz}} + \phi _{2} B_{r} }} & {{ - zY_{{rz}} + rY_{{rr}} + \phi _{3} B_{r} }} \\ {{ - nY_{{zr}} + \lambda B_{{zr}} + \phi _{1} B_{z} }} & {{nY_{{zz}} + \lambda B_{{zz}} + \phi _{2} B_{z} }} & {{ - zY_{{rz}} + zY_{{zz}} + \phi _{3} B_{z} }} \\ \end{array} } \right)} $$

and

$$ \begin{array}{*{20}c} {\lambda = {\left( {\beta W_{n} + a{\left( {V + nV_{n} + \rho V_{b} } \right)} - rY_{r} + zY_{z} } \right)}/\rho ,} \\ {\phi _{1} : = {\left( { - Y_{r} + nrY_{{rr}} - nzY_{{zr}} } \right)}/\rho _{1} ,} \\ {\phi _{2} : = {\left( {Y_{z} - nrY_{{zr}} + nzY_{{zz}} } \right)}/\rho _{1} ,} \\ {\phi _{3} = {\left( {\beta W_{{nn}} + a{\left( {2V_{n} + nV_{{nn}} } \right)} + r^{2} Y_{{rr}} + z^{2} Y_{{zz}} - 2rzY_{{rz}} } \right)}/\rho .} \\ \end{array} $$

Solving the equation system 53 by using Cramer's rule we obtain

$$ {\text{d}}r = \frac{{{\text{d}}\beta W_{n} + da{\left( {V + nV_{n} + \rho V_{b} } \right)}}} {{{\left| J \right|}}} \cdot B_{z} \cdot {\left[ {{\left( {zY_{{zz}} - zY_{{rz}} } \right)}B_{r} - {\left( {rY_{{rr}} - zY_{{zr}} } \right)}B_{z} } \right]}, $$

(54a)

$${\text{d}}z = \frac{{{\text{d}}\beta W_{n} + da{\left( {V + nV_{n} + \rho V_{b} } \right)}}}{{{\left| J \right|}}} \cdot B_{r} \cdot {\left[ {{\left( {zY_{{rz}} - zY_{{zz}} } \right)}B_{r} - {\left( {zY_{{zr}} - rY_{{rr}} } \right)}B_{z} } \right]},$$

(54b)

$$dn = \frac{{{\text{d}}\beta W_{n} + {\text{d}}a{\left( {V + nV_{n} + \rho V_{b} } \right)}}}{{{\left| J \right|}}} \cdot {\left[ {B^{2}_{r} {\left( {nY_{{zz}} + \lambda B_{{zz}} } \right)} + B^{2}_{z} {\left( {nY_{{rr}} + \lambda B_{{zr}} } \right)} - 2B_{r} B_{z} {\left( { - nY_{{rz}} + \lambda B_{{rz}} } \right)}} \right]}.$$

(54c)

Derivation of Eqs. 37a–37d: While Eqs. 37a, 37c and 37d follow directly from total differentiation of Eq. 24, nr and nz, respectively, totally differentiating $Y{\left( {\overline{r} - nr,nz} \right)}$ yields

$$ {\text{d}}y = nY_{z} {\text{d}}z - nY_{r} {\text{d}}r + {\left( {zY_{z} - Y_{r} } \right)}{\text{d}}n. $$

(55)

B _r =Y _r /λ and B _z =−Y _z /λ from Eqs. 34a and 34b, respectively, inserted into B _r dr+B _z d _z =0 yields Y _r dr−Y _z d _z =0 such that Eq. 55 simplifies to Eq. 37b.

Proof of Proposition 2

Model A: The results in lines 1 and 5 of Table 1 follow directly from Eqs. 36a–36c and 37a–37d by taking into account that ∣J∣<0, B _r >0, B _z <0, Y _rr <0, Y _zz <0, B _rz ≤0, Y _rz ≥0 and λ>0.

