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4.4 The Logistic Equation

2 min read•june 24, 2024

Logistic equations model population growth with limited resources. They show how populations increase rapidly at first, then slow down as they approach a maximum sustainable size called the .

and solutions help visualize patterns. These tools reveal how populations behave over time, approaching or diverging from the based on initial conditions and growth rates.

The Logistic Equation

Carrying capacity in logistic growth

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  • Maximum sustainable population size in a given environment
    • Depends on available resources (food, water, shelter)
    • Limited by factors such as space and competition (territory, mates)
  • Population growth slows as it nears carrying capacity (KK)
    • Increased competition for scarce resources
    • regulate growth (disease, predation)
  • At carrying capacity, population stabilizes
    • Births and deaths are balanced
    • Population remains constant at equilibrium ()

Direction fields for logistic equations

  • Visual representation of solution behavior for
    • Arrows show rate of change (slope) at each point
    • Horizontal arrows signify constant population
    • Vertical arrows indicate increasing (up) or decreasing (down) population
  • : dPdt=rP(1−PK)\frac{dP}{dt} = rP(1 - \frac{P}{K})
    • PP: population size
    • rr:
    • KK: carrying capacity
  • Constructing a direction field:
    1. Select points (t,P)(t, P) in the tt-PP plane
    2. Calculate slope at each point using logistic equation
    3. Plot arrows with corresponding slopes
  • Interpreting direction fields:
    • Locate carrying capacity KK (horizontal arrows)
    • Examine solution behavior near KK
      • Arrows pointing to KK: population approaching carrying capacity
      • Arrows pointing away from KK: population diverging from carrying capacity
  • Identify where population is zero

Solutions of logistic equations

  • Solving logistic equations using
    1. Rearrange equation: dPP(1−PK)=rdt\frac{dP}{P(1 - \frac{P}{K})} = rdt
    2. Integrate both sides
    3. Solve for P(t)P(t) to find general solution
  • General solution: P(t)=K1+Ce−rtP(t) = \frac{K}{1 + Ce^{-rt}}
    • CC: constant determined by P(0)P(0)
  • Interpreting solutions in population growth context:
    • As t→∞t \to \infty, P(t)→KP(t) \to K (population approaches carrying capacity)
    • CC determines initial population relative to KK
      • C>1C > 1: initial population below K/2K/2
      • 0<C<10 < C < 1: initial population above K/2K/2
    • rr affects speed of population growth towards KK
  • Real-world applications of logistic model:
    • Bacterial growth in limited nutrient medium (petri dish)
    • Invasive species dynamics in new environments (kudzu, zebra mussels)
    • Epidemic spread within a susceptible population (COVID-19, influenza)
    • in ecology and environmental science

Characteristics of logistic growth

  • represents population growth over time
  • Initial phase resembles when resources are abundant
  • Growth rate slows as population approaches carrying capacity
  • Logistic equation is a fundamental model in population dynamics

Key Terms to Review (27)

