Fig 1.
Overview of the mechanisms assumed in the model.
Cancer cells divide at a rate α(1 + g), where g is the local growth factor (GF) concentration, and die at a constant rate μ. The GF is produced at rate ρ by all cancer cells and decays at rate δ. Lastly, cells migrate at rate ν. Parameter values are provided in Table 1.
Table 1.
Model parameters.
Fig 2.
(A) The population growth rate (3) as a function of the population density for three different values of the death rate μ. The inset shows the entire range of growth rates. (B) The per-capita growth rate f(n)/n as a function of the population density for three different values of the death rate μ. Here the Allee effect is evident as an increasing per-capita growth rate at low densities. All parameter values are given in Table 1 and the critical death rate is calculated according to (8).
Fig 3.
Comparison of the ODE-model and IB-model under long range dispersal.
(A) The dynamics of the IB-model (dashed) and the numerical solution of (3) (solid) under long range dispersal for three different initial conditions. All parameter values are given in Table 1 and the death rate μ = 3.75 × 10−5. (B) The population density after 11 days of the IB-model (circles) and the numerical solution of (3) (solid line) under long range dispersal and no cell migration when the death rate μ is varied. All other parameter values are given in Table 1.
Fig 4.
The population density after 11 days as a function of the death rate μ.
Results from the IB-model (circles) and the numerical solution of (3) (solid line) under (A) short range dispersal and no cell migration and (B) short range dispersal and a cellular diffusion coefficient of Dc = 1.6 × 10−10 cm2/s. All other parameter values are given in Table 1.
Fig 5.
Least squares fit of the Allee ODE-models to in vitro growth of the glioblastoma cell lines U3013MG, U3123MG and U3289MG.
Panels A-C show the best fit of the Allee model whereas D-F show the best fit for a logistic growth model (where ρ = 0). Visual inspection suggests that the Allee model outperforms the logistic model. This was confirmed using Akaike Information Criterion (see Table 2).
Table 2.
The parameter values (A, B, μ) refer to the optimal fit for the Allee model. For a visual comparison see Fig 5.