Question

Use the solution you found in Part 1f to show that the Gompertz model can be rewritten as dP/dt=−λe^(−rt)P, where λ is a positive constant.

We have examined several population models and in this question we will explore yet another model that has proved to be an accurate description of the growth of solid tumours. Let P(t) represent the population of tumour cells and consider the following differential equation known as the Gompertz model: r and K are positive constants that you will need to interpret. Part 1. Exploring the model. a) Find the equilibria, sketch the phase diagram of this DE, determine the stability of equilibria. b) Discuss the growth dynamics of P. What happens to df as P nears an equilibrium (on either side)? Interpret the meaning of equilibria in terms of tumour growth. c) What is the maximum growth rate and where does it occur? d) What is the average growth rate of the population over its range of sizes (zero to equilibrium)? e) Using various values for r and k, sketch several slope fields for this DE rather using, for example, https://bluffton.edu/homepages/facstaff/nesterd/java/slopefields.html (note that you enter log in place of In). f) Solve the equation using separation of variables and an initial size of Po. Plot the solution for various initial values Po. g) How does r affect the shape of the solution? (Use parts e) and f) to answer). h) Compare the Gompertz model to the logistic model for population growth. Discuss any similarities or differences that you find.

j) Consider grouping the factors in the equation like this: dP/dt=-(λe^(-rt))P. Make an interpretation of this equation. In other words, what assumption about tumour growth would lead us to write down such an equation?

k) Now consider grouping the factors in the equation like this: dP/dt=−λ(e^(-rt)P). Again, explain what assumption about tumour growth would lead to such an equation.

Answer 1