I have solved a $3\times 3$ Non-linear system using numerical methods (post here), and now would like to argue for the uniqueness of my solution. My approach would be to make use of the Implicit Function Theorem. For this I would consider the function $\mathbf{F}:[0,1]^3\to \mathbb{R}^3$, such that $\mathbf{F}(\mathbf{x})=(F_1(\mathbf{x}),F_2(\mathbf{x}),F_3(\mathbf{x}))$, where:
$$F_1(\mathbf{x})=K c_1 x_1-(1-x_2)(1-x_3)-\beta_1 \gamma_1(1-x_2)x_3-\beta_1 \gamma_2(1-x_3)x_2-\beta_1\gamma_3x_3x_2$$ $$F_2(\mathbf{x})=K c_2 x_2-(1-x_1)(1-x_3)-\beta_2 \gamma_4(1-x_1)x_3-\beta_2(1-\gamma_2)(1-x_3)x_1-\beta_2\gamma_5x_3x_1$$ $$F_3(\mathbf{x})=K c_3 x_3-(1-x_2)(1-x_1)-\beta_3(1-\gamma_1)(1-x_2)x_1-\beta_3(1-\gamma_4)(1-x_1)x_2-\beta_3 (1-\gamma_3-\gamma_5)x_1x_2$$
For $K,c_i>0$, $\gamma_i\in[0,1]$ and $\beta_i \in (0,1]$ for all $i=1,2,3$.
With my numerical method I have obtained a solution, let's say $\mathbf{F}(\hat{x})=(F_1(\hat{x}_1),F_2(\hat{x}_2),F_3(\hat{x}_3))=(0,0,0)$. Notice how $\mathbf{F}$ has continuous derivatives and is defined in a compact set $[0,1]^3$. The Jacobian associated with this system is: $$ \mathcal{J}= \left( \begin{matrix} \frac{\partial F_{1}}{\partial x _{1}} & \frac{\partial F_{1}}{\partial x_{2}} & \frac{\partial F_{1}}{\partial x _{3}}\\ \frac{\partial F_{2}}{\partial x_{1}} & \frac{\partial F_{2}}{\partial x_{2}} & \frac{\partial F_{2}}{\partial x_{3}}\\ \frac{\partial F_{3}}{\partial x_{1}} & \frac{\partial F_{3}}{\partial x_{2}} & \frac{\partial F_{3}}{\partial x_{3}} \end{matrix} \right) $$
My question: would $\det \mathcal{J}(\hat{x}_1,\hat{x}_2,\hat{x}_3)\neq 0$ be sufficient to argue that $(\hat{x_1},\hat{x_2},\hat{x}_3)$ is unique, provided $\mathbf{F}$ is continuously differentiable? Would there be a way to use the Inverse Function Theorem to do analysis of the solution $(\hat{x}_1,\hat{x}_2,\hat{x}_3)$ with respect to parameters, or would numerical methods be the best way to do so?