9.2 Exponential and logistic growth models

Population growth models are crucial for understanding demographic changes. Exponential models assume constant growth without limits, while logistic models factor in carrying capacity. These concepts help explain how populations expand and stabilize over time.

Studying these models reveals insights into resource use, competition, and environmental constraints. By comparing exponential and logistic growth, we can better predict population trends and potential challenges in various ecological and human contexts.

Exponential vs Logistic Growth

Assumptions and Characteristics

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• Exponential growth models assume populations grow at a constant rate over time without limiting factors
  • Growth rate remains the same regardless of population size
  • Population doubles at regular intervals based on growth rate (doubling time calculated as 70 divided by annual percentage growth rate)
• Logistic growth models incorporate carrying capacity, the maximum population size an environment can sustain given available resources
  • Growth slows as population approaches carrying capacity due to resource limitations and increased competition
  • Population initially grows exponentially but levels off, forming an S-shaped curve
• Exponential growth typically seen in populations with abundant resources and no competition (bacteria in a petri dish with ample nutrients)
• Logistic growth more common in real-world populations facing environmental constraints (deer population in a forest with limited food and space)

Real-World Applications

• Understanding exponential growth helps predict rapid population increases that may lead to resource overexploitation and potential crashes if unchecked (invasive species like zebra mussels in the Great Lakes)
• Logistic growth suggests populations have a maximum sustainable size determined by carrying capacity, limiting indefinite growth
  • Intraspecific competition for resources increases as populations approach carrying capacity, reducing birth rates and increasing death rates, stabilizing population size (predator-prey dynamics between wolves and moose on Isle Royale)
• Logistic growth implies potential for population oscillations around carrying capacity due to time lags in resource availability and population responses (boom-and-bust cycles in hare and lynx populations)

Calculating Population Growth Rates

Exponential Growth Formulas

• Exponential growth model represented by equation $N(t) = N₀e^{rt}$
  • $N(t)$ is population size at time $t$
  • $N₀$ is initial population size
  • $r$ is growth rate
  • $e$ is mathematical constant approximately equal to 2.71828
• Calculate growth rate ($r$) in exponential model using formula $r = (ln(N(t)) - ln(N₀)) / t$, where $ln$ is natural logarithm

Logistic Growth Formulas

• Logistic growth model represented by equation $N(t) = K / (1 + ((K - N₀) / N₀)e^{-rt})$
  • $K$ is carrying capacity
  • $N(t)$ is population size at time $t$
  • $N₀$ is initial population size
  • $r$ is growth rate
  • $e$ is mathematical constant
• Calculate intrinsic growth rate ($r$) in logistic model using formula $r = (1 / t) * ln((K - N₀) / (K - N(t)))$
• Determine time required to reach specific population size in logistic model using formula $t = (1 / r) * ln((K - N₀) / (K - N(t)))$

Implications of Growth Models

Exponential Growth Consequences

• Rapid population increases may result in overexploitation of resources and potential population crashes if left unchecked (algal blooms in eutrophic lakes)
• Larger growth rate leads to shorter doubling time and more rapid population increases

Logistic Growth Consequences

• Populations have maximum sustainable size determined by carrying capacity of environment, limiting potential for indefinite growth
• Intraspecific competition for resources increases as populations approach carrying capacity, reducing birth rates and increasing death rates, stabilizing population size (density-dependent regulation in plant populations)
• Potential for population oscillations around carrying capacity due to time lags in resource availability and population responses (predator-prey cycles)

Limitations of Growth Models

Simplifying Assumptions

• Exponential growth models assume unlimited resources and no density-dependent factors affecting growth, rarely the case in real-world populations
• Exponential models do not account for age structure, migration, or environmental variability, which can significantly impact population dynamics
• Logistic growth models assume constant carrying capacity over time, but carrying capacity can vary due to changes in resource availability, habitat quality, or other environmental factors (impact of climate change on species' ranges)

Unaccounted Factors

• Logistic models do not consider potential for populations to overshoot carrying capacity and experience crashes or extinctions due to resource depletion or other factors (collapse of Easter Island civilization)
• Both models assume closed populations with no immigration or emigration, which may not accurately represent real-world populations open to exchange with other populations (metapopulation dynamics)
• Models do not account for effects of demographic stochasticity, which can have significant impacts on small populations or those near extinction thresholds (genetic drift in small, isolated populations)

Cautionary Use

• Exponential and logistic growth models are simplified representations of complex ecological processes
• Models should be used cautiously when making predictions or management decisions, considering their limitations and assumptions (fisheries management based on population growth models)