# logistic differential equation problem setup

The Pacific halibut fishery has been modeled by the differential equation $$\frac{dy}{dt} = ky(1-\frac{y}{m})$$ where $$y(t)$$ is the biomass in kg at time $$t$$ years. The carrying capacity is estimated to be $$M = 8x10^7$$kg and $$k = 0.71/yr$$. If $$y(0) = 2x10^7$$kg, find the biomass a year later.

I am having a difficult time calculating this problem so I would just like some input on whether I setup the problem correctly.

$$y(t) = \frac{8x10^7}{1 + 3x10^7e^{-0.71}}, ~~A = \frac{8x10^7 - 2x10^7}{2x10^7} = 3x10^7$$

I think it is just a simple plugging into the equation given in the reading which is $$P(t) = \frac{M}{1+Ae^{-kt}}, ~~ A = \frac{M-P_0}{P_0}$$

Let me k now if this is correct or not, thanks

It is $$y(t)=\frac{m e^{c_1 m+k t}}{e^{c_1 m+k t}-1}$$