Name: finding fish biomass with integrals
Uploaded: 2026-01-12T12:24:52.000Z
Duration: 1 min 27 s
Channel: JoVE

Dans un plan d’eau, le poids total de la population de poissons est appelé la biomasse de poissons. Le taux de croissance d’une population de poissons peut être modélisé par la fonction G(t), où t est le nombre d’années et G(t) est le taux de croissance mesuré en kilogrammes par an.La biomasse initiale pour l’an 2000 est indiquée.L’objectif est de trouver la biomasse de la population de poissons en 2005.Pour cela, la fonction de taux est intégrée pour trouver la fonction de biomasse B(t).La méthode de substitution est utilisée pour résoudre l’intégrale en prenant u et sa dérivée du.L’intégrale résultante dans u est simplifiée, et l’expression est ensuite réécrite pour obtenir B(t) avec une constante inconnue c.Les quantités connues sont prises comme conditions initiales. Après avoir substitué ces valeurs et résolu, on trouve la constante inconnue c. Ensuite, en substituant ce c dans l’équation, B(t) est réécrit.Pour trouver la biomasse en 2005, t égal à 5 est substitué à cette équation de biomasse.Après simplification de l’équation, la fonction de biomasse est trouvée, montrant la biomasse totale de la population de poissons en 2005.

modèles de croissance avec intégration : résolution de problèmes

En mod&#xE9;lisation des populations, l&#x2019;int&#xE9;gration constitue une m&#xE9;thode syst&#xE9;matique permettant de d&#xE9;terminer des grandeurs cumul&#xE9;es &#xE0; partir de taux de variation connus. Elle trouve notamment une application en &#xE9;cologie, o&#xF9; la biomasse d&#x2019;une population de poissons dans un plan d&#x2019;eau d&#xE9;signe son poids total. Lorsque le taux de croissance de cette biomasse est connu en fonction du temps, le calcul int&#xE9;gral permet de d&#xE9;terminer la biomasse totale &#xE0; une date ult&#xE9;rieure.
Taux de croissance et fonction de biomasse
Soit G(t) la fonction repr&#xE9;sentant le taux de croissance de la population de poissons, o&#xF9; t repr&#xE9;sente le temps mesur&#xE9; en ann&#xE9;es et o&#xF9; G(t) est ce taux de variation de la biomasse, exprim&#xE9; en kilogrammes par an. La biomasse elle-m&#xEA;me est d&#xE9;crite par une fonction B(t). Puisque le taux de croissance caract&#xE9;rise l&#x2019;&#xE9;volution de la biomasse au cours du temps, la fonction de biomasse s&#x2019;obtient par l&#x2019;int&#xE9;gration de la fonction G par rapport au temps.
Utilisation de la m&#xE9;thode de substitution
Pour &#xE9;valuer l&#x2019;int&#xE9;grale d&#xE9;finissant la fonction de biomasse, vous avez recours &#xE0; la m&#xE9;thode de substitution. Celle-ci consiste &#xE0; choisir une variable de substitution appropri&#xE9;e u ainsi que sa diff&#xE9;rentielle, du, afin de simplifier l&#x2019;int&#xE9;grande. Apr&#xE8;s avoir effectu&#xE9; la substitution, l&#x2019;int&#xE9;grale est r&#xE9;&#xE9;crite en fonction de u, ce qui en facilite l&#x2019;&#xE9;valuation. Une fois l&#x2019;int&#xE9;gration r&#xE9;alis&#xE9;e, l&#x2019;expression obtenue est reconvertie en la variable initiale t, ce qui conduit &#xE0; une forme g&#xE9;n&#xE9;rale de la fonction de biomasse qui inclut une constante d&#x2019;int&#xE9;gration inconnue.
Application des conditions initiales
La biomasse initiale de la population de poissons en 2000 est donn&#xE9;e et sert de condition initiale. En substituant la valeur correspondante du temps ainsi que la biomasse associ&#xE9;e dans la fonction de biomasse g&#xE9;n&#xE9;rale, vous pouvez d&#xE9;terminer la constante d&#x2019;int&#xE9;gration. Cette &#xE9;tape garantit que le mod&#xE8;le refl&#xE8;te fid&#xE8;lement l&#x2019;&#xE9;tat connu du syst&#xE8;me &#xE0; l&#x2019;instant initial. Une fois cette constante d&#xE9;termin&#xE9;e, la fonction de biomasse est r&#xE9;&#xE9;crite sous sa forme d&#xE9;finitive.
D&#xE9;termination de la biomasse en 2005
Pour d&#xE9;terminer la biomasse de la population de poissons en 2005, on fixe la variable temporelle &#xE0; cinq ans apr&#xE8;s 2000. En substituant cette valeur dans la fonction de biomasse finale et en simplifiant, on obtient la biomasse totale de la population de poissons en 2005.

Modèles de croissance avec intégration : résolution de problèmes

En mod&#xE9;lisation des populations, l&#x2019;int&#xE9;gration constitue une m&#xE9;thode syst&#xE9;matique permettant de d&#xE9;terminer des grandeurs ...

growth-models-with-integration-problem-solving

Education

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Calculus

Calcul

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Integrals

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TERMES ET CONCEPTS CLÉS

Growth Models with Integration: Problem Solving

finding fish biomass with integrals

Fish biomass can be modeled with integration when the growth rate is known over time. In ecology, biomass means the total weight of a fish population in a body of water. If the growth rate of that biomass is given as a function of time, calculus can be used to find the total biomass at a later date.

The growth rate is written as G(t), where t is time in years and G(t) is the rate of change of biomass in kilograms per year. The biomass itself is written as B(t). Because the growth rate shows how biomass changes, the biomass function is found by integrating the growth rate with respect to time.

To evaluate the integral for B(t), the substitution method is used. A new variable, u, and its differential, du, are chosen to make the integrand simpler. After the integral is rewritten in terms of u, it is easier to solve. Then the expression is changed back into the original variable t, which gives a general biomass function with a constant of integration.

An initial biomass value from the year 2000 is then used to find that constant. This starting value acts as an initial condition. By substituting the known time and biomass into the general function, the constant of integration can be determined. The biomass function is then written in its final form.

To find the biomass in 2005, the time variable is set to five years after 2000. Substituting t = 5 into the final biomass function and simplifying gives the total fish biomass for that year.

Finding Fish Biomass with Integrals

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growth models with integration: problem solving

Fish biomass can be modeled with integration when the growth rate is known over time. In ecology, biomass means the total weight of a fish population in a body of water. If the growth rate of that biomass is given as a function of time, calculus can be used to find the total biomass at a later date. The growth rate is written as G(t), where t is time in years and G(t) is the rate of change of biomass in kilograms per year. The biomass itself is written as B(t). Because the growth rate shows...

key terms and concepts

Use integration and substitution to model fish biomass from a growth rate, apply an initial condition, and calculate the biomass in 2005.

Finding Fish Biomass with Integrals | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Finding Fish Biomass with Integrals

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