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


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