Correction: Yang et al. Operational Decisions on Remanufacturing under the Product Innovation Race. Sustainability 2023, 15, 4920

The authors would like to make the following corrections about the published paper [1]. The changes are as follows:

(1)

Replacing the “Remanufacturing Scenario” in Table 3.

Remanufacturing Scenario

(i) For $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}<c_r<\frac{\gamma c_n}{1+a}$,

Equilibrium decisions: $q_1^{{Ri}^{\ast }}=\frac{1-c_n-bc}{2}$, $q_n^{{Ri}^{\ast }}=\frac{1+a-c_n+c_r-\gamma }{2\left(1+a-\gamma \right)}$, $q_r^{{Ri}^{\ast }}=\frac{\gamma c_n-c_r-a{c}_r}{2\gamma (1+a-\gamma )}$, $K^{Ri}=\frac{\rho (1+a-2\gamma +2c_n{c}_r-a\gamma -c_n^2-c_r^2+{\gamma }^2)}{2\left(a+2\right)\left(1-\gamma \right)\left(1+a-\gamma \right)}$.

Equilibrium profits:

${\pi }_T^{{Ri}^{\ast }}=\frac{\left(\begin{array}{l}2\rho cb\gamma c_n-2\rho \gamma c_n-2\rho cb{\gamma }^2{c}_n+2\rho c^2{b}^2\gamma -2\gamma c_n+\rho c_r^2\\ +a\gamma c_n^2-2a\gamma c_n-2\rho cb\gamma -b^2{c}^2\gamma -b^2{c}^{2}a\gamma +b^2{c}^2{\gamma }^2+\gamma \\ +\rho \gamma a^2-\rho {\gamma }^{2}a+2\rho {\gamma }^2{c}_n-{\gamma }^2{c}_n^2+2\rho \gamma a-2\rho \gamma a{c}_n+a\gamma \\ +\rho \gamma c_n^2+\rho c{\gamma }^{2}a+2\rho cb{\gamma }^2+2\rho cb\gamma a{c}_n+\rho \gamma +2\rho c^2{b}^2\gamma a\\ +\gamma c_n^2-2\rho c^2{b}^2{\gamma }^2-\rho {\gamma }^2-2\rho cb\gamma a+2{\gamma }^2{c}_n-{\gamma }^2-2\rho \gamma c_n{c}_r\end{array}\right)}{4\gamma \left(a-\gamma +1\right)}-K\left({\delta }_1^2-1\right)/2$.

(ii) For $\frac{\gamma c_n}{1+a}<c_r<\frac{1+a-c_n-c_{n}a+b\gamma c_n-cb-cba}{b(1+a)}$,

Equilibrium decisions: $q_1^{Rii}=\frac{1+a-c_n-c_{n}a+c_n\gamma b-c_{r}b-ba{c}_r-cb-cba}{2(1+a-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a)}$, $q_n^{{Rii}^{\ast }}=\frac{1+a+\gamma b^{2}a+c_n\gamma b-\gamma b+\gamma b^2+\gamma b^2{c}_r+\gamma b^{2}c-c_n-\gamma c_n{b}^2-{\gamma }^2{b}^2}{2(1+a-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a)}$, $q_r^{Rii\ast }=b{q}_1^{Rii}$, $K^{Rii}=\frac{\rho \left[\begin{array}{l}1+a+2{\gamma }^2{b}^2{c}_n+{\gamma }^4{b}^4-2{\gamma }^3{b}^4+{\gamma }^2{b}^4+2c{b}^3{\gamma }^2\\ -{\gamma }^3{b}^{4}a+{\gamma }^2{b}^{4}a-2c_n{\gamma }^2{b}^3-c_n^2{\gamma }^2{b}^2-c_r^2{b}^4{\gamma }^2\\ -c^2{b}^4{\gamma }^2+2c_n^2{\gamma }^2{b}^3-c_n^2{\gamma }^2{b}^4+2c_r{b}^3{\gamma }^2-{\gamma }^2{b}^{2}a\\ +2\gamma b^{2}a-2c_n\gamma b-c_n^2-3{\gamma }^2{b}^2+2\gamma b^2+2c_n^2\gamma b\\ +2c_n\gamma b^2{c}_r+2c_n\gamma b^{2}c-2c_n^2\gamma b^2-2c_r{b}^4{c}_r^2\\ +2c_n{\gamma }^2{b}^4{c}_r+2c_n{\gamma }^2{b}^{4}c-2c_n{c}_r{b}^3{\gamma }^2-2c_{n}c{b}^3{\gamma }^2\end{array}\right]}{2\left(a+2\right)\left(1-{\gamma }^2{b}^2+\gamma b^2\right)\left(1+a-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a\right)}$.

