Solving the logistic differential equation part 2 | Khan Academy

Khan Academy
24 Jul 201409:58
EducationalLearning
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TLDRThe video script presents a detailed mathematical derivation of the logistic differential equation, a model used to predict population growth when limited by a carrying capacity. The presenter begins by approaching the solution for N(t), the population size at time t, given an initial condition between zero and the maximum carrying capacity K. Through algebraic manipulation and the use of logarithmic properties, the presenter simplifies the equation to express N(t) in terms of exponential functions. The solution is then further refined to isolate N(t), resulting in the logistic function, which is a sigmoidal curve that increases rapidly at first and then slows as it approaches the carrying capacity. The logistic function is particularly useful for making predictions about population dynamics and is demonstrated to have the desired properties when plotted. The presenter encourages viewers to explore the function further using tools like Wolfram Alpha or a graphing calculator.

Takeaways
  • 🧮 The video discusses finding the solution to a logistic differential equation, which models population growth with a carrying capacity K.
  • 📈 The logistic function starts with an initial condition N(0) between 0 and K, representing the population at time zero.
  • ✍️ Algebraic manipulation is used to express the logistic differential equation in terms of logarithms and exponentials.
  • 🔁 By applying properties of logarithms, the equation is transformed to isolate variables and constants, leading to a solution for N, the population size at time T.
  • 🌱 The logistic function is derived to show how population growth slows as it approaches the carrying capacity K of the environment.
  • 📉 The growth rate of the population increases initially but then decreases as the population size nears K, preventing it from exceeding the carrying capacity.
  • 🧵 The solution involves taking the reciprocal of both sides of the equation and introducing constants, which are later solved for using the initial condition.
  • 🔍 The constant C is determined by using the initial condition N(0) = N_knot, allowing for the complete formulation of the logistic function.
  • 📊 The logistic function, when graphed, demonstrates the expected population growth pattern with an S-shaped, or sigmoid, curve.
  • ⏱️ As time T progresses, the population N(T) increases, but the rate of increase diminishes as the carrying capacity is approached.
  • 🔮 The logistic function can be used to make predictions about future population sizes at different times, which is valuable for ecological modeling and understanding population dynamics.
  • 🌐 The video encourages viewers to plot the logistic function using tools like Wolfram Alpha or graphing calculators to observe its properties and verify the model's behavior.
Q & A

  • What is the logistic differential equation?

    -The logistic differential equation is a mathematical model used to describe population growth that accounts for limited resources. It modifies the exponential growth model by including a carrying capacity, K, which is the maximum population size that the environment can sustain.

  • What is the initial condition for the logistic function discussed in the script?

    -The initial condition for the logistic function is that the population size, N, at time T=0, denoted as N_0, is between zero and the carrying capacity, K.

  • How does the logistic function account for the carrying capacity in population growth?

    -The logistic function incorporates the carrying capacity by slowing the growth rate as the population size approaches K. This is achieved through a term that reduces the growth rate proportionally to the ratio of the current population size to the carrying capacity.

  • What mathematical property is used to simplify the logistic differential equation?

    -The property of logarithms is used to simplify the logistic differential equation. Specifically, the use of logarithm rules allows the equation to be rewritten in a form that separates the variables, facilitating the solution process.

  • What does the variable R in the logistic function represent?

    -In the logistic function, R represents the intrinsic growth rate of the population, which is the rate at which the population would grow if there were no limiting factors such as carrying capacity.

  • How is the constant C used in the logistic function derived?

    -The constant C in the logistic function is derived by using the initial condition N(0) = N_0. By substituting T=0 into the logistic equation and equating it to N_0, the constant C can be solved for.

  • What is the significance of the term e^(-RT) in the logistic function?

    -The term e^(-RT) represents the exponential decline in the growth rate of the population as time progresses and the population size approaches the carrying capacity K. It ensures that the growth rate decreases to zero when the population reaches K.

  • What does the logistic function predict about the population size over time?

    -The logistic function predicts that the population size will increase at a rate that starts fast and then slows down as it approaches the carrying capacity K. The population size will stabilize around K in the long term, assuming no external changes to the environment or the population.

  • How can one plot the logistic function to visualize the population growth over time?

    -One can plot the logistic function using graphing software like Wolfram Alpha or a graphing calculator. By inputting the logistic function with a specific carrying capacity K and growth rate R, one can visualize how the population size changes over time, starting with an initial condition N_0.

  • What are some real-world applications of the logistic function?

    -The logistic function is used in various fields to model growth phenomena that are subject to limitations, such as population ecology, epidemiology, and economics. It helps in making predictions about future population sizes, spread of diseases, and market penetration of new products.

  • Why is the logistic function considered a more accurate model for population growth than the exponential growth model?

    -The logistic function is considered more accurate because it takes into account the finite resources of the environment, which leads to a slowdown in population growth as the carrying capacity is approached. This contrasts with the exponential growth model, which predicts unlimited growth that is unrealistic in natural systems.

Outlines
00:00
🧮 Completing the Logistic Differential Equation

The video script begins by picking up from part one, where the presenter is close to finding the solution for N of T that satisfies the logistic differential equation with an initial condition between zero and K. The presenter uses algebra and logarithm properties to manipulate the equation, eventually expressing it in the form of E^(RT+C), where E is the base of the natural logarithm, R is the growth rate, T is time, and C is a constant. The solution process involves taking reciprocals and applying algebraic manipulations to isolate N, the population size at time T. The presenter encourages viewers to attempt solving for N themselves and provides a step-by-step guide through the algebraic process.

