Logistic growth Knowledge-Based Study

Study notes

1. In natural ecosystems, populations rarely grow unchecked for long periods. While exponential growth assumes unlimited resources and yields a J-shaped curve, real environments have finite supplies of food, space, water, and shelter.

   As population density increases, resource limitation slows down the rate of growth. This pattern of resource-limited population regulation is known as logistic growth.

   A central concept in logistic growth is carrying capacity, denoted by the symbol K. Carrying capacity represents the maximum population size that a specific environment can sustainably support over time without degrading the habitat.

   When population size is plotted against time during logistic growth, it produces a characteristic S-shaped (sigmoid) curve. This curve can be divided into three distinct phases:

   • Lag phase: Population size is small, and individuals adapt to their environment. Growth is slow because there are few reproducing individuals.
   • Exponential (log) phase: The birth rate significantly exceeds the death rate. Resources remain abundant relative to population size, leading to rapid, accelerating growth.
   • Stationary (plateau) phase: As population density nears carrying capacity (K), environmental resistance intensifies. The birth rate declines, the death rate increases, and the net growth rate approaches zero, stabilizing the population around K.

   The logistic model mathematically modifies the exponential growth equation by incorporating a density-dependent term reflecting resource limitation:

   $$\frac{dN}{dt} = rN \left(\frac{K - N}{K}\right)$$

   The variables in this equation represent the following biological parameters:

   • $\frac{dN}{dt}$: The instantaneous rate of population increase over time.
   • r: The intrinsic per capita growth rate of the population.
   • N: The current population size.
   • K: The carrying capacity of the habitat.
   • $\frac{K - N}{K}$: The fraction of carrying capacity that remains available for future growth.

   When N is very small relative to K, the term $\frac{K - N}{K}$ is close to 1, causing the population to grow almost exponentially. However, as N approaches K, this term nears 0, causing $\frac{dN}{dt}$ to decline to zero.

   Logistic growth is mediated primarily by density-dependent limiting factors. These factors exert an increasingly strong regulating effect as the population density rises:

   • Intraspecific competition: Individuals of the same species compete for limited resources such as food, nesting sites, and sunlight.
   • Predation and disease: High population densities facilitate pathogen transmission and attract higher concentrations of predators.
   • Waste accumulation: High metabolic waste concentrations can lower reproductive output or increase mortality.

   In natural conditions, populations rarely stay at an exact, static value of K. Instead, they typically oscillate around carrying capacity due to seasonal changes, environmental fluctuations, and delayed density dependence.