Equilibrium profits:

${\pi }_T^{{Rii}^{\ast }}={\pi }_1^{{Rii}^{\ast }}+{\pi }_2^{{Rii}^{\ast }}=\left(p_1-c_n\right)q_1+\left(p_n-c_n\right)q_n+\left(p_r-c_r\right)q_r-cb{q}_1-K\left({\left(1+a\right)}^2-1\right)/2$.

(2)

Replacing the statement of Proposition 3.

Proposition 3.

The profits under the remanufacturing scenario are higher than that of no remanufacturing (i.e., ${\pi }_T^{R\ast }>{\pi }_T^{N\ast }$) if, and only if, the cost of $c_r$ is not pronounced (i.e., $c_r<c_r^{\Delta }$); otherwise, the opposite is true.

(3)

Replacing the statement of Proposition 5.

Proposition 5.

Under the partial remanufacturing scenario, the total environmental impact increases with the remanufacturing cost, that is, $\partial E^R/\partial c_r>0$; however, under the full remanufacturing scenario, the opposite is true, that is, $\partial E^R/\partial c_r<0$.

(4)

Replacing the statement of Proposition 6.

Proposition 6.

The threshold values of $c_r^{\Delta }$ increase with the collection target of $b$, i.e., $\partial c_r^{\Delta }/\partial b>0$; moreover, the equilibrium quantities $\partial q_1/\partial b<0$.

(5)

Replacing the “Analysis of the Scenario with Remanufacturing” in Appendix A.

Applying backward, starting from Period 2 again. In Stage 1 of Period 2, the manufacturer sets ($q_n,q_r$) to maximize the profit in Period 2. That is,

$${\mathrm{\Pi }}_2^R=\left(p_n-c_n\right)q_n+\left(p_r-c_r\right)q_r-cb{q}_1$$

subject to $q_r\le b{q}_1$, and $q_n,q_r>0$.

To maximize the problems with constraints, like [52] and [41], we write the above problem as following the Lagrangean:

$$L=\left(p_n-c_n\right)q_n+\left(p_r-c_r\right)q_r-cb{q}_1+\lambda (b{q}_1-q_r)$$

Using the KKT optimality conditions:

$$\begin{array}{c}\frac{\partial L}{\partial q_n}=1+a-2q_n-2q_{n}a-2\gamma q_r-c_n\\ \frac{\partial L}{\partial q_r}=\gamma -2\gamma q_n-2\gamma q_r-c_r+\lambda \\ \lambda (b{q}_1-q_r)=0\end{array}$$

The above equations have two solutions. They depend on whether $\lambda$ are greater than or equal to zero.

(i) $\lambda =0$. Solving the gradient conditions with $\lambda =0$, we have

$$q_n=\frac{1+a-c_n-\gamma +c_r}{2(1+a-\gamma )},q_r=\frac{\gamma c_n-a{c}_r-c_r}{2\gamma (1+a-\gamma )}$$

(1)

(ii) $\lambda >0$. Solving the system, we obtain

$$q_n=\frac{1+a-2\gamma b{q}_1-c_n}{2\left(1+a\right)},q_r=b{q}_1,\lambda =\frac{2\gamma b{q}_1-2r^{2}b{q}_1+2\gamma b{q}_{1}a-\gamma c_n+c_r+a{c}_r}{(1+a)}$$

(2)

(6)

Replacing the “Appendix B. Proof of Proposition 1”.

(i) For $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}=\underset{¯}{c_r}<c_r<{\overline{c}}_r=\frac{\gamma c_n}{1+a}$,

Solving $K^{Ri}-K^N=\frac{\rho (a{c}_r^2-2c_r{c}_{n}a+c_n^2\gamma a-2c_r{c}_n-c_n^2{\gamma }^2+2c_n^2\gamma +c_r^2)}{2(a^2+3a+2)(\gamma -1)(1+a-\gamma )}$, we find that whether $K^{Ri}-K^N>0$ depends on the numerator of the above equation. Then, let $t=a{c}_r^2-2c_r{c}_{n}a+c_n^2\gamma a-2c_r{c}_n-c_n^2{\gamma }^2+2c_n^2\gamma +c_r^2$, we can obtain:

$c_{r1}=\frac{c_n+a{c}_n+c_n\sqrt{-\left(a+1\right)\left(\gamma -1\right)\left(a-\gamma +1\right)}}{a+1},$ $c_{r2}=\frac{c_n+a{c}_n-c_n\sqrt{-\left(a+1\right)\left(\gamma -1\right)\left(a-\gamma +1\right)}}{a+1}$. Comparing $c_{r1}$, $c_{r2}$ $\underset{¯}{c_r}$ and ${\overline{c}}_r$, we find that $\underset{¯}{c_r}<{\overline{c}}_r<c_{r1}<c_{r2}$, which means $K^N>K^{Ri}$.

