Maximilian Strobl, Jeffrey West, Yannick Viossat, Mehdi Damaghi, Mark Robertson-Tessi, Joel Brown, Robert Gatenby, Philip Maini, Alexander Anderson

Read the manuscriptAs you will notice this equation has a slightly different form to the \(r\)-\(K\) formulation we are so familiar with today (more on that in a bit). It also does not yet include the \(K\). The \(r\)-\(K\) formulation originated in the 1930s and was first popularised in a textbook by Raymond Pearl

A final important development was the introduction of the \(r\)-\(K\) selection theory by MacArthur and Wilson in 1967

**Figure 1:** Why \(K\) in the logistic growth model is not just an environmental constraint. A 2-D on lattice, birth-death, cellular automaton model implemented in HAL^{3}. A) No death. Cells proliferate at rate \(\rho=0.027d^{-1}\) until they fill up all 10,000 sites. Setting \(r=\rho\) and \(K=10,000\) in a logistic growth model gives the same predictions. B) With a death rate of \(\delta=30\% \, \rho\) the population can never fully saturate the domain. Only changing \(r\) in the logistic growth fails to capture this. C) Reducing \(\rho\) by 50% further decreases the number of cells at equilibrium. Again this is missed if just changing \(r\) in the logistic model. A population's carrying capacity is not determined just by the environment!

**Figure 2:** Interaction of two competing species modelled with the Lotka-Volterra model. Species 2 is assumed to have a reduced fitness. A) If we assume \(r\) and \(K\) are independent, the frequency of Species 2 decreases with its fitness, but Species 2 will never go extinct. This is because in assuming this independence, we are assuming that there is not turnover. B & C) When turnover is included, so that \(r\) and \(K\) are linked, the less fit species will go extinct over time. Parameters: \(r_1=0.027 d^{-1}, N_1(0) = N_2(0) = 0.05 K\). Reproduced from^{15}.

Explicitly adding the death term (or making the carrying capacity parameter of each population a function of its growth rate) also fixes the above inconsistency in the Lotka-Volterra model. If we assume non-zero death, the growth rates contribute to the steady states of this model, and the less fit population will go extinct over time (Figures 2B & C; for the maths see e.g. [8]). Finally, this also explains why it has been difficult to find evidence for for the \(r\)-\(K\) selection hypothesis - \(r\) and \(K\) are not independent (for more details see [8] and [4])!

Around that time, Fred Adler posted a blog post

Tumours are a disease of up-regulated cell division. But how about cell death? How frequently does a tumour cell die - likely not very often, right? While we found it hard to find data on this topic (and if you have any recommendations, please post them below), to our surprise it seems like cell production and cell loss in tumours are closely balanced (e.g. [14]). On second thought, this actually makes sense: in homeostatic tissue cell birth and death is exactly balanced! This paints a rather dynamic picture of the tumour ecosystem - one in which cells are continuously replacing each other, and this rate will determine the speed of evolution towards, and away from resistance.

Finally, as a small disclaimer, I am not trying to advocate logistic growth as a tumour growth model. If you intend to quantitatively describe tumour growth, there is strong evidence that models such as the Gompertzian model are more suited (e.g. [10, 2]). And again the cancer world isn’t alone here: in 1920, Pearl and Reed used the logistic growth model to predict that the American population would saturate at 197 Million

1. Technically speaking, the situation is a bit more complicated. In the above form of the Lotka-Volterra model, every point along the line \(N_1+N_2 = K\) is a steady state. Which point along this line a trajectory will converge to, depends on the ratio \(r_1/r_2\). However, unless we make this ratio infinite, Species 2 will never go extinct (Figure 2A) (jump back)

2. Interestingly the CA does not settle at \(\hat{K}\) but at \(\left(1-\frac{\delta}{\rho}\right)^{1/4}K\); see an upcoming pre-print by Gregory Kimmel et al. (jump back)

3. Thanks very much to Jeffrey West and Andrew Krause for many useful discussions, and helping me with this post. This wouldn’t have happened without you!

- Frederick Adler. All pithy maxims about modeling are wrong, but some are useful - The Mathematical Oncology Blog.
- Sebastien Benzekry, Clare Lamont, Afshin Beheshti, Amanda Tracz, John M.L. Ebos, Lynn Hlatky, and Philip Hahnfeldt. Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth. PLoS Computational Biology, 10(8):e1003800, aug 2014.
- Rafael R. Bravo, Etienne Baratchart, Jeffrey West, Ryan O. Schenck, Anna K. Miller, Jill Gallaher, Chandler D. Gatenbee, David Basanta, Mark Robertson-Tessi, and Alexander R. A. Anderson. Hybrid Automata Library: A flexible platform for hybrid modeling with real-time visualization. PLOS Computational Biology, 16(3):e1007635, mar 2020.
- Jeremy Fox. Zombie ideas in ecology: r and k selection, 2011.
- G. F. Gause and A. A. Witt. Behavior of Mixed Populations and the Problem of Natural Selection. The American Naturalist, 69(725):596–609, nov 1935.
- Alfred J Lotka. Elements of physical biology. 1925.
- Robert H MacArthur and Edward O Wilson. The theory of island biogeography, volume 1. Princeton university press, 2001.
- James Mallet. The struggle for existence: how the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution, and speciation. Evolutionary Ecology Research, 14(August):627–665, 2012.
- T. R. Malthus. An Essay on the Principle of Population. Or a View of its Past and Present Effects on Human Happiness; with an Inquiry into our Prospects Respecting the Future Removal or Mitigation of the Evils which it Contains. John Murray, London, 1826.
- M. Marusic, Bajzer, J. P. Freyer, and S. Vuk-Pavlovic. Analysis of growth of multicellular tumour spheroids by mathematical models. Cell Proliferation, 27(2):73–94, feb 1994.
- AG McKendrick and M Kesava Pai. Xlv.—the rate of multiplication of micro-organisms: a mathematical study. Proceedings of the Royal Society of Edinburgh, 31:649–653, 1911.
- Raymond Pearl. Introduction to medical biometry and statistics. WB Saunders, 1923.
- Raymond Pearl and Lowell J Reed. On the rate of growth of the population of the united states since 1790 and its mathematical representation. Proceedings of the National Academy of Sciences of the United States of America, 6(6):275, 1920.
- G G Steel. Cell loss as a factor in the growth rate of human tumours. European Journal of Cancer (1965), 3(4-5):381–387, 1967.
- Maximilian Andreas Roland Strobl, Jeffrey West, Joel Brown, Robert Gatenby, Philip Maini, and Alexander Anderson. Turnover modulates the need for a cost of resistance in adaptive therapy. BioRxiv, 2020.
- Pierre-Francois Verhulst. Notice sur la loi que la population suit dans son accroissement. Corresp. Math. Phys., 10:113–126, 1838.
- Yannick Viossat and Robert John Noble. The logic of containing tumors. bioRxiv, page 2020.01.22.915355, jan 2020.
- Wikipedia. r/k selection, 2020.