Gompertz growth model problem

Question

The growth of tumor cells is characterized with Gompertz model.

$N'=-aNln(bN),$

where N(t) is proportional to the number of cells in the tumor, while a and b denote positive parameters. Question:Which biological interpretation would you give to a and b? What do you think makes a Gompertz model better than a logistic growth? Any kind of hint is appreciated.

Answer 1

The parameter $a$ scales the time. The parameter $b$ reflects the carrying capacity $K$ through the identity $$K=1/b.$$ Recall that $K$ is the limit of $N(t)$ when $t\to\infty$ for every initial population, and that $K$ is also a threshold such that $N'(t)\gt0$ if $N(t)\lt K$ and $N'(t)\lt0$ if $N(t)\gt K$.

What do you think makes a Gompertz model better than a logistic growth?

Are Gompertz models "better than" logistic ones?

Some differences concern the regimes where $N$ is very small or very large.

When $N$ is very small, the logistic growth $N'=aN(1-N/K)$ reduces to $N'\approx aN$ while Gompertz reads $N'\approx-aN\log N$ hence Gompertz populations grow more quickly than logistic ones, at very low initial populations. On the contrary, when $N$ is large, the logistic growth reduces to $N'\approx-(a/K)N^2$ while Gompertz reads $N'\approx-aN\log N$ hence Gompertz populations decrease less quickly than logistic ones, at high initial populations.