(ii) For $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}=\underset{¯}{c_r}<c_r<{\overline{c}}_r=\frac{\gamma c_n}{1+a}$, solving $K^{Rii}-K^N=\frac{\rho \left[\begin{array}{l}2c_n^2{\gamma }^2{b}^3+2\gamma c_n{b}^2{c}_r+2\gamma c_n{b}^{2}c+3c_n^2\gamma b+2\gamma c_n{b}^2-2\gamma b^{2}ca\\ -2\gamma b{c}_{n}a+c_n^2{\gamma }^2{b}^{3}a+2c_n^2\gamma ba-2c_n^2\gamma b^{2}a-2c_n\gamma b^{3}c-2c_r{b}^2\gamma a\\ +2c_n\gamma b^{2}a+c_r^2{b}^3\gamma a+c^2{b}^3\gamma a+2c_r{b}^{3}c\gamma -2c_n\gamma b^3{c}_r+2\gamma c_n{b}^2{c}_{r}a\\ +2\gamma c_n{b}^{2}ca-2c_r{b}^2\gamma -c_n^2{\gamma }^3{b}^3+c_r^2{b}^3\gamma +c^2{b}^3\gamma -2c_{n}b{c}_{r}a-2c_n^2\gamma b^2\\ -2c_{n}bca-2c_n\gamma b^{3}ca+2c_r{b}^{3}c\gamma a-2c_n\gamma b^3{c}_{r}a+2c_n+2c_{n}a\\ -2c_n^{2}a-2c_{n}bc-2b\gamma c_n-2\gamma b^{2}c-2c_{n}b{c}_r+b\gamma -2c_n^2+\gamma ba\end{array}\right]}{2\left(a+2\right)\left(1-{\gamma }^2{b}^2+\gamma b^2\right)\left(1+a-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a\right)}$ we find that whether $K^{Rii}-K^N>0$ depends on the numerator of the above equation. Then, similar to scenario (i), we can yield the roots of:

$$c_{r1}=\frac{\left[\begin{array}{l}2b\gamma -2b\gamma c_n-2\gamma b^{2}c-2\gamma b^{2}ca+2\gamma c_n{b}^2-2\gamma b{c}_{n}a+2c_n\gamma b^{2}a+2c_{n}a+2c_n\\ +2\gamma ba+2{(c_n^2(1-{\gamma }^2{b}^2+\gamma b^2)(1+a)(1+a-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a))}^{1/2}\end{array}\right]}{2(b\gamma +\gamma ba)b}$$

$$c_{r2}=\frac{\left[\begin{array}{l}2b\gamma -2b\gamma c_n-2\gamma b^{2}c-2\gamma b^{2}ca+2\gamma c_n{b}^2-2\gamma b{c}_{n}a+2c_n\gamma b^{2}a+2c_{n}a+2c_n\\ +2\gamma ba-2{(c_n^2(1-{\gamma }^2{b}^2+\gamma b^2)(1+a)(1+a-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a))}^{1/2}\end{array}\right]}{2(b\gamma +\gamma ba)b}$$

Comparing $c_{r1}$, $c_{r2}$ $\underset{¯}{c_r}$ and ${\overline{c}}_r$, we find that $\underset{¯}{c_r}<{\overline{c}}_r<c_{r1}<c_{r2}$, which means $K^N>K^{Rii}$.

In sum, we can conclude that remanufacturing is always detrimental to the incentives for new product innovation (i.e., $K^R<K^N$).

(7)

Replacing the “Appendix C. Proof of Proposition 2”.

