then exponentially for and where is a certain solution of a logistic equation.
However, the solution of the logistic equation can be solved explicitly as u ( t ) = e at a u 0 a − b u 0 + b u 0 e at.
By using a fluctuation lemma, Li and Chen [4] obtained sufficient conditions which ensure the second species will be driven to extinction while the first one will stabilize at a certain solution of a logistic equation.
Since it is easy to compute the solution of the logistic equation directly on each of the subintervals, we do precisely only that in these following examples: Consider the following equations: u 1 ″ = 8 u − 4 u 2, u ( 0 ) = 6, 0 ≤ t ≤ 2, u 2 ″ = 8 u − 4 u 2, u ( 2 + ) = 2. 5, 2 + ≤ t ≤ T. Open image in new window.
After protracted negotiations between the governments of Australia, Britain, Canada, India, New Zealand and South Africa, agreement had been reached to establish an integrated formation with the aim of increasing the political significance of their contribution, as well as facilitating the solution of the logistic and operational problems faced by the various Commonwealth contingents.
Meanwhile, after protracted negotiations between the governments of Australia, Britain, Canada, India, New Zealand and South Africa, agreement had been reached to establish an integrated formation with the aim of increasing the political significance of their contribution, as well as facilitating the solution of the logistic and operational problems faced by the various Commonwealth contingents.
Using this information we calibrate the solution of the logistic equation, using our previous estimate of K, to that data, which provides an estimate of λ.
To estimate λ, we found the value of λ that minimised the least squares error between our experiments and the solution of the logistic equation.
To estimate the cell proliferation rate, we found the value of λ that minimised an estimate of the least–squares error between our experiments and the solution of the logistic equation [see methods Assessing goodness of fit, Equation 6 and Additional File 1].
Here the least squares error is given by, (6) Error P = ∑ i = 1 2 (Ep i − Sp i ) 2 ∑ i = 1 2 (Ep i ) 2, where, E p i corresponds to the non dimensional cell density averaged over four experimental replicates, Sp i is the corresponding non dimensional cell density from the solution of the logistic equation and i corresponds to the two time points, t=24 and t=48 hours.
