How to justify left this system of equations?

The reason the three equations are centered horizontally in your code is because you're using a matrix environment, which is programmed to center the contents of each entry, using inline math mode.

• Switching to a cases environment takes care of left-aligning the cell contents. However, the equations continue to be typeset in inline math mode and the spacing between the rows is very tight. This makes it a bit hard to take in the material. Moreover, the rows are fully left-aligned instead of being aligned on the = symbols.

• I suggest you employ an aligned environment inside the cases environment and align the equations on the = symbols. This change not only improves the visual alignment; importantly, it also switches to display math mode, making the material (IMNSHO) much easier to take in.

\documentclass{article}
\usepackage{amsmath} % for 'cases' and 'aligned' environments
\begin{document}

%% OP's original code: 'matrix' env.  inside \left\{ ... \right.
\[
\left\{
\begin{matrix}
u_{t}=\nabla\cdot D(u_{t})\nabla u_{t}+\mu_{1} \frac{c_1}{k_1+c_1}u_1+\mu_{2}\frac{c_2}{k_2+k_{12}c_1+c_2}u_1\\
c_{1,t}=\Delta c_1-\frac{\mu_{1}}{y_1}u_{1\infty} \frac{c_1}{k_1+c_1}u_{1}\\ 
c_{2,t}=\Delta c_2-\frac{\mu_{2}}{y_2}u_{1\infty}\frac{c_2}{k_2+c_2+k_{12}c_1}u_1\\
\end{matrix}
\right. \tag{I}
\]

%% 'cases' instead of 'matrix'
\[
\begin{cases}
u_{t}=\nabla\cdot D(u_{t})\nabla u_{t}+\mu_{1} \frac{c_1}{k_1+c_1}u_1+\mu_{2}\frac{c_2}{k_2+k_{12}c_1+c_2}u_1\\
c_{1,t}=\Delta c_1-\frac{\mu_{1}}{y_1}u_{1\infty} \frac{c_1}{k_1+c_1}u_{1}\\ 
c_{2,t}=\Delta c_2-\frac{\mu_{2}}{y_2}u_{1\infty}\frac{c_2}{k_2+c_2+k_{12}c_1}u_1
\end{cases} \tag{II}
\]

%% employ an 'aligned' environment inside the 'cases' environment 
\[
\begin{cases}
\begin{aligned}
u_{t}   &= \nabla\cdot D(u_{t})\nabla u_{t}+\mu_{1} \frac{c_1}{k_1+c_1}u_1+\mu_{2}\frac{c_2}{k_2+k_{12}c_1+c_2}u_1\\
c_{1,t} &= \Delta c_1-\frac{\mu_{1}}{y_1}u_{1\infty} \frac{c_1}{k_1+c_1}u_{1}\\ 
c_{2,t} &= \Delta c_2-\frac{\mu_{2}}{y_2}u_{1\infty}\frac{c_2}{k_2+c_2+k_{12}c_1}u_1
\end{aligned}
\end{cases} \tag{III}
\]

\end{document}

You can use the cases environment from amsmath.

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{equation}
  \begin{cases}
    u_{t}=\nabla\cdot D(u_{t})\nabla u_{t}+\mu_{1}
    \frac{c_1}{k_1+c_1}u_1+\mu_{2}\frac{c_2}{k_2+k_{12}c_1+c_2}u_1\\ 
    c_{1,t}=\Delta c_1-\frac{\mu_{1}}{y_1}u_{1\infty} \frac{c_1}{k_1+c_1}u_{1}\\ 
    c_{2,t}=\Delta
    c_2-\frac{\mu_{2}}{y_2}u_{1\infty}\frac{c_2}{k_2+c_2+k_{12}c_1}u_1    
  \end{cases}\tag{I}
\end{equation}

\end{document}

P.S. You can use the \dfrac macro if you wish displayed fractions.