Model B: In case of the Leontief net offspring function $b = \min {\left[ {r,c{\left( {\overline{z} - z} \right)}} \right]} - \varepsilon$ efficiency requires $r = c{\left( {\overline{z} - z} \right)}$ such that Eq. 33 turns into

$${\user1{\mathcal{H}}} = \beta W{\left( n \right)} + Y{\left[ {\overline{r} - nr,n{\left( {\overline{z} - \frac{r} {c}} \right)}} \right]} + anV{\left( {n,r - \varepsilon } \right)} + \mu n{\left( {r - \varepsilon } \right)}$$

(56)

and we obtain the first-order conditions

$$ {\user1{\mathcal{H}}}_{r} = - nY_{r} - n\frac{{Y_{z} }} {c} + anV_{b} + n\mu = 0, $$

(57a)

$${\mathop \mu \limits^. } = \rho \mu - \beta W_{n} - a{\left( {V + nV_{n} } \right)} + rY_{r} - {\left( {\overline{z} - \frac{r}{c}} \right)}Y_{z} - \mu {\left( {r - \varepsilon } \right)}.$$

(57b)

Then the long-run equilibrium is characterized by nb=0 which implies $r = c{\left( {\overline{z} - z} \right)} = \varepsilon$ and by

$$ \beta W_{n} + a{\left( {V + nV_{n} + \rho V_{b} } \right)} - {\left( {r + \rho } \right)}Y_{r} + {\left( {z - \frac{\rho } {c}} \right)}Y_{z} = 0. $$

(58)

Since the long-run levels of r and z are constant Eq. (58) determines the long-run level of n. Implicit differentiation of Eq. (58) yields

$$ \frac{{{\text{d}}n}} {{{\text{d}}\beta }} = - \frac{{W_{n} }} {{\widetilde{J}}}\quad {\text{and}}\;\;\frac{{{\text{d}}n}} {{{\text{d}}a}} = - \frac{{V + nV_{n} + \rho V_{b} }} {{\widetilde{J}}} $$

(59)

where $\widetilde{J}: = \beta W_{{nn}} + a{\left( {2V_{n} + nV_{{nn}} } \right)} + r{\left( {r + \rho } \right)}Y_{{rr}} + z{\left( {z - \frac{\rho } {c}} \right)}Y_{{zz}} - {\left( {2rz + \rho z - \frac{{\rho r}} {c}} \right)}Y_{{rz}}$ is assumed to be negative (stability of the equilibrium) implying that dn/dbeta;>0 and dn/da>0. Since dr/dβ=dr/da=dz/dβ=dz/da=0, the results in lines 2 and 6 are straightforward.

Model C: In case of $u = \min {\left[ {\beta W{\left( n \right)},Y{\left( {\overline{r} - nr,nz} \right)}} \right]}$ the social planner's objective is to maximize

$${\user1{\mathcal{L}}} = \beta W{\left( n \right)} + anV{\left( {n,b} \right)} + \mu nb + \lambda {\left[ {\beta W{\left( n \right)} - Y{\left( {\overline{r} - nr,nz} \right)}} \right]} + \lambda _{b} n{\left[ {B{\left( {r,z} \right)} - b} \right]}$$

(60)

and the first-order conditions read

$$ {\user1{\mathcal{L}}}_{r} = \lambda _{b} nB_{r} + \lambda nY_{r} = 0, $$

(61a)

$$ {\user1{\mathcal{L}}}_{z} = \lambda _{b} nB_{z} - \lambda nY_{z} = 0, $$

(61b)

$$ {\user1{\mathcal{L}}}_{b} = anV_{b} + n\mu - n\lambda _{b} = 0, $$

(61c)

$${\user1{\mathcal{L}}}_{\lambda } = \beta W{\left( n \right)} - Y{\left( {\overline{r} - nr,nz} \right)} = 0,$$

(61d)

$${\mathop \mu \limits^. } = \rho \mu - \beta W_{n} - a{\left( {V + nV_{n} } \right)} + \lambda {\left( {zY_{z} - rY_{r} - \beta W_{n} } \right)} - \mu b.$$

(61e)

Observe that the efficient long-run allocation (r, z, n) is characterized by

$$n_{B} {\left( {r,z} \right)} = 0,$$

(62a)

$$\beta W{\left( n \right)} - Y{\left( {\overline{r} - nr,nz} \right)} = 0,$$

(62b)

$$B_{r} Y_{z} + B_{z} Y_{r} = 0.$$

(62c)

