In all ecosystems, living beings are related to each other in a similar way. Whether it’s a desert or a pond, we can always find species that maintain connections through competition, symbiosis, parasitism, predation, and so on. These interactions can be simulated with systems of differential equations whose solutions describe the expected behavior over time. An example of these biological models is that proposed by biophysicist Alfred J. Lotka and mathematician Vito Volterra – independently – in the 1920s, which is able to describe the relationships between predators and prey.
The so-called Lotka-Volterra model was first used to answer a question asked by marine biologist Umberto d’Ancona. He had observed that fishermen in the Adriatic Sea caught a higher than usual percentage of sharks, rays and other large predators during World War I. D’Ancona attributed this anomaly to the decline in fishing activity caused by the war. However, it was strange that this reduction no longer benefited the medium-sized species, which are most consumed by humans. Intrigued, he consulted the issue with Volterra. The mathematician wanted to use two equations to describe how this change would affect the average numbers of prey and predators.
To do this, he devised a system of two equations that reflect the connection between the two species, whose unknowns are the number of prey – say, medium-sized fish – represented by the variable x, and the number of predators – sharks. , represented by Y. The equations contain four fixed parameters: A, which represents the reproductive rate of the prey; B, which relates to the likelihood of a prey being hunted; C, the mortality rate of predators; and D, related to the proportion of catches required for predator reproduction. The equations establish the values of the derivatives x’ and y’, which represent the variation of the populations over time with respect to the previous variables and parameters.
1. Equations without fishing
The first equation indicates that the variation in the number of prey, given a prey population of x individuals and predators of y, equals Ax, the number of prey hatched, minus Bxy, which represents the number of prey caught in the hunt . On the other hand, the second equation states that the variation of predators is Dxy, the predators that are born thanks to the food received, minus Cy, the predators that died.
Equation for calculating variation in prey numbers and predator numbers.
In this model, when there are no predators, prey reproduces at an exponential rate without limit. On the other hand, the lack of prey leads to the extinction of predators, and the more individuals have to compete for scarce food, the faster the population will decline.
2. Graphics
In every other case, the equations show that both populations fluctuate periodically over time around averages given by C/D for prey and A/B for predators—denoted by broken lines in the image. If fishing activity is introduced into the equations with a new parameter E, we get an effect corresponding to a reduction in the birth rate of prey – change from A to AE – and an increase in the mortality rate of predators – from C to C + E – Auf this way, a decrease in fishing activity, ie in E, leads to an increase in the average number of predators —(AE)/B— and a decrease in the number of prey animals —(C+E)/D— , that’s exactly what d’Ancona observed.
3. Equations with fishing
The utility of the predator-prey model is not limited to ecology. In 1967, the economist Richard M. Goodwin used these equations to explain economic fluctuations as a result of mismatches between labor and wages. In particular, he proposed that the employment rate and labor costs are variables that cycle, much like prey and predator numbers. Goodwin’s proposal to describe the labor market introduced a new idea into theoretical economics: his mathematical model provided an interpretation of capitalism’s own cycles through causes endogenous to the system, without having to resort to external shocks.
Goodwin used these equations to explain economic fluctuations as a result of discrepancies between work and wages.
Despite their simplicity, the Lotka-Volterra equations are useful for modeling various complex systems and are still used in many cases today. In addition, several variations have been introduced in recent years to simulate more complex situations, such as: B. interactions between a larger number of species, cannibalism among predators or defense strategies of prey. The Lotka-Volterra system was one of the first in the history of mathematical modeling, a path of success followed by many of the models used today in fields as seemingly distant from each other as meteorology or epidemiology.
Alba Garcia Ruiz Y Enrique Garcia Sanchez They are PhD researchers from the Higher Council for Scientific Research at the Department of Mathematical Sciences.
coffee and theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, meeting points between mathematics and others share social and cultural expressions and remember those who shaped their development and knew how to turn coffee into theorems. The name recalls the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that converts coffee into theorems.”
Editing and coordination: Ágata A. Timón G. Longoria (ICMAT).
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