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Revista de Matemática Teoría y Aplicaciones

versão impressa ISSN 1409-2433

Rev. Mat vol.21 no.2 San José Jul./Dez. 2014

 

Modelo matemático del transporte de una toxina en una red trófica marina

Mathematical model for toxin transport in a marine food chain

Daniel Arbelaez A.*+ Jorge Mauricio Ruiz V. *



Resumen

Algunos casos de intoxicación por consumo de peces contaminados, como la intoxicación por ciguatera, ocurren inesperadamente y no son fáciles de detectar previamente, dado que los peces que portan la toxina no presentan aspecto y/o síntomas de enfermos. En este trabajo proponemos un modelo matemático para el transporte y acumulación de una toxina a través de una cadena alimentaria. El modelo se plantea mediante un sistema rígido de ecuaciones diferenciales que describen la dinámica. Se analiza la estabilidad local de la solución de equilibrio. Se discuten diferentes escenarios de aparición de brotes de una toxina a partir de simulaciones numéricas obtenidas mediante un esquema de discretización que combina un método de Runge-Kutta de tercer orden y la regla del trapecio, evitando la rigidez del sistema. Los resultados muestran que el tiempo que tarda en desaparecer la toxina en la red trófica depende del estado en que se encuentra la dinámica poblacional al momento del brote. Esta información puede emplearse para establecer un tiempo de veda en la pesca de tal manera que la toxina se reduzca a niveles inocuos para la salud humana.

Palabras clave: cadena trófica; modelación matemática; transporte de biotoxinas; ecuaciones diferenciales rígidas.

Abstract

Some cases of poisoning by consumption of contaminated fish, like the Ciguatera poisoning, occur unexpectedly and they are not easy to detect previously since the fish that carry the toxin do not have appearance and/or symptoms of illness. In this paper a mathematical model for transport and acumulation of a toxin through a food chain is proposed. The model is a stiff system of ordinary differential equations that describes the dynamic. We propose a numerical scheme that combines a third-order Runge-Kutta method and trapezoidal rule to avoid the stiffness of the system. Several scenarios of toxin outbreaks are simulated; the results show that the time it takes to the toxin disappear in the trophic chain, depends on the state of the dynamics population at the time of the outbreak. This information can be used to set a ban on fishing until the toxin be reduced to harmless levels for the human health.

Keywords: food chain; mathematical modeling; biotoxin transport; stiff differential equations.

Mathematics Subject Classification: 92D25, 93A30, 37N25, 65L04.



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Referencias

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* Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá D.C., Colombia. E-Mail: darbelaeza@unal.edu.co

Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá D.C., Colombia. E-Mail: jmruizv@unal.edu.co

Received: 9/Sep/2013; Revised: 4/Jun/2014; Accepted: 12/Jun/2014

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