It is interesting to note that Eqs. 62a–62c are independent of the parameter a. Total differentiation of 62a–62c yields¹⁷

$$ {\left( {\begin{array}{*{20}c} {{B_{r} }} & {{B_{z} }} & {0} \\ {0} & {0} & {{\beta W_{n} - rY_{r} + zY_{z} }} \\ {{\phi _{1} }} & {{\phi _{5} }} & {{\phi _{6} }} \\ \end{array} } \right)} \cdot {\left( {\begin{array}{*{20}c} {{{\text{d}}r}} \\ {{{\text{d}}z}} \\ {{{\text{d}}n}} \\ \end{array} } \right)} = {\left( {\begin{array}{*{20}c} {0} \\ {{ - {\text{d}}\beta W}} \\ {0} \\ \end{array} } \right)} $$

(63)

where

$$\begin{array}{*{20}c} {\phi _{1} : = B_{{rr}} Y_{z} - B_{r} Y_{{zr}} n + B_{{zr}} Y_{r} - B_{z} Y_{{rr}} n < 0,} \\{\phi _{5} : = B_{{rz}} Y_{z} + B_{r} Y_{{zz}} n + B_{{zz}} Y_{r} + B_{z} Y_{{zr}} n < 0,} \\{\phi _{6} : = B_{r} {\left( {zY_{{zz}} - rY_{{zr}} } \right)} + B_{z} {\left( {zY_{{rz}} - rY_{{rr}} } \right)} < 0.} \\\end{array}$$

Solving the equation system 63 we obtain

$$ \frac{{{\text{d}}r}} {{{\text{d}}\beta }} = \frac{{B_{z} W\phi _{6} }} {{{\left| {\overline{J} } \right|}}} < 0,\frac{{{\text{d}}z}} {{{\text{d}}\beta }} = - \frac{{B_{z} W\phi _{6} }} {{{\left| {\overline{J} } \right|}}} < 0,\frac{{{\text{d}}n}} {{{\text{d}}\beta }} = \frac{{W{\left( {B_{r} \phi _{5} - B_{z} \phi _{4} } \right)}}} {{{\left| {\overline{J} } \right|}}} > 0, $$

(64)

where ${\left| {\overline{J} } \right|}$ is assumed to be negative. Finally we differentiate u=βW(n)=y to get

$$du = Wd\beta + \beta W_{n} dn = dy$$

(65)

which establishes du/dβ>0 and dy/dβ>0.

Model D: Now suppose u and b are specified as $b = \min {\left[ {r,c{\left( {\overline{z} ,z} \right)}} \right]} - \varepsilon$ and $u = \min {\left[ {\beta W{\left( n \right)},Y{\left( {\overline{r} - nr,nz} \right)}} \right]}$. Then it is easy to see that the long-run allocation is determined by $r = c{\left( {\overline{z} - z} \right)} = \varepsilon$ and

$$\beta W{\left( n \right)} - Y{\left( {\overline{r} - nr,nz} \right)} = 0.$$

(66)

Implicit differentiation of Eq. 66 yields

$$\frac{{dn}} {{d\beta }} = - \frac{W} {{\beta W_{n} + rY_{r} - zY_{z} }} > 0,$$

(67)

since βW _n +rY _r −zY _z is assumed to be negative. Summarizing, we have dr/dβ=0, and dz/dβ=0, d(rn)/dβ>0, d(zn)/dβ>0, and du/dβ=dy/dβ=W+β W _n (dn/dβ)>0.

Appendix C: Comparative dynamics in the two-species model

First, observe that

$$B^{2}_{z} dz_{{21}} = 0.$$

(68)

Next, total differentiation of Eqs. 49a, 49c–49e leads to

$$ \widetilde{J} \cdot {\left( {\begin{array}{*{20}c} {{{\text{d}}r}} \\ {{{\text{d}}z_{{11}} }} \\ {{{\text{d}}n_{1} }} \\ {{{\text{d}}n_{2} }} \\ \end{array} } \right)} = {\left( {\begin{array}{*{20}c} {0} \\ {0} \\ {{ - {\text{d}}\beta W_{{n_{1} }} B^{1}_{r} - {\text{d}}a_{1} {\left( {V^{1} + n_{1} V^{1}_{n} + \rho V^{1}_{b} } \right)}B^{1}_{r} + {\text{d}}\gamma {\left( {rB^{1}_{r} + \rho + z_{{11}} B^{1}_{z} } \right)}}} \\ {{ - {\text{d}}\beta W_{{n_{2} }} B^{1}_{r} - {\text{d}}a_{2} {\left( {V^{2} + n_{2} V^{2}_{n} + \rho V^{2}_{b} } \right)}B^{1}_{r} + {\text{d}}\gamma {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)}}} \\ \end{array} } \right)} $$

