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

نمذجة نمو السكان باستخدام المعادلات التفاضلية تربط الافتراضات حول كيفية تغير السكان مع مرور الوقت. وبافتراض موارد غير محدودة، فإن معدل نمو السكان يتناسب مع حجمها.ترجمة هذا الافتراض إلى معادلة تعطي معادلة تفاضلية من الدرجة الأولى مع حل أسي.في الواقع، الموارد مثل الغذاء والماء والمأوى والطاقة محدودة، مما يحد من نمو السكان إلى حجم مستدام أقصى يسمى القدرة على التحمل M.إذا كان عدد السكان أقل من M، تكون الموارد كافية، ويحدث النمو نحو M. إذا تجاوز العدد M، فإن النقص يسبب معدلات وفيات أعلى أو هجرة، مما يقلل من عدد السكان.لنمذجة هذه الاتجاهات، يتم وضع افتراضين: أولا، بالنسبة لقيمة P(t) صغيرة، يكون معدل النمو متناسبا مع P(t)، وثانيا، إذا تجاوز P(t) مقدار M، يبدأ عدد السكان في الانخفاض.الجمع بين هذه الافتراضات يعطي نموذج النمو اللوجستي. معدل النمو يتناسب مع حجم السكان الحالي P(t) والفرق بين القدرة الاستيعابية M و P(t)، والذي يمكن التعبير عنه أيضا بشكل عام.

النمذجة باستخدام المعادلات التفاضلية

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Education

JoVE Core

Calculus

حساب التفاضل والتكامل

Unit

وحدة

Differential Equations

المعادلات التفاضلية

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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

introduction to differential equations

Modeling Motion with Second-Order Equations

Differential equations connect a function with its derivatives. They are used to describe changing systems in science and engineering. A differential equation is classified by order, which is the highest derivative it contains. A first-order equation includes only the first derivative, while a second-order equation includes the second derivative of the unknown function. A common second-order differential equation comes from a spring-mass system. In this idealized model, a mass is attached to a...

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...

separable differential equations

Solving Separable Differential Equations

A separable differential equation is a first-order differential equation that can be written with dy/dx as a product of one function of x and one function of y. This form makes it possible to separate the variables so that all y terms stay on one side and all x terms stay on the other. After the variables are separated, each side can be integrated with respect to its own variable. The result is a relationship between x and y. Depending on the functions involved, that relationship may be...

orthogonal trajectories

Perpendicular Curves in Calculus

Orthogonal trajectories are a pair of curve families that cross at right angles. In this example, the starting family is made of parabolas that open sideways along the x-axis. They share the same shape, but a scaling parameter changes how wide each curve is. Every curve in the family passes through the origin and spreads out at a different rate. To find the orthogonal trajectories, the slope of each parabola must be calculated first. The slope of a tangent shows how steep the curve is at a...

linear differential equations

Integrating Factors for Motion with Drag

The integrating factor method solves first-order linear differential equations that are not separable. It is especially useful when a system has a constant driving force and a resistive force that depends on the changing variable. In these problems, the equation can be rearranged so the left-hand side becomes a product rule after multiplication by an integrating factor. A clear example is a car moving under a constant engine force while air resistance acts against its motion. Newton’s second...

differential equations: problem solving

Differential Equations and Falling Motion

Differential equations can model the motion of a falling object when gravity and air resistance act at the same time. In this example, a heavy test weight is released from rest during a ship safety check. Gravity pulls the weight downward, and air resistance pushes upward as the speed increases. Newton’s Second Law links the net force on the weight to its acceleration. Gravity provides a constant downward force equal to mass times gravitational acceleration. Air resistance is modeled as a...

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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