I am having difficulty understanding a particular solution in a research paper I am reading.
Title: "Analysis of nonlinear duopoly game with heterogeneous players"(2006)
Authors: Jixiang Zhang, Qingli Da, Yanhua Wang
I would appreciate your help in clarifying the steps involved.
Problem:
On page 142 of the paper, there is an inequality with the following condition:
\begin{equation}
\text{condition 2: } 1+\text{Tr}(\mathbf{J})+\text{Det}(\mathbf{J}) > 0
\end{equation}
The goal is to find the range of alpha values that violate this condition. And substitute the \begin{align*} Tr(J), \ Det(J), \ q_1^*, \ q_2^*.
\end{align*} previously proposed by the researcher.:
\begin{equation} 2 + \alpha a - \frac{ab(b + 2c_1)\alpha}{3b^2 + 4c_1c_2 + 4b(c_1 + c_2)} - \frac{4a(b + 2c_2)(b + c_1 + \frac{b^2}{8(b + c_2)})\alpha}{3b^2 + 4c_1c_2 + 4b(c_1 + c_2)} \end{equation}
Then, differentiate with respect to alpha. My result: \begin{equation} -\frac{a(b + 2c_2)(5b^2 + 4c_1c_2 + 4b(c_1 + c_2))}{2(b + c_2)(3b^2 + 4c_1c_2 + 4b(c_1 + c_2))} \end{equation}
*** the range of alpha that the researcher considered is as follows:
\begin{equation} \alpha < \frac{4(b + c_2)(3b^2 + 4c_1c_2 + 4b(c_1 + c_2))}{a(b + 2c_2)(5b^2 + 4c_1c_2 + 4b(c_1 + c_2))} \end{equation}
I tried to solve this inequality using Mathematica, but the numerator and denominator of my solution were different from those in the paper. The constants also did not match. I am not sure where I made a mistake.
Questions:
Why does the inequality in the paper have different signs and constants compared to Mathematica's solution? Is it okay to differentiate an inequality when it has a squared term and is difficult to solve otherwise? I noticed that other studies I have read did not use differentiation, but this paper did.