05:01
📈 Deriving the Logistic Function

The second paragraph continues the algebraic process to derive the logistic function, which models population growth. The presenter starts by simplifying the equation further and applying initial conditions to find the constant C. The logistic function is then fully expressed, showing how it starts at N knot (the initial population size) and increases at a decreasing rate as it approaches the maximum carrying capacity K of the environment. The presenter suggests that viewers plot the function using tools like Wolfram Alpha or a graphing calculator to observe its properties. The logistic function is noted for its utility in making predictions about population sizes at different times, providing a satisfying conclusion to the mathematical modeling of population dynamics.

Mindmap
Keywords
💡Logistic Differential Equation
The logistic differential equation is a mathematical model used to describe the growth of a population that is limited by the carrying capacity of the environment. It is defined by the equation dN/dt = rN(1 - N/K), where N is the population size, r is the intrinsic growth rate, and K is the carrying capacity. In the video, the logistic differential equation is the central theme as the presenter works through solving it to find the population size N over time (N(t)).
💡Carrying Capacity (K)
Carrying capacity, denoted as K, is the maximum population size that an environment can sustain indefinitely. It is a key concept in the logistic differential equation and represents the limit at which the population growth rate begins to decline. In the video, K is used to establish the upper bound for the population size N, which is essential for solving the logistic function.
💡Intrinsic Growth Rate (r)
The intrinsic growth rate, often symbolized as r, is the natural rate of increase of a population in the absence of limiting factors. It is a parameter in the logistic differential equation that influences how quickly the population grows. In the script, r is used in the context of the logistic equation to illustrate how the growth rate changes as the population size N approaches the carrying capacity K.
💡Logarithm Properties
Logarithm properties are mathematical rules that help simplify and manipulate logarithmic expressions. In the video, logarithm properties are used to rewrite the logistic differential equation in a form that is easier to solve. Specifically, the presenter uses the property that the logarithm of a quotient is the difference of the logarithms to transform the equation into a form that can be exponentiated to solve for N.
💡Natural Logarithm (ln)
The natural logarithm, denoted as ln, is the logarithm to the base e (approximately 2.71828). It is used in the video to transform the logistic differential equation into a form that allows for the application of exponential properties. The natural log is used to express the equation in a way that can be exponentiated to solve for the population size N as a function of time t.
💡Exponential Function
An exponential function is a mathematical function of the form f(x) = a * b^x, where a and b are constants and b > 0. In the context of the video, the exponential function e^(RT) is used to express the solution to the logistic differential equation, where e is the base of the natural logarithm and R is the intrinsic growth rate.
💡Reciprocal
The reciprocal of a number is the value which, when multiplied by the original number, results in the product of 1. In the video, the presenter uses the concept of reciprocals to manipulate the logistic equation and solve for the population size N. The reciprocal is used to express the relationship between N and the constants in the logistic function.
💡Constant of Integration (C)
In calculus, the constant of integration, often denoted as C, is a constant added to the integral of a function to account for the constant of integration when finding the antiderivative. In the video, C is used to represent an arbitrary constant that arises when integrating the logistic differential equation to find the function N(t).
💡Logistic Function
The logistic function is a common S-shaped curve (sigmoid function) that describes growth that starts slowly, accelerates, and then slows again as it approaches an upper limit. It is used in the video as the solution to the logistic differential equation, which models population growth. The logistic function is defined as N(t) = K / (1 + (C/K) * e^(-RT)), where C is a constant determined by the initial conditions.
💡Initial Condition
An initial condition is a fixed value or condition that applies at the start of a problem or process. In the video, the initial condition is given as N(0) = N_0, where N_0 is the initial population size. The initial condition is crucial for solving differential equations as it provides the necessary boundary condition to determine the constant of integration.
💡Population Growth
Population growth refers to the increase in the number of individuals in a population over time. The video focuses on modeling population growth using the logistic differential equation, which accounts for the carrying capacity and intrinsic growth rate to predict how a population will grow and stabilize over time.
Highlights

The logistic differential equation is used to model population growth, assuming an initial condition between 0 and K.

Algebraic manipulation is used to simplify the equation and find the solution for N of T.

Logarithm properties are applied to rewrite the left hand side of the equation.

The equation is solved for N by taking the reciprocal of both sides.

Constants are introduced and solved for using the initial condition N of 0 equals N knot.

The final solution for the logistic differential equation is derived and called the logistic function.

The logistic function has the desired properties of starting at N knot, increasing at an increasing rate, and then slowing down as the maximum population is reached.

The logistic function can be used to make predictions about future population sizes.

The solution involves raising E to various powers and simplifying using properties of exponents.

Constants are introduced and later solved for using the initial condition.

The reciprocal of both sides of the equation is taken multiple times to isolate terms and solve for N.

The numerator and denominator are divided by N to simplify the equation.

The final solution is obtained by multiplying the numerator and denominator by N knot K to eliminate fractions.

The logistic function is derived step by step using algebraic manipulation and properties of logarithms and exponents.

The solution to the logistic differential equation is found, which can model population growth and make predictions.

The logistic function starts at N knot, increases rapidly, then slows down as it approaches the maximum population K.

The logistic function has practical applications in modeling and predicting population sizes over time.

The derivation of the logistic function provides good algebra practice and a deeper understanding of the model.

Transcripts

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Logistic EquationMathematicsPopulation ModelingDifferential EquationsAlgebraLogarithmsExponential FunctionsEnvironmental ScienceEcologyGraphingWolfram Alpha