(69)

where

$$ \widetilde{J}: = {\left( {\begin{array}{*{20}c} {{B^{1}_{r} }} & {{B^{1}_{z} }} & {0} & {0} \\ {0} & {{n_{1} }} & {{z_{{11}} }} & {{ - z_{{21}} }} \\ {{\phi _{7} B^{1}_{{rr}} - \gamma B^{1}_{r} }} & {{ - \gamma B^{1}_{z} - \gamma z_{{11}} B^{1}_{{zz}} }} & {{{\left[ {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right]}B^{1}_{r} }} & {0} \\ {{\phi _{8} B^{1}_{{rr}} }} & {{\gamma B^{1}_{{zz}} {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)}}} & {0} & {{{\left[ {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right]}B^{1}_{r} }} \\ \end{array} } \right)} $$

and

$$ \phi _{7} : = \beta W_{{n_{1} }} + a_{1} {\left( {V^{1} + n_{1} V_{{n_{1} }} + \rho V^{1}_{b} } \right)} - \gamma r < 0\;{\text{and}}\;\phi _{8} : = \beta W_{{n2}} + a_{2} {\left( {V^{2} + n_{2} V^{2}_{n} + \rho V^{2}_{b} } \right)} > 0. $$

Then solving Eq. 69 by Cramer's rule gives

$$ \frac{{{\text{d}}r}} {{{\text{d}}a_{1} }} = - \frac{1} {{{\left| {\widetilde{J}} \right|}}}z_{{11}} {\left( {B^{1}_{r} } \right)}^{2} B^{1}_{z} {\left( {V^{1} + n_{1} V^{1}_{n} } \right)}{\left[ {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right]}, $$

(70)

$$ \frac{{dz_{{11}} }} {{da_{1} }} = \frac{1} {{{\left| {\widetilde{J}} \right|}}}z_{{11}} {\left( {B^{1}_{r} } \right)}^{3} {\left( {V^{1} + n_{1} V^{1}_{n} } \right)}{\left[ {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right]}, $$

(71)

$$ \frac{{{\text{d}}n_{1} }} {{{\text{d}}a_{1} }} = - \frac{{{\left( {V^{1} + n_{1} V^{1}_{n} } \right)}B^{1}_{r} }} {{{\left| {\widetilde{J}} \right|}}}{\left( {B^{1}_{r} } \right)}^{2} n_{1} {\left[ {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right]} + z_{{21}} \gamma B^{1}_{r} B^{1}_{{zz}} {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)} - z_{{21}} \phi _{8} B^{1}_{z} B^{1}_{{rr}} , $$

(72)

$$ \frac{{{\text{d}}n_{2} }} {{{\text{d}}a_{1} }} = \frac{{{\left( {V^{1} + n_{1} V^{1}_{n} } \right)}B^{1}_{r} }} {{{\left| {\widetilde{J}} \right|}}} - z_{{11}} \gamma B_{r} B^{1}_{{zz}} {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)} + z_{{11}} \phi _{8} B^{1}_{z} B^{1}_{{rr}} , $$

(73)

$$ \frac{{{\text{d}}r}} {{{\text{d}}a_{2} }} = \frac{1} {{{\left| {\widetilde{J}} \right|}}}z_{{21}} {\left( {B^{1}_{r} } \right)}^{2} B^{1}_{z} {\left( {V^{2} + n_{2} V^{2}_{n} } \right)}{\left[ {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right]}, $$

(74)

$$ \frac{{{\text{d}}z_{{11}} }} {{{\text{d}}a_{2} }} = - \frac{1} {{{\left| {\widetilde{J}} \right|}}}z_{{21}} {\left( {B^{1}_{r} } \right)}^{3} {\left( {V^{2} + n_{2} V^{2}_{n} } \right)}{\left[ {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right]}, $$

