Rate of Production of Methane by Methanosarcina

ksdhart2

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Hi all. I'm having some difficulties with a problem from my Diff. Eq. class. Here's the full problem text:

The reference [1] provides a model of the rate of production of methane (CH4) by the archeobacteria Methanosarcina in the presence of nickel (Ni). Table 0.1 identifies the quantities involved in the model. (Myr = "Millions of years")

\(\displaystyle A(t)\)Production of methane (CH4), proportional to M
\(\displaystyle \dfrac{dA}{dt}\)1/MyrRate of change (derivative) of A(t) relative to time t
\(\displaystyle K(t)\)Carrying capacity of Methanosarcina
\(\displaystyle k_0\)Positive constant
\(\displaystyle k_1\)Positive constant
\(\displaystyle \beta_0\)1/MyrPositive constant
\(\displaystyle \beta_1\)1/MyrPositive constant
\(\displaystyle \beta(t)\)1/MyrRatio \(\displaystyle \dfrac{dA}{dt} \div K\)
\(\displaystyle M\)Accumulated dimensionless mass of Carbon (C)
\(\displaystyle t\)MyrTime, millions of years

The model relates such quantities by the following equations:

\(\displaystyle K(t) = k_0 + k_1 \cdot A(t)\)

\(\displaystyle \beta(t) = \beta_0 + \beta_1 \cdot A(t)\)

\(\displaystyle \dfrac{dA}{dt} = \beta(t) \cdot K(t)\)

1.1) Find a formula for A(t) in terms of \(\displaystyle t, \beta_0, \beta_1, k_0, \text{ and } k_1\), with A(0) = 0, for t in an interval about 0.

1.2) Prove that there exists a time \(\displaystyle t_\dagger > 0\) such that for each Z > 0 there exists a time \(\displaystyle t_z < t_\dagger\) when \(\displaystyle A(t_z) = Z\).

References

[1] Daniel H. Rothman. Mathematical expression of a global environmental catastrophe. Notice of the American Mathematical Society, 64(2): 138-140, February 2017. http://www.ams.org/publications/notices/201702/rnoti-p138.pdf

I've made a tiny bit of progress on 1.1 but haven't yet started on 1.2. The trivial solution is if A(t) = 0 for every t, then \(\displaystyle \beta(t) \text{ and } K(t) = 0\) for every t. But if \(\displaystyle A(t) \ne 0\) for some \(\displaystyle t \ne 0\), then we have:

\(\displaystyle \dfrac{dA}{dt} = \beta_0 \cdot K_0 + (\beta_0 \cdot K_1 + \beta_1 \cdot K_0) \cdot A(t) + \beta_1 \cdot K_1 \cdot A^2(t)\)

But I'm not really sure where to go from here. My professor says I need to separate the variables to end up with all the A(t) and A'(t) on one side and the constants on the other. But, I'm not sure how to do that. I've been really scrambling to take down notes this week and very little makes of it makes any sense to me. I recall something about this being an exponential decay problem (?) and I should use \(\displaystyle e^{\int \: A(t) dt}\) at some point, but uh... I'm not even sure about that. Any help would be much appreciated.
 
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