Mathematics and Biology

Each organic being is striving to increase in a geometrical ratio. Thus, it wouldn't be surprising when asked why one species hasn’t been victorious over another in the great battle of life. With the aid of carefully arranged quantitative experiments in which an animal was isolated in a special chamber, all the complicating circumstances being removed, Pavlov discovered the laws of the formation, preservation and extinction of the conditional reflexes, which constitute the basis for an objective conception of higher nervous activity.

He said, “The rate of multiplication is different for every kind of organism in close connection with their size. Small organisms, that is organisms weighing less, at the same time multiply much more rapidly than large organisms.” This infinity of the possible multiplication of organisms can be considered as the subordination of the increase of living matter in the biosphere to the rule of inertia. It can be regarded as empirically established that the process of multiplication is retarded in its manifestation only by external forces; it dies off with a low temperature, ceases and becomes weaker with an insufficiency of food or respiration.

Moreover, botanists characterize the struggle for existence by the percentage decrease in the number of individuals on a unit of surface in a certain unit of time.The smaller the number of trunks remaining on the unit of surface, the greater the percentage of those which perish. In better conditions of existence competition proceeds with greater intensity, and the percent of individuals which perish is greater. A biotype which shows itself to be the most resistant in an intrabiotic struggle for existence, may turn out to be the weakest one in an interbiotic struggle between different biotypes of the same species.

It is believed competition is keenest when the individuals are most similar. The more unlike plants are, the greater the difference in their needs and hence some adjust themselves to the reaction of others. Relative stability of separate biotypes tells us that the biotypes yielding the greatest percentage of survivors under a small density of cultivation may occupy the last place in this respect in conditions of a dense culture.

However, the application of quantitative methods to experimental biology presents such difficulties and has more than once led to such erroneous results that the reader would have the right to consider the material very skeptical. It is very well known that the differential equations derived from the curves observed in an experiment can be only regarded as empirical expressions and they do not throw any real light on the underlying factors which control the growth of the population.

Thus the right way to go about the investigation is, as Professor Gray ('29) says, a direct study of factors which control the growth rate of the population and the expression of these factors in a quantitative form. In this way real differential equations will be obtained and in their integrated form they will harmonize with the results obtained by observation.

Generally speaking, biologists usually have to deal with empirical equations. The essence of such equations is admirably expressed in the following words of Raymond Pearl, “The worker in practically any branch of science is more or less frequently confronted with this sort of problem: he has a series of observations in which there is clear evidence of a certain orderliness, on the one hand, and evident fluctuations from that order, on the other hand. What he obviously wishes to do, is to emphasize the orderliness and minimize the fluctuations about it. He would like an expression, exact if possible, or, failing that, approximate, of the law if there be one. This means a mathematical expression of the functional relation between the variables.”

It should be made clear at the start that there is, unfortunately, no methods known to mathematics which will tell anyone in advance of the trial what is either the correct or even the best mathematical function with which to graduate a particular set of data. The choice of the proper mathematical function is essentially, at its very best, only a combination of good judgment and good luck.

Furthermore, the struggle for the essential substances is not the struggle for the existence of all living things. For instance, in the process of photosynthesis, the transforming surface of green living matter exclusively employs light of a specific wavelength. Therefore, experiments were conducted to determine how the microcosm's energy would be allocated between the two competing species' populations.

Now we'll look at how Protozoa fight for survival in tightly curated populations (Paramecium caudatum and Paramecium aurelia). In this case, growth will be limited by a lack of organic nutritive components, which is similar to a lack of accessible energy. The microcosm's energetical resources will be kept at a constant level during the experiment, which is the second unique feature. This is similar to what occurs in nature, where the energy level is maintained by a continuous influx of solar energy.

Despite what appears to be a complete change of conditions in passing from one period of growth to another, a certain law of the struggle for existence, which can be expressed by a system of differential equations of competition, remains constant throughout. The law states that each species has definite multiplication coefficients.

As we move on to the relationship between the predator and the prey and how one specie destroys another, mathematical investigations have shown that the process of interaction between the predator and the prey leads to periodic oscillations in numbers of both species, and all this of course ought to be verified under carefully controlled laboratory conditions. This case presents a considerable interest from a purely biological viewpoint. The amount of food required by Didinium is very great and it demands a fresh Paramecium every three hours. Satisfactory results have been obtained on Osterhout's medium, but here also Didinium has grown worse than on the oaten medium.

Mathematicians arrived at this conclusion by studying the properties of the differential equation for the predator-prey relations. Such periodic oscillations can continue for a long time. This experiment was repeated many times, being sometimes made in a large vessel in which there were many hundreds of thousands of infusoria. The predator was introduced at different moments of the growth of population of the prey, but nevertheless the same result was always produced.

A Literature Review by-

Hiloni Sanghavi

Niharika Garg

Shreyash Kanojia

Jitesh Bohra

Tanmay Nayak

Gungun Singh

Cover Source- http://sites.science.oregonstate.edu/~deleenhp/teaching/fall15/MTH427/Gause-The-Struggle-for-Existence.pdf