(75)

$$ \frac{{{\text{d}}n_{1} }} {{{\text{d}}a_{2} }} = - \frac{{{\left( {V^{2} + n_{2} V^{2}_{n} } \right)}B^{1}_{r} }} {{{\left| {\widetilde{J}} \right|}}}{\left\lceil {\gamma z_{{11}} B^{1}_{{zz}} + z_{{21}} \phi _{7} B^{1}_{z} B^{1}_{{rr}} } \right\rceil }, $$

(76)

$$ \frac{{{\text{d}}n_{2} }} {{{\text{d}}a_{2} }} = - \frac{{{\left( {V^{2} + n_{2} V^{2}_{n} } \right)}B^{1}_{r} }} {{{\left| {\widetilde{J}} \right|}}}{\left\lceil {{\left( {B^{1}_{r} } \right)}^{2} n_{1} {\left( {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right)} + \gamma {\left( {z_{{11}} } \right)}^{2} B^{1}_{{zz}} + z_{{11}} \phi _{7} B^{1}_{z} B^{1}_{{rr}} } \right\rceil }, $$

(77)

$$ \frac{{{\text{d}}r}} {{{\text{d}}\beta }} = - \frac{{{\left( {B^{1}_{r} } \right)}^{2} B^{1}_{z} }} {{{\left| {\widetilde{J}} \right|}}}{\left[ {W_{{n_{1} }} z_{{11}} {\left( {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right)} - W_{{n_{2} }} z_{{21}} {\left( {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right)}} \right]}, $$

(78)

$$ \frac{{{\text{d}}z_{{11}} }} {{{\text{d}}\beta }} = \frac{{{\left( {B^{1}_{r} } \right)}^{3} }} {{{\left| {\widetilde{J}} \right|}}}{\left\lceil {W_{{n_{1} }} z_{{11}} {\left( {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right)} - W_{{n_{2} }} z_{{21}} {\left( {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right)}} \right\rceil }, $$

(79)

$$ \frac{{{\text{d}}n_{1} }} {{{\text{d}}\beta }} = \frac{{B^{1}_{r} }} {{{\left| {\widetilde{J}} \right|}}}{\left[ { - W_{{n_{1} }} {\left\{ {{\left( {B^{1}_{r} } \right)}^{2} n_{i} {\left[ {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right]} + z_{{21}} \gamma B^{1}_{r} B^{1}_{{zz}} {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)} - z_{{21}} \phi _{8} B^{1}_{z} B^{1}_{{rr}} } \right\}} - W_{{n_{2} }} {\left\{ {z_{{11}} \gamma B^{1}_{r} B^{1}_{{zz}} + z_{{21}} \phi _{7} B^{1}_{z} B^{1}_{{rr}} } \right\}}} \right]}, $$

(80)

$$\frac{{{\text{d}}n_{2} }}{{{\text{d}}\beta }} = - \frac{{B^{1}_{r} }}{{{\left| {\widetilde{J}} \right|}}}{\left[ {W_{{n1}} {\left\{ {z_{{11}} \gamma B^{1}_{r} B^{1}_{{zz}} {\left( {z_{{21}} + \frac{\rho }{{B^{2}_{z} }}} \right)} - z_{{11}} \phi _{8} B^{1}_{z} B^{1}_{{rr}} } \right\}} + W_{{n_{2} }} {\left\{ {{\left( {B^{1}_{r} } \right)}^{2} n_{1} {\left[ {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right]} + {\left( {z_{{11}} } \right)}^{2} \gamma B^{1}_{r} B^{1}_{{zz}} + z_{{11}} \phi _{7} B^{1}_{z} B^{1}_{{rr}} } \right\}}} \right]},$$

(81)

$$\frac{{{\text{d}}r}}{{{\text{d}}\gamma }} = \frac{{B^{1}_{r} B^{1}_{z} }}{{{\left| {\widetilde{J}} \right|}}}{\left[ {{\left( {rB^{1}_{r} + \rho + z_{{11}} B^{1}_{z} } \right)}z_{{11}} {\left( {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right)} - {\left( {z_{{21}} + \frac{\rho }{{B^{2}_{z} }}} \right)}z_{{21}} {\left( {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right)}} \right]}$$