Asymptotic Behavior: Asymptotic behavior refers to the long-term tendency of a function or process to approach a particular value or state as the independent variable approaches infinity. This concept is crucial in understanding the behavior of various mathematical models, including those related to exponential growth and decay, as well as the logistic equation.
Carrying capacity: Carrying capacity is the maximum population size of a species that an environment can sustain indefinitely, given the food, habitat, water, and other necessities available in the environment. In differential equations, it is a key parameter in the logistic growth model.
Carrying Capacity: Carrying capacity is the maximum population size of a species that an environment can sustainably support without significant negative impacts on the environment or the population itself. It is a fundamental concept in ecology that describes the balance between a population's growth and the limitations of its habitat.
Density-Dependent Factors: Density-dependent factors are environmental influences that affect the population growth of a species in relation to its population density. These factors play a crucial role in regulating the size and dynamics of a population, particularly in the context of the logistic equation.
Differential Equations: Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model and analyze various phenomena in science, engineering, and other fields where the rate of change of a quantity is of interest.
Direction Fields: A direction field, also known as a slope field or vector field, is a visual representation of the solutions to a first-order differential equation. It provides information about the behavior and characteristics of the solutions without explicitly solving the equation.
DP/dt: dP/dt is the rate of change of a variable P with respect to time t. It represents the instantaneous rate of change or the derivative of the variable P over time, and is a fundamental concept in calculus and its applications.
Equilibrium Points: Equilibrium points, also known as critical points or fixed points, are specific values of a variable where the rate of change of a function is zero. These points represent the values at which a system is in a state of balance or stability, with no net change occurring over time.
Exponential Growth: Exponential growth is a type of growth pattern where a quantity increases at a rate proportional to its current value. This means that the quantity grows by a consistent percentage over equal intervals of time, leading to a rapid and accelerating increase in its value.
Gompertz equation: The Gompertz equation is a type of mathematical model for time series, often used to describe growth processes. It is particularly useful in modeling populations, tumors, or other biological systems where growth slows over time.
Growth rate: Growth rate quantifies the change in a population or quantity over time, often expressed as a percentage. In mathematical models, it is a crucial parameter that influences the behavior of solutions to differential equations.
Inflection Point: An inflection point is a point on a curve at which the curve changes from being concave (curving downward) to convex (curving upward), or vice versa. It is a critical point where the direction of the curve's curvature changes, indicating a shift in the function's behavior.
Initial population: Initial population ($P_0$) is the starting number of individuals in a given population before any time has passed. It serves as a key parameter in solving differential equations modeling population growth.
Intrinsic Growth Rate: The intrinsic growth rate, also known as the natural growth rate, is a measure of the maximum potential growth of a population in a given environment when there are no limitations on resources or space. It represents the rate at which a population would grow if it had unlimited access to resources and was not limited by factors such as competition, predation, or environmental constraints.
Logistic differential equation: A logistic differential equation models population growth by incorporating a carrying capacity, which limits the growth as the population size increases. The general form is $\frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right)$, where $P$ is the population size, $r$ is the intrinsic growth rate, and $K$ is the carrying capacity.
Logistic Equation: The logistic equation is a mathematical model used to describe the growth of a population over time, taking into account the limiting effects of resources and the environment. It is a fundamental concept in the field of population dynamics and has applications in various disciplines, including biology, ecology, and economics.
Logistic Growth: Logistic growth is a mathematical model that describes the growth of a population or system over time, where the growth rate slows down as the population or system approaches a maximum capacity or carrying capacity. This model is commonly used in various fields, including biology, ecology, and economics, to understand and predict the dynamics of systems with limited resources or constraints.
Phase line: A phase line is a graphical tool used to study the qualitative behavior of one-dimensional autonomous differential equations. It visually represents equilibrium points and their stability.
Pierre François Verhulst: Pierre François Verhulst was a Belgian mathematician who, in 1838, developed the logistic equation, a mathematical model that describes the growth of a population over time. His work on the logistic equation has become a fundamental concept in the study of population dynamics and has applications in various fields, including biology, ecology, and economics.
Population Dynamics: Population dynamics is the study of how and why the size and composition of populations change over time. It examines the factors that influence the growth, decline, and stability of populations, including birth rates, death rates, migration patterns, and the interactions between different species within an ecosystem.
Population Growth Model: A population growth model is a mathematical representation that describes the change in the size of a population over time. It is a fundamental concept in the field of ecology and population biology, used to understand and predict the dynamics of various populations, from microorganisms to large animals and even human populations.
S-Shaped Curve: An S-shaped curve, also known as a sigmoid curve, is a graphical representation of a function that exhibits an 'S'-like shape. This curve is commonly observed in various natural and social phenomena, including population growth, the spread of innovations, and the adoption of new technologies.
Separation of Variables: Separation of variables is a method used to solve ordinary differential equations by breaking down the equation into two or more simpler equations, each involving only one of the variables. This technique allows for the integration of the individual components to obtain the solution to the original differential equation.
Sigmoid Function: The sigmoid function, also known as the logistic function, is a mathematical function that takes on an S-shaped curve. It is widely used in various fields, including machine learning, biology, and economics, to model processes that exhibit a gradual increase or decrease over time, often reaching a saturation point.
Steady State: Steady state refers to a condition in a dynamic system where key variables, such as population size or resource levels, remain constant over time. This concept is particularly important in the context of the logistic equation, which models the growth of a population under limited resources.
Threshold population: The threshold population is the minimum population size required for a specific behavior or phenomenon to occur within a model. In differential equations, it is often associated with the point at which growth rates change.
Tumor Growth Model: A tumor growth model is a mathematical representation that describes the growth and progression of a tumor over time. These models are used to understand the dynamics of tumor development and the factors that influence its growth, which is crucial for cancer research and treatment planning.
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Glossary
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4.5 First-order Linear Equations