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

In einem Gewässer wird das Gesamtgewicht der Fischpopulation als Biomasse der Fische bezeichnet. Die Wachstumsrate einer Fischpopulation lässt sich durch die Funktion G(t) modellieren, wobei t die Anzahl der Jahre und G(t) die Wachstumsrate in Kilogramm pro Jahr darstellt.Die anfängliche Biomasse für das Jahr 2000 wird angegeben.Das Ziel ist es, die Biomasse der Fischpopulation im Jahr 2005 zu bestimmen.Um dies zu finden, wird die Ratenfunktion integriert, um die Biomassefunktion B(t) zu bestimmen.Die Substitutionsmethode wird verwendet, um das Integral zu lösen, indem U und seine Ableitung du genommen werden.Das resultierende Integral in u wird vereinfacht, und der Ausdruck wird dann zurückgeschrieben, um B(t) mit einer unbekannten Konstanten c zu erhalten.Die bekannten Größen werden als Anfangsbedingungen betrachtet. Nach dem Einsetzen dieser Werte und dem Lösen wird die unbekannte Konstante c gefunden. Dann wird dieses c in die Gleichung eingefügt und B(t) umgeschrieben.Um die Biomasse im Jahr 2005 zu bestimmen, wird t gleich 5 in diese Biomassegleichung eingesetzt.Nach Vereinfachung der Gleichung ergibt sich die Biomassefunktion, die die Gesamtbiomasse der Fischpopulation im Jahr 2005 zeigt.

wachstumsmodelle mit integration: problemlösung

In der Populationsmodellierung bietet die Integration eine systematische M&#xF6;glichkeit, aus bekannten &#xC4;nderungsraten akkumulierte Gr&#xF6;&#xDF;en zu bestimmen. Eine solche Anwendung findet sich in der &#xD6;kologie, wo das Gesamtgewicht einer Fischpopulation in einem Gew&#xE4;sser als deren Biomasse bezeichnet wird. Ist die Wachstumsrate dieser Biomasse als Funktion der Zeit bekannt, l&#xE4;sst sich die Gesamtbiomasse zu einem zuk&#xFC;nftigen Zeitpunkt mithilfe der Integralrechnung berechnen.
Wachstumsrate und Biomassefunktion
Die Wachstumsrate der Fischpopulation wird durch die Funktion G(t) dargestellt, wobei t die Zeit in Jahren und G(t) die Wachstumsrate der Biomasse in Kilogramm pro Jahr angibt. Die Biomasse selbst wird durch die Funktion B(t) beschrieben. Da die Wachstumsrate die zeitliche Ver&#xE4;nderung der Biomasse angibt, erh&#xE4;lt man die Biomassefunktion durch Integration der Wachstumsratenfunktion &#xFC;ber die Zeit.
Anwendung der Substitutionsmethode
Zur Auswertung des die Biomassefunktion definierenden Integrals wird die Substitutionsmethode verwendet. Dabei w&#xE4;hlt man eine geeignete Substitutionsvariable u und deren Differential du, um den Integranden zu vereinfachen. Nach der Substitution wird das Integral in Abh&#xE4;ngigkeit von u umgeschrieben, was die Auswertung erleichtert. Nach Abschluss der Integration wird der Ausdruck wieder in die urspr&#xFC;ngliche Variable t zur&#xFC;ckgef&#xFC;hrt. Dies ergibt eine allgemeine Form der Biomassefunktion, die eine unbekannte Integrationskonstante enth&#xE4;lt.
Anwendung der Anfangsbedingungen
Die Anfangsbiomasse der Fischpopulation im Jahr 2000 ist bekannt und dient als Anfangsbedingung. Durch Einsetzen des entsprechenden Zeitwertes und der Biomasse in die allgemeine Biomassefunktion l&#xE4;sst sich die Integrationskonstante bestimmen. Dieser Schritt gew&#xE4;hrleistet, dass das Modell den bekannten Zustand des Systems zum Startzeitpunkt korrekt widerspiegelt. Nach der Bestimmung dieser Konstante wird die Biomassefunktion in ihrer endg&#xFC;ltigen Form angegeben.
Bestimmung der Biomasse im Jahr 2005
Um die Biomasse der Fischpopulation im Jahr 2005 zu berechnen, wird die Zeitvariable auf f&#xFC;nf Jahre nach 2000 gesetzt. Durch Einsetzen dieses Wertes in die finale Biomassefunktion und anschlie&#xDF;ende Vereinfachung erh&#xE4;lt man die Gesamtbiomasse der Fischpopulation im Jahr 2005.

Wachstumsmodelle mit Integration: Problemlösung

In der Populationsmodellierung bietet die Integration eine systematische M&#xF6;glichkeit, aus bekannten &#xC4;nderungsraten akkumulierte Gr&#xF6;&#xDF;en ...

growth-models-with-integration-problem-solving

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Calculus

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Integrals

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KEY TERMS AND CONCEPTS

WICHTIGE BEGRIFFE UND KONZEPTE

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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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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