(82)

$$ \frac{{{\text{d}}z_{{11}} }} {{{\text{d}}\gamma }} = \frac{{{\left( {B^{1}_{r} } \right)}^{2} }} {{{\left| {\widetilde{J}} \right|}}}{\left[ { - {\left( {rB^{1}_{r} + \rho + z_{{11}} B^{1}_{z} } \right)}z_{{11}} {\left( {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right)} + {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)}z_{{21}} {\left( {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right)}} \right]}, $$

(83)

$$ \frac{{{\text{d}}n_{1} }} {{{\text{d}}\gamma }} = \frac{1} {{{\left| {\widetilde{J}} \right|}}}{\left[ {{\left( {rB^{1}_{r} + \rho + z_{{11}} B^{1}_{z} } \right)}{\left\{ {n_{1} {\left( {B^{1}_{r} } \right)}^{2} {\left( {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right)} + z_{{21}} \gamma B^{1}_{r} B^{1}_{{zz}} {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)} - z_{{21}} \phi _{8} B^{1}_{z} B^{1}_{{rr}} } \right\}} + {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)}{\left( {z_{{11}} z_{{21}} \gamma B^{1}_{r} B^{1}_{{zz}} + z_{{21}} \phi _{7} B^{1}_{z} B^{1}_{{rr}} } \right)}} \right]}, $$

(84)

$$ \frac{{{\text{d}}n_{2} }} {{{\text{d}}\gamma }} = \frac{1} {{{\left| {\widetilde{J}} \right|}}}{\left[ {{\left( {rB^{1}_{r} + \rho + z_{{11}} B^{1}_{z} } \right)}{\left\{ {z_{{11}} \gamma B^{1}_{r} B^{1}_{{zz}} {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)} - z_{{11}} \phi _{8} B^{1}_{z} B^{1}_{{rr}} } \right\}} + {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)}{\left\{ {n_{1} {\left( {B^{1}_{r} } \right)}^{2} {\left( {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right)} + {\left( {z_{{11}} } \right)}^{2} \gamma B^{1}_{r} B^{1}_{{zz}} + z_{{11}} \phi _{7} B^{1}_{z} B^{1}_{{rr}} } \right\}}} \right]}, $$

(85)

where

$$ {\left| {\widetilde{J}} \right|} = B^{1}_{r} {\left\{ {n_{1} {\left[ {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right]}{\left[ {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right]}{\left( {B^{1}_{r} } \right)}^{2} + {\left[ {z_{{11}} {\left( {\gamma B^{1}_{z} + \gamma z_{{11}} B^{1}_{{zz}} } \right)} + z_{{21}} \gamma B^{1}_{{zz}} {\left( {z_{{21}} + \frac{\rho } {{B^{2}_{z} }}} \right)}} \right]}{\left[ {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right]}B^{1}_{r} } \right\}} - B^{1}_{z} {\left\{ { - z_{{11}} {\left( {\phi _{7} B^{1}_{{rr}} - \gamma B^{1}_{r} } \right)}{\left[ {\beta W_{{n_{2} n_{2} }} + a_{2} {\left( {2V^{2}_{n} + n_{2} V^{2}_{{nn}} } \right)}} \right]}B^{1}_{r} + z_{{21}} \phi _{8} B^{1}_{{rr}} {\left[ {\beta W_{{n_{1} n_{1} }} + a_{1} {\left( {2V^{1}_{n} + n_{1} V^{1}_{{nn}} } \right)}} \right]}B^{1}_{r} } \right\}}. $$

(86)

The results of Table 2 follow from Eqs. 70–86 and the properties of the functions W, B ¹, B ², V ¹ and V ². In addition, we assume $\beta W_{{n_{i} n_{i} }} + a_{i} {\left( {2V^{i}_{n} + ^{i}_{{nn}} } \right)} < 0$ for i=1,2 and ϕ₇ B _rr ¹−γB _r ¹<0 which are sufficient for the negative definiteness of $\widetilde{J}{\left( {{\left| {\widetilde{J}} \right|} > 0} \right)}$.