In all ecosystems living things are related in similar ways. Whether it is a desert or a pond, we can always find species that maintain links of competition, symbiosis, parasitism, predation… 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 the one proposed by biophysicist Alfred J. Lotka Y the mathematician Vito Volterra —independently—in the 1920s, which is able to describe relationships between predators and prey.
The so-called Lotka-Volterra model was first used to answer a question posed by the marine biologist Umberto d’Ancona. He had observed that, during World War I, fishermen in the Adriatic Sea were catching a higher percentage than usual of sharks, rays, and other large predators. D’Ancona attributed this anomaly to the decrease in fishing activity caused by the war. However, it was strange that this reduction did not benefit more the medium-sized species, most consumed by humans. Intrigued, he consulted the problem with Volterra. The mathematician wanted to describe, using a pair of equations, how this change affected the average number of prey and predators.
To do this, he devised a system of two equations that reflects the interconnection between the two species, whose unknowns are the number of prey —for example, medium-sized fish—, represented by the variable x, and of predators —sharks—, represented by Y. The equations include four fixed parameters: A, which represents the reproductive rate of the prey; B, which is related to the probability that a prey will be hunted; C, the mortality rate of predators; and D, related to the proportion of catches necessary for the reproduction of predators. The equations establish the values of the derivatives x’ and y’, which represent the variation of the populations in 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, starting from a prey population of x individuals and predators of y, is equal to Ax, the number of prey hatched, minus Bxy, which represents the number of prey captured in The hunt. On the other hand, the second equation establishes that the variation of the predators is Dxy, the predators that are born thanks to the food obtained, minus Cy, the deceased predators.
In this model, when there are no predators, prey reproduce at an exponential rate, with no limit. On the other hand, the absence of prey leads predators to become extinct, and the more individuals have to compete for the scarce food available, the faster the population will decline.
2. Graphics
In any other case, the equations establish that, over time, both populations fluctuate periodically around some average values, given by C/D for prey and by A/B for predators —in the image, marked with broken lines. Yes fishing activity is introduced in the equations with a new parameter E, an effect equivalent to reducing the birth rate of preys —changing A to AE— and increasing the mortality rate of predators —changing from C to C+E— is obtained. In this way, a decrease in fishing activity, that is, in E, translates into an increase in the average number of predators —(AE)/B— and a reduction in the number of prey —(C+E)/D— , which is just 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 consequence of mismatches between labor and wages. Specifically, he argued that the employment rate and the cost of wages are variables that evolve cyclically, similar to the number of prey and predators. Goodwin’s proposal to describe the labor market introduced a new idea in theoretical economics: his mathematical model gave an interpretation of capitalism’s own cycles through endogenous causes to the system, without the need to resort to external shocks.
Despite their simplicity, the Lotka-Volterra equations are useful for modeling various complex systems and are still applied in many cases today. Furthermore, in recent years there have been various variations in order to simulate more complex situations, such as interactions between a greater number of species, phenomena of cannibalism among predators or defensive strategies of prey. The Lotka-Volterra system was one of the first in the history of mathematical modelling, a path of success followed by many of the models used today in branches apparently as far apart as meteorology or epidemiology.
Alba Garcia Ruiz Y Enrique Garcia Sanchez They are pre-doctoral researchers of the Higher Council for Scientific Research at the Institute of Mathematical Sciences.
Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Edition and coordination: Ágata A. Timón G Longoria (ICMAT).
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