Decoding the math of competition

Competition is everywhere in the natural world. Any attempt made to observe the natural system makes this evident. Carnivores compete with other carnivores over the best hunting grounds. Males compete for dominance over other males for reproductive advantage.

Even something as simple as throwing a few grains of rice to a group of pigeons can demonstrate to us how inbuilt competition is in the natural world — a concept that has fascinated ecologists for many decades. These complex workings of ‘interspecific competition’ — competition between two different species of animals in natural systems — has caught the attention of two undergraduate students from the Indian Institute of Science (IISc), Bengaluru, and is the focus of their recent research.

Ecologists bank on various mathematical formulae to model the competition that exists in the natural world. Such models can help them determine the well-being of the ecosystem, identify species that may soon give in to the competition and become endangered or extinct, and understand predator-prey relationships.

Complexity of natural systems

Lotka-Volterra equations are the most commonly used mathematical equations to study competition in the wild. The equations were first introduced by Alfred J Lotka to study chemical reactions. He did not apply them until 1925 to study the interaction between predator and prey. In a 1926 paper entitled ‘Fluctuations in the abundance of species considered mathematically’, Vito Volterra used these equations to quantify the interactions between two or more biologically associated species. This publication built on the already existing equations by making various special inquiries into how two species in the wild interact.

The Lotka-Volterra equations are a pair of first-order, nonlinear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one as predator and the other as prey. The equation takes into account the total number of interacting species, the population sizes of each of the species, the carrying capacity of the species (the maximum number of individuals of the species that can be supported by its environment) and the intrinsic growth rate of the species (birth rate subtracted from the death rate).

Though there are so many variables, the equation doesn’t always do justice to the deep complexity of natural systems. These equations assume that the prey population finds ample food at all times, that this food supply depends entirely on the population of the prey, that the rate of change of population is proportional to its size, and that predators have unlimited appetite. The equation disregards the genetic adaptation due to change in environment and also does not take into account that individuals of different ages have different food requirements.

Over the years, many researchers have modified the equation to take into account all parameters of the natural system in which the species are competing, and in the process, they addressed some assumptions the mathematical equations made about the natural systems, while some others remained unchanged. Some such assumptions include not considering the age and/or size of the competitors, all individuals consuming the same amount of resource, the resources never being consumed to extinction, not considering the interaction among resources and the growth rate of the consumers being dependent on the consumption of the resource.

Far-fetching effects

The researchers at IISc were quick to notice that some of these assumptions do not comply with a realistic system, and have focused on the competition coefficient — a term in Lotka-Volterra type equations that represent the effect one species has on another species. “We are trying here to find a new formulation for the competition coefficient, which is a major parameter in these equations. Usually it is put by hand or deduced mathematically from some resource growth models. Here, we are trying to deduce it from real field data,” says Anshuman Swain, one of the study’s authors. Their study was published recently in the journal Current Science.

The researchers have remodelled the competition coefficient using new terms that are relevant to field data. For example, consider a study that focuses on how jackals compete with tigers in the wild. Both the animals are carnivores and have a definite overlap in the prey they consume, resulting in competition for the resources. Jackals hunt in groups but a tiger searches for the prey alone. The modified formula allows the researchers of such study to take into account some of such behavioural characteristics of the study organism while calculating competition coefficients.

The modifications introduced by the researchers have far-fetching effects. The new model overcomes the challenges posed by assumptions of previous models and fits natural systems better. It will also help other researchers to use the data they have collected in the field at specific time intervals to estimate population sizes — something that was not previously possible to great accuracy as the population growth was assumed to be linear for the equation. The new model takes into account the fact that resources can be exhausted by a species, and the genetic background and age of the competitors can determine the minimum amount of the resource consumed by an individual. Finally, it also tackles the assumption that resources do not interact with one another.

“This model can be used to study most natural systems involving competition according to us, but hasn’t yet been used to study any specific system. We have just given a new analytic formulation,” adds Anshuman. The model can not only be used to study interspecific competition, but also intraspecific competition which consists of different groups of the same species of animal competing with one another, taking into account the effect of age. With the help of this improved formulation researchers can now, with a greater accuracy than before, estimate competition coefficients and population sizes in their study system. The new formulations will help us gain a more in depth understanding of how competition takes place in the wild.

(The author is with Gubbi Labs, a Bengaluru-based research collective)