It's been a while since I've had to do any optimization problems and I've been having trouble with the following question while trying to review calc 3.
Two competing microbes in a given region can be modeled with the equations $$\frac{dm_1}{dt} = 0.24m_1-0.00006m_1^2-0.00002m_1m_2$$ $$\frac{dm_2}{dt} = 0.31m_2-0.000062m_2^2-0.0000207m_1m_2,$$ where $m_1$ and $m_2$ represent amount of microbes. Determine the amount of microbe which will result in the biggest total annual microbe growth.
I guess I'm just having trouble figuring out what I'm maximizing. Am I supposed to maximize $dm_1dt+dm_2dt$? I assume once I have an equation then I just take the partials and set equal to 0 right? Any tips would be appreciated, thanks!