Based on Table 1, $\frac{\partial K^{Ri}}{\partial c_r}=\frac{\rho (c_n-c_r)}{\left(a+2\right)\left(1-\gamma \right)\left(a-\gamma +1\right)}>0$ and $\frac{\partial K^{Rii}}{\partial c_r}=\frac{\rho \gamma b\left[b\gamma +c_n-c_r{b}^2\gamma -b\gamma c_n-\gamma b^{2}c+\gamma c_n{b}^2\right]}{2\left(a+2\right)\left(1-{\gamma }^2{b}^2+\gamma b^2\right)\left(1+a-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a\right)}>0$.

In sum, we can conclude that under the remanufacturing scenario, the incentives for new product innovation increase with the remanufacturing cost, that is, $\partial K^R/\partial c_r>0$.

(8)

Replacing the “Appendix D. Proof of Proposition 3”.

Based on the outcomes in Table 1, we can find that ${\pi }_T^{Ri\ast }-{\pi }_T^{N\ast }=\frac{{(c_r-r{c}_n+a{c}_r)}^2}{4r(1+a-r)(1+a)}>0$,

This means that ${\pi }_T^{Ri\ast }-{\pi }_T^{N\ast }>0$ is always true.

(ii) For $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}=\underset{¯}{c_r}<c_r<{\overline{c}}_r=\frac{\gamma c_n}{1+a}$,

After the observation of

${\pi }_T^{Rii\ast }-{\pi }_T^{N\ast }=\frac{\left[\begin{array}{l}\gamma b{a}^2-2c_n{c}_r-4c_n{c}_{r}a+2\gamma c_{n}b{c}_{r}a-2c_n{b}^{2}c{\gamma }^{2}a-c^2{b}^3{\gamma }^2\\ -{\gamma }^{2}b+2\gamma c_n{b}^{2}c+c_n^2\gamma b+2c_n^2\gamma -4\gamma b^{2}ca-4\gamma b{c}_{n}a+2{\gamma }^{2}bc\\ +2cb\gamma c_{n}a+2c_n^2\gamma ba+2c^2{b}^3\gamma a+4\gamma c_n{b}^{2}ca-2\gamma c_{n}a+2a^2{c}_r\\ +c^2{b}^3\gamma -2c_n{b}^{2}c{\gamma }^2-2c_{r}bc{a}^2-2c_n\gamma b{a}^2-2c_n{c}_r{a}^2\\ +2{\gamma }^{2}b{c}_{n}a+2\gamma c_{n}b{c}_r-4c_{r}bca-c_r^{2}b+2c_r-2b\gamma c_n-2\gamma b^{2}c\\ -2\gamma c_n+4a{c}_r+2cb\gamma c_n+b\gamma +2\gamma ba-2c_{r}bc-2c_r^{2}b{a}_n\\ +2c_n^2\gamma a-{\gamma }^{2}ba+2c{b}^2{\gamma }^2-2c_n^2{\gamma }^{2}b-c_r^{2}b{a}^2+2c_n\gamma b^{2}c{a}^2\\ +c_n^2\gamma b{a}^2+c^2{b}^3\gamma a^2-2b^{2}c\gamma a^2+2c{b}^2{\gamma }^{2}a-c_n^2{\gamma }^{2}ba-c^2{b}^3{\gamma }^{2}a\end{array}\right]}{4({\gamma }^2{b}^2-1-a-\gamma b^2-\gamma b^{2}a)(1+a)}$ we find that whether ${\pi }_T^{Rii\ast }-{\pi }_T^{N\ast }>0$ depends on the numerator of the above equation. Then, solving the above equation, we can yield the roots as follows:

$c_{r1}=\frac{\left[\begin{array}{l}2a-2c_{n}a-2cba-2cb-2c_n+2+2b\gamma c_n\\ +2(1-c_n-cb)\sqrt{(1+a)(1+a-r^2{b}^2+r{b}^2+r{b}^{2}a)}\end{array}\right]}{2b(1+a)}$ $c_{r2}=\frac{\left[\begin{array}{l}2a-2c_{n}a-2cba-2cb-2c_n+2+2b\gamma c_n\\ -2(1-c_n-cb)\sqrt{(1+a)(1+a-r^2{b}^2+r{b}^2+r{b}^{2}a)}\end{array}\right]}{2b(1+a)}$ comparing $c_{r1}$, $c_{r2}$ $\underset{¯}{c_r}$ and ${\overline{c}}_r$, we find that, for any $c>0$, then $\underset{¯}{c_r}<c_{r2}<{\overline{c}}_r<c_{r1}$. That is, if, and only if, the cost of $c_r<c_{r2}$, then ${\pi }_T^{Rii\ast }>{\pi }_T^{N\ast }$, otherwise, ${\pi }_T^{Rii\ast }<{\pi }_T^{N\ast }$.

