The aim of this assignment is to study the birth-death processes presented by Johnson and colleagues in the article,

Cancer cell population growth kinetics at low densities deviate from the exponential growth model and suggest an Allee effect (2019) PLOS Biol Biol 17(8):3000399

Exercise 1 - The model

1. Describe the concepts of "Allee effect", "Gompertzian models" and "demographic stochasticity". Use the description from the article, but also from other sources (give the references).

2. Describe the 7 growth models (for each one: deterministic and stochastic version, and a short description)

Exercise 2 - Moment closure

The master equation can be used to computed the first moments of $N(t)$.

1. What are the assumptions made by the authors to perform moment closure.

2. Check that the expressions for $\langle N \rangle$ and the variance $\Sigma$.

Exercise 3 - Deterministic vs stochastic formulation

The moment closure was performed so that $\langle N \rangle$ satisfies the same ODE as the deterministic formulation. Compute the coefficient of variation $$C_V = \frac{Var}{\langle N \rangle},$$ when

1. $\langle N \rangle$ is small
2. $\langle N \rangle$ is large

where "small" and "large" are defined by the range of cell numbers experimentally observed. Is the stochastic formulation warranted?

References

Cancer cell population growth kinetics at low densities deviate from the exponential growth model and suggest an Allee effect (2019) PLOS Biol Biol 17(8):3000399

Assignment in progess, more to come!