Monday, January 30, 2023

A formula for sharks: equations shared by ecologists and economists

In all ecosystems, living beings are related in the same way. 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 modeled with systems of differential equations whose solutions vary over time Describes expected behavior. An example of these biological models is the biophysicist Alfred J. Lotka and the mathematician Vito Volterra independently in the 1920s, capable of describing the relationship between predators and prey.

The so-called Lotka–Volterra model was first used to answer a question posed by marine biologist Umberto d’Ancona. He had noticed 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 discrepancy to a decrease in fishing activity due to the war. It was strange, however, that this reduction did not benefit the medium-sized species most commonly consumed by humans. Intrigued, he consulted on the problem with Volterra. The mathematicians wanted to describe how this change affected the average numbers of prey and predators, using a pair of equations.

To do this, they devised a system of two equations that represents the interrelationship between two species that have an unknown number of prey items—for example, medium-sized fish—, represented by the variable x, and predators—sharks—, represented by Y. The equations include four fixed parameters: A, which represents the prey’s reproductive rate; b, which is related to the probability that a prey will be hunted; c, mortality of predators; and D is related to the proportion of the catch required for the reproduction of predators. The equations establish the values ​​of the derivatives x’ and y’, which represent the variation of the population in time with respect to previous variables and parameters.

1. Equation without fishing

The first equation indicates that the variation in the number of prey, starting from a prey population of x individuals and y predators, is equal to 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 establishes that the variation of predators is Dxy, the predators born thanks to the food obtained, minus Cy, the dead 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 drives predators to extinction, and the more individuals that have to compete for the scarce food available, the more rapid the population decline.

2. Graphics

In any other case, the equations establish that, over time, both populations fluctuate periodically around mean values, given by C/D for prey and A/B for predators. – In the image, the closure is marked with lines -. If fishing activity is introduced into the equations with a new parameter E, we obtain the equivalent effect of reducing the birth rate of prey – changing A to AE – and increasing the mortality of predators – going from C to C+E. In this way, a decrease in fishing activity, i.e. in E, translates into an increase in the average number of predators – (AE)/B – and a decrease in the number of prey – (C + E)/D –. , which d’Ancona had seen.

3. Equation with fishing

The utility of the predator-prey model is not limited to ecology. In 1967, economist Richard M. Goodwin used these equations to explain economic fluctuations resulting from the mismatch between labor and wages. Specifically, he proposed that the employment rate and wage costs 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 explained capitalism’s own cycles through endogenous causes of the system, without the need to resort to external shocks.

Despite their simplicity, the Lotka–Volterra equations are useful for modeling a variety of complex systems and are still applied in many cases today. Furthermore, various variations have been introduced in recent years to simulate more complex situations, such as interactions between a greater number of species, cannibalism between predators, or defensive strategies of prey. The Lotka–Volterra system was one of the first in the history of mathematical modelling, a path to success that has followed many models used today in branches clearly distinct from meteorology or epidemiology.

alba garcia ruiz why Enrique Garcia Sanchez He is a Pre-Doctoral Researcher of the Higher Council for Scientific Research at the Institute of Mathematical Sciences.

coffee and theorem Coordinated by the Institute of Mathematical Sciences (ICMAT), there is a section dedicated to mathematics and the environment in which they are created, in which researchers and members of the Center describe the latest advances in this discipline, sharing meeting points between mathematics and others. Social and cultural expressions and remember the people who marked their development and knew how to turn coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Renyi: “A mathematician is a machine that turns coffee into theorems.”

Version and Sync: Agate A. Rudder G Longoria (ICMAT),

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