In sum, we can conclude that the profits under the remanufacturing scenario are higher than that of no remanufacturing (i.e., ${\pi }_T^{R\ast }>{\pi }_T^{N\ast }$) if, and only if, the cost of $c_r$ is not pronounced (i.e., $c_r<c_r^{\Delta }=c_{r2}$); otherwise, the opposite is true.

(9)

Replacing the “Appendix E. Proof of Proposition 4”.

Since $E^R=i_d\left(q_1^{R\ast }+q_n^{R\ast }\right)$, then we can obtain

(i) For $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}=\underset{¯}{c_r}<c_r<{\overline{c}}_r=\frac{\gamma c_n}{1+a}$,

$E^{Ri}-E^N=i_d\frac{\left(a{c}_r+c_r-\gamma c_n\right)}{2\left(a+1\right)(1+a-\gamma )}<0$ is always true.

(ii) For $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}=\underset{¯}{c_r}<c_r<{\overline{c}}_r=\frac{\gamma c_n}{1+a}$,

solving $E^{Rii}-E^N=i_d\frac{b\left[\begin{array}{l}2c_n\gamma a+\gamma b{c}_{r}a+\gamma bca+{\gamma }^{2}b+c_n\gamma b{a}^2-c{b}^2{\gamma }^{2}a\\ +c{b}^{2}r{a}^2-c_n{\gamma }^{2}ba+\gamma bc+\gamma b{c}_r-a^2{c}_r-\gamma b{a}^2\\ -\gamma a+{\gamma }^{2}ba-c{b}^2{\gamma }^2-2a{c}_r-2c_n{\gamma }^{2}b-2b\gamma a-\gamma \\ +c_n\gamma b+2c_n\gamma +2\gamma b^{2}ca+2c_n\gamma ba-c_r-b\gamma +\gamma b^{2}c\end{array}\right]}{2(a+1-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a)(a+1)}$, we find that whether $E^{Rii}-E^N<0$ depends on the numerator of the above equation. Then, solving the above equation, we can yield the root of: $c_{r1}=\frac{r\left[\begin{array}{l}bca-1+c_{n}b{a}^2-r{b}^{2}ca+c{b}^2{a}^2-c_n\gamma ba+2c{b}^{2}a\\ +2c_{n}ba+c_{n}b+c{b}^2+b\gamma a+2c_{n}a-b{a}^2-\gamma b^{2}c\\ -2c_n\gamma b-2ba+2c_n-b+b\gamma -a+bc\end{array}\right]}{1-b\gamma a-b\gamma +a^2+2a}$. When $c_r>c_{r1}$ $E^{Rii}-E^N<0$ is true.

Comparing $c_{r1}$, $\underset{¯}{c_r}$ and ${\overline{c}}_r$, we find that $c_{r1}<\underset{¯}{c_r}<{\overline{c}}_r$.

Then, we can conclude that, for any $\underset{¯}{c_r}<c_r<{\overline{c}}_r$, $E^{Rii}-E^N<0$ is always true.

In sum, we can conclude that the presence of remanufacturing is always beneficial to the environment, that is, $E^N>E^R$.

(10)

Replacing the “Appendix F. Proof of Proposition 5”.

(i) For $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}=\underset{¯}{c_r}<c_r<{\overline{c}}_r=\frac{\gamma c_n}{1+a}$

$\partial E^{Ri}/\partial c_r=\frac{1}{2(a-\gamma +1)}>0$ is always true.

(ii) For $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}=\underset{¯}{c_r}<c_r<{\overline{c}}_r=\frac{\gamma c_n}{1+a}$,

$\partial E^{Rii}/\partial c_r=\frac{-b(1+a-b\gamma )}{2(a+1-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a)}<0$ is always true.

Under the partial remanufacturing scenario, the total environmental impact increases with the remanufacturing cost, that is, $\partial E^R/\partial c_r>0$; however, under the full remanufacturing scenario, the opposite is true, that is, $\partial E^R/\partial c_r<0$.

(11)

Replacing the “Appendix G. Proof of Proposition 6”.

