Hello,
the formula to calculate the number of individuals given the initial population size is:
Nₜ = rᵗN₀
where Nₜ is the number of individuals at time t, N₀ is the initial number of individuals and r is the net reproductive rate. For bacteria, I reckon I can change it to:
Nₜ = μᵗN₀
where μ is the specific growth rate of a given bacterium. For Escherichia coli, I have found a value of 0.72. Now, bacteria are modeled to grow in a logistic fashion, thus I need to introduce the term 1-N/K where K is the carrying capacity of the environment. Hence, I need to convert these equations in equivalents containing the logistic term. I have found the Verhulst equation:
Nₜ₊₁ = μNₜ(1-Nₜ/K)
and the Ricker's curve:
Nₜ₊₁ = Nₜexp[μ(1-Nₜ/K)]
but these calculate the increase at the next time interval, e.g. from the beginning of the experiment to 1 hour after. However, these do not contain the term t that allows me to calculate the population expansion at any given time, for instance 24 hours after the beginning of the experiment. My questions are:
- how can I convert Nₜ = μᵗN₀ to an equation containing the logistic term?
- how can I solve the Ricker's curve for Nₜ?
Thank you
I got an answer from Prof. James Murray (Emeritus Professor of Mathematical Biology, University of Oxford).
Essentially it can be solved algebraically, but the whole series has to be written down.
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If you write down the first few populations N1, N2, etc. you get:
N2 = rN1(1 - N1/K) = r[rN0(1 - N0/K){1- rN0(1 - N0/K)]/K}
So if you have N2 to get N0 you have to solve the last algebraic equation for N0 with N2 given. It requires the solution of a quadratic.
If you write down N3 you get:
N3 =rN2(1-N2/K) = r[r[rN0(1 - N0/K){1- rN0(1 - N0/K)]/K}][1 - {r[rN0(1 - N0/K){1- rN0(1 - N0/K)]/K}}/K]
So, now, if you now have the measured N3 you have to solve the last nonlinear algebraic equation for N0. I doubt it can be done algebraically! You can do it by moving back 1 at a time until you get N0
"""
Well, there is not much to add. I only have the values for μ and K. I would like to find the logistic's equivalent to Nₜ = rᵗN₀ (Nₜ ~ μN₀(1-N₀/K)) ... I don't have data points.
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