Name: population growth: exponential vs. logistic
Uploaded: 2026-01-12T12:24:36.000Z
Duration: 1 min 25 s
Channel: JoVE

Modelar el crecimiento poblacional con ecuaciones diferenciales conecta suposiciones sobre cómo cambian las poblaciones a lo largo del tiempo. Asumiendo recursos ilimitados, la tasa de crecimiento poblacional es proporcional a su tamaño.Traduciendo esta suposición en una ecuación, se obtiene una ecuación diferencial de primer orden con una solución exponencial.En realidad, recursos como la comida, el agua, el refugio y la energía son finitos, lo que limita el crecimiento poblacional a un tamaño máximo sostenible llamado capacidad de carga M.Si la población es menor que M, los recursos son suficientes y el crecimiento se produce hacia M. Si supera M, la escasez provoca tasas de mortalidad o migración más altas, reduciendo la población.Para modelar estas tendencias, se hacen dos suposiciones: primero, para un valor pequeño de P(t), la tasa de crecimiento es proporcional a P(t), y segundo, si P(t) supera M, la población comienza a disminuir.Combinando estas suposiciones, se obtiene el modelo de crecimiento logístico. La tasa de crecimiento es proporcional al tamaño actual de la población P(t) y a la diferencia entre la capacidad de carga M y P(t), que también puede expresarse de forma general.

modelado con ecuaciones diferenciales

La din&#xE1;mica poblacional puede describirse matem&#xE1;ticamente considerando el tama&#xF1;o de la poblaci&#xF3;n P(t) en funci&#xF3;n del tiempo. La tasa de cambio de la poblaci&#xF3;n se representa entonces mediante la derivada de P(t). Un supuesto sencillo es que la tasa de crecimiento es proporcional al tama&#xF1;o de la propia poblaci&#xF3;n. Esto da lugar a un modelo de crecimiento exponencial, donde la poblaci&#xF3;n crece r&#xE1;pidamente sin l&#xED;mite. Si bien esta es una primera aproximaci&#xF3;n &#xFA;til, no refleja un comportamiento realista a largo plazo.
Crecimiento exponencial y sus limitaciones
En el modelo de crecimiento exponencial, las poblaciones crecen continuamente a una tasa determinada por su tama&#xF1;o actual y una constante de proporcionalidad. Esto da lugar a un crecimiento que se acelera con el tiempo. Sin embargo, en entornos naturales, el crecimiento ilimitado no es posible debido a limitaciones de recursos, como el alimento, el espacio y la competencia. Esta discrepancia entre la teor&#xED;a y la realidad requiere un modelo refinado.
Crecimiento log&#xED;stico y capacidad de carga
Para abordar estas limitaciones, el modelo de crecimiento log&#xED;stico incorpora el concepto de capacidad de carga, denotada por M. Se trata de la poblaci&#xF3;n m&#xE1;xima que el entorno puede mantener a lo largo del tiempo. El modelo parte de dos supuestos clave: cuando la poblaci&#xF3;n es peque&#xF1;a, el crecimiento se asemeja mucho a un comportamiento exponencial, pero a medida que la poblaci&#xF3;n se acerca a la capacidad de carga, la tasa de crecimiento se ralentiza y finalmente se estabiliza. Si la poblaci&#xF3;n supera este l&#xED;mite, las presiones ambientales hacen que la poblaci&#xF3;n disminuya y tienda de nuevo al equilibrio.
Modelo de crecimiento combinado
Al integrar estos supuestos, el modelo log&#xED;stico equilibra la tendencia de las poblaciones a crecer r&#xE1;pidamente cuando su tama&#xF1;o es reducido con las restricciones ambientales que impiden la expansi&#xF3;n indefinida. El patr&#xF3;n de crecimiento resultante comienza con un aumento exponencial, pero se desacelera gradualmente hasta estabilizarse, formando una curva en forma de S. Este modelo proporciona una representaci&#xF3;n m&#xE1;s precisa y ampliamente aplicable de la din&#xE1;mica poblacional en ecolog&#xED;a y campos relacionados.

Modelado con ecuaciones diferenciales

La din&#xE1;mica poblacional puede describirse matem&#xE1;ticamente considerando el tama&#xF1;o de la poblaci&#xF3;n P(t) en funci&#xF3;n del tiempo. La ...

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JoVE Core

Calculus

Cálculo

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Unidad

Differential Equations

Ecuaciones diferenciales

KEY TERMS AND CONCEPTS

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Modeling with Differential Equations

population growth: exponential vs. logistic

Population growth can be modeled with differential equations by treating population size, P(t), as a function of time. The derivative of P(t) shows how fast the population changes. A simple starting idea is that the growth rate is proportional to the population size.

That assumption gives an exponential growth model. In this model, the population rises continuously and faster over time. It can be a useful first approximation, but it does not match long-term behavior in nature.

Real populations cannot grow forever because resources are limited. Food, space, and competition all slow growth. To describe that limit, the logistic growth model uses carrying capacity, M, which is the largest population the environment can support over time.

The logistic model also matches two key stages of growth. When the population is small, it behaves much like exponential growth. As the population gets close to carrying capacity, the growth rate slows and the population levels off. If the population goes above that limit, environmental pressure can push it back down toward equilibrium.

Together, these ideas produce a more realistic growth curve. The model starts with rapid increase and then bends into an S-shaped curve. This makes logistic growth useful for describing population dynamics in ecology and related fields.

Population Growth: Exponential vs. Logistic

population-growth-exponential-vs-logistic

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modeling with differential equations

Population growth can be modeled with differential equations by treating population size, P(t), as a function of time. The derivative of P(t) shows how fast the population changes. A simple starting idea is that the growth rate is proportional to the population size. That assumption gives an exponential growth model. In this model, the population rises continuously and faster over time. It can be a useful first approximation, but it does not match long-term behavior in nature. Real...

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key terms and concepts

Learn how differential equations model population growth, from exponential increase to logistic growth with carrying capacity and an S-shaped curve.

Population Growth: Exponential vs. Logistic | 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

Population Growth: Exponential vs. Logistic

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