Based on the analysis of Proposition 3, we find that, for $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}=\underset{¯}{c_r}<c_r<{\overline{c}}_r=\frac{\gamma c_n}{1+a}$, there is a threshold of $c_r^{\Delta }=c_{r2}=\frac{\left[\begin{array}{l}2a-2c_{n}a-2cba-2cb-2c_n+2+2b\gamma c_n\\ -2(1-c_n-cb)\sqrt{(1+a)(1+a-r^2{b}^2+r{b}^2+r{b}^{2}a)}\end{array}\right]}{2b(1+a)}$ then for $\raisebox{1ex}{$\partial c_r^{\Delta }$}\!\left/ \!\raisebox{-1ex}{$\partial b$}\right.=\frac{\left[\begin{array}{l}a-c_{n}a+\gamma b^{3}ca+1+\gamma b^{3}c-c_n-{\gamma }^2{b}^{3}c\\ +(c_n-1)\sqrt{(1+a)(1+a-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a)}\end{array}\right]}{b^2\sqrt{(1+a)(1+a-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a)}}$ we find that for any $c>0$, $\raisebox{1ex}{$\partial c_r^{\Delta }$}\!\left/ \!\raisebox{-1ex}{$\partial b$}\right.>0$ is always true.

We are now in a position to prove $\partial q_1/\partial b<0$ as follows.

(i) For $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}=\underset{¯}{c_r}<c_r<{\overline{c}}_r=\frac{\gamma c_n}{1+a}$

$$\partial q_1^{Ri}/\partial b=-\frac{c}{2}<0$$

(ii) For $\frac{\gamma (b\gamma +b^{2}c+b^{2}ca-b-ba-b^2\gamma c-b\gamma c_n+c_n+b{c}_{n}a+c_n)}{1+a}=\underset{¯}{c_r}<c_r<{\overline{c}}_r=\frac{\gamma c_n}{1+a}$,

$$\partial q_1^{Rii}/\partial b=q_1^{Rii}=\frac{\left[\begin{array}{l}2{\gamma }^{2}ba-c_n{\gamma }^2{b}^{2}a-a^2{c}_r-2\gamma b{a}^2+\gamma c{b}^2-c{\gamma }^2{b}^2\\ +c_r\gamma b^2+2b\gamma c_n-c_r{\gamma }^2{b}^2+c_n\gamma a-c_n{\gamma }^2{b}^2+c_n{\gamma }^3{b}^2\\ +4b\gamma c_{n}a+c{a}^2\gamma b^2-a{c}_r{\gamma }^2{b}^2+2c_r\gamma b^{2}a+2c\gamma b^{2}a\\ -c{a}^2-c+a^2{c}_r\gamma b^2-ca{\gamma }^2{b}^2-2{\gamma }^{2}b{c}_{n}a-c_r+c_n\gamma \\ +2\gamma b{c}_n{a}^2+2{\gamma }^{2}b-2\gamma b-2b{\gamma }^2{c}_n-2a{c}_r-2ca-4\gamma ba\end{array}\right]}{2{(1+a-{\gamma }^2{b}^2+\gamma b^2+\gamma b^{2}a)}^2}$$

Then, solving the above equation, we can yield the root of: $c_{r1}=\frac{\left[\begin{array}{l}4b\gamma c_{n}a-c-c_n{\gamma }^2{b}^{2}a+c{a}^2\gamma b^2+2\gamma c{b}^{2}a-ca{\gamma }^2{b}^2\\ -2{\gamma }^{2}b{c}_{n}a+2\gamma b{c}_n{a}^2-4\gamma ba+c_n{\gamma }^3{b}^2-c{\gamma }^2{b}^2\\ -c_n{\gamma }^2{b}^2+c\gamma b^2+2b\gamma c_n+2{\gamma }^{2}ba-2{\gamma }^{2}bcn-2\gamma b{a}^2\\ +c_n\gamma a-2b\gamma +2{\gamma }^{2}b+c_n\gamma -c{a}^2-2ca\end{array}\right]}{1-a^2\gamma b^2-2\gamma b^{2}a+a{\gamma }^2{b}^2+{\gamma }^2{b}^2-\gamma b^2+a^2+2a}$. When $c_r>c_{r1}$ $\partial q_1^{Rii}/\partial b<0$ is true.

Comparing $c_{r1}$, $\underset{¯}{c_r}$ and ${\overline{c}}_r$, we find that $c_{r1}<\underset{¯}{c_r}<{\overline{c}}_r$.

Then, we can conclude that, for any $\underset{¯}{c_r}<c_r<{\overline{c}}_r$, $\partial q_1^{Rii}/\partial b<0$ is always true.

In sum, we can conclude the equilibrium quantities as $\partial q_1/\partial b<0$.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.