Malthus

When microbial cells are incubated into a batch culture containing fresh culture media, their increase in concentration can be monitored. It is common to use the cell dry weight as a measurement of cell concentration. The simplest relationships describing exponential cell growth are unstructured models. Unstructured models view the cell as an entity in solution, which interacts with the environment. One of the simplest models is that of Malthus 19... 

Models of population growth are analogous to chemical reaction rate equations. In the model developed by Malthus in 1798, the rate of change of the population N of Earth is dN/dt = births — deaths. The numbers of births and deaths are proportional to the population, with proportionality constants b and d. Derive the integrated rate law for population change. How well does it fit the approximate data for the population of Earth over time given below ... 

Now consider one-step processes. As an example we take the Malthus-Verhulst problem mentioned in VI.9, because it has been the subject of some discussion. The M-equation for the population growth is... 

Exercise. For the following Malthus-Verhulst equation with random coefficient,... 

The use of antimony sulfide, Sb2S3, designated in the early writings simply as antimony, along with the saltpeter, sulfur, and charcoal, which were the standard ingredients of all pyrotechnic compositions, appears to have been introduced in the early part of the seventeenth century. John Bate s Book of Fireworks, 1635, containing information derived from the noted Professors, as Mr. Malthus, Mr. Norton, and the French Authour, Des Recreations Mathematiques, 2 mentions no mixtures which contain antimony. Typical of his mixtures are the following. 

F. Malthus (Frangois de Malthe), Treatise of Artificial Fireworks, 1629 Robert Norton, The Gunner, 1628. 

In nature there is an automatic selection which is continuously going on simply because variations exist between individual organisms (the breeders fact), and because not all can survive in a limited environment (Malthus fact). Darwin described the principle of natural selection at the end of Chapter 4 of On the Origin of Species, and it may be useful to read it in his own words ... 

Malthus, T.R. 1798. An Essay on the Principle of Population, as It Affects the Future Improvement of Society. J. Johnson, London. 

The population density at steady state is indeterminate, and no steady state is possible unless Eq. (27) is fulfilled exactly. As a matter of fact Eq. (27) grossly contradicts experience, since in continuous propagation there is a range of holding times in which steady state can be achieved. Hence something is wrong with the Malthus model. 

One sees that N approaches 1/)S as r -> oo. Equation (29) is a more realistic form than Malthus law, since there is now a stationary phase following exponential growth. 

According to Malthus, populations tend to increase exponentially, unless checked by some factor, such as a shortage of nutrients. A survey of some of the early literature on population growth is given by Whittaker (W2) and by Pearl (PI). 

Clearly the foregoing result is better than that obtained from Malthus law. However, any model such as Eq. (23) or (24) is suspect because it does not account for the effect of environment on growth, or the reciprocal effect of growth on environment this will be considered shortly. Also experimental data exist which cannot be reconciled with such models. Finn and Wilson (F2), in a study of the continuous propagation of yeast, found that under certain conditions there did not appear to be any steady state rather, the population density oscillated about some mean value, with a well-defined frequency. This brings the question of stability into the discussion we will show that Eq. (23) is incompatible with the existence of the phenomenon observed by Finn and Wilson. 

Thomas Malthus relied on an exponential-growth model to make his famous prediction about human population growth. 

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