Models in Biology.

Here is my mini-assignment on the models in biology in general.

Models in Biology: Population growth model

1. An example of a model in biology

Mathematical models can be applied to the study of population dynamics. Population dynamics have been studied for the last couple of years and a number of models have been developed over that time. One of the first approaches was developed by Thomas Malthus, who became widely known for his theories about population and his model is now known as the simple logistic model, also called the Malthus model.

2. What is the purpose of the model

In general, the population growth model helps understand processes that occur in the ecological system. In particular, if applied to human population, for example, the model may be used to predict the population growth and approach the potential problems that the growth will cause, such as overpopulation, shortage of housing or fresh water, pollution and similar.

3. What does the model represent

In fact, there are multiple models of the population growth and the complete list is probably outside the scope of this exercise. Generally, the population growth model is a total number of species in the population as a function of time t, and there are multiple factors that influence the result of the function. More complex models consider more factors and are more accurate.

4. How is it represented

The simplest model may be the arithmetic model. In this case the population is defined as follows:

And the population size is a simple function of births, immigration, deaths and emigration. While this model may be accurate in retrospect, it is not very useful in predicting the population growth because the variables on the right side of the equation are generally not known beforehand.

Another well-known model is an exponential model

This model assumes that the population grows at a certain rate r. This model is a simplification because it makes several assumptions, such as the rate being constant, ignoring emigration and immigration and ignoring restrictions on population growth which will inevitably apply.

A logistic growth model appears to be more advanced.

Compared to the exponential model, this model takes into account the carrying capacity K, which is the maximum sustainable population size or, simply, the largest amount of species the environment can support. As the population size approaches K, the population grows and the population size N can never be larger than K. This model still makes certain assumptions, such as that K remains constant, and influence of immigration and emigration is ignored too.

5. Is the representation accurate?

None of the models mentioned above appears to be exactly accurate. For example, the exponential growth model is fairly accurate at the initial phase. However, at a later stage a lot of other effects become significant which are not considered by the model.

6. How could you validate your model?

The model can be simulated by computational methods but that, of course, does not say anything about the validity of the model. Intuitively, it appears that the model can not be proven valid; the best possible outcome would be to estimate the range of the possible error. Even in a relatively simple case, for example a model of a bacterial growth under known conditions, it is unlikely that the population size will be exactly equal to the predicted value. If we repeat the experiment a number of times, the resulting population size will probably be in a certain range, following a normal distribution. In complex models, such as the human population size, the estimates may be in much wider range. For example, the projections of human population in 2050 made by UN range from low 8 billion to high 10.5 billion. The best case for validating the model is to be able to explain the actual observed data within the experimental error.

7. Additional information

The population growth models are just the basics of the population biology. The models above only apply to the population of a single species. However, most species on the planet interact with other species and mutually influence their population sizes. One of the examples of a model involving two species is the parasite-host system. Such model will consider additional factors compared to single-species model: hosts that carry parasites will give rise to the next generation of parasites, while host that do not carry parasites will produce their own offspring, the fraction of the hosts that are parasitized depend on the rate of the encounters of the two species etc. Another possible example is the interaction between a plant species and a herbivore. Such models generally require differential equations to describe them.

by Evgeny. Also posted on my website