Comprehensive Lecture Notes on Density-Dependent Population Growth and Regulation

Unbounded Growth vs. Real-World Population Limits

• Theoretical Unbounded Growth (E. coli Example):

  • Consider a single cell of E. coli placed in an unbounded environment with a doubling time of $30\,\text{min}$.
  • Starting with $N_0 = 1$ cell, after $6\,\text{days}$, exponential population growth yields an astronomically high population size ($N$ far exceeding realistic planetary limits).
  • Real-world populations do not exhibit perpetual unbounded exponential growth because environmental factors check population expansion.

• Empirical Observations in Microorganisms:

  • Experimental growth curves of Streptococcus species under varying antibiotic concentrations show distinct phases:
  • Early Phase: The population growth closely matches the exponential model $\frac{dN}{dt} = rN$. Increased antibiotic concentrations lower the per capita growth rate ($r$).
  • Late Phase: As time progresses, populations do not grow infinitely. Growth slows down, reaches a maximum plateau, and eventually enters a decline phase in closed culture systems (e.g., bacterial flasks).

• Population Growth Rate Dynamics Over Time:

  • Initial Stage: Growth rate (slope of population size over time, $\frac{dN}{dt}$) is positive but low due to small initial population size ($N$).
  • Intermediate Stage: Growth rate accelerates and achieves its maximum slope.
  • Late Stage: Growth rate remains positive but decelerates toward zero.
  • Equilibrium Stage: Net population growth rate becomes zero ($\frac{dN}{dt} = 0$).

Density-Independent vs. Density-Dependent Regulation

• Density-Independent Factors:

  • Environmental factors that limit population size regardless of population density ($N$).
  • Mechanism: Recurrent abiotic disturbances (e.g., environmental storms) periodically reduce population size regardless of how many individuals are present, resetting growth continuously without altering per capita birth ($b$) or death ($d$) rate functions.

• Density-Dependent Factors (Population Regulation):

  • Factors where the intensity of the effect on per capita birth or death rates directly depends on population density ($N$).
  • Mechanisms: Intraspecific competition for limited resources, disease transmission dynamics, and predator-prey interaction frequencies.

• Per Capita Growth Rate Model ($r = b - d$):

  • In a density-independent model, birth rate ($b$) and death rate ($d$) are constant functions with a slope of zero relative to population density ($N$).
  • Consequently, the intrinsic rate of increase $r = b - d$ remains constant across all population densities.
  • In a density-dependent model, crowded conditions cause per capita death rates ($d$) to increase or per capita birth rates ($b$) to decrease as $N$ increases.
  • When $b > d$, per capita growth rate is positive and population increases.
  • When $d > b$, per capita growth rate is negative and population declines.

• Empirical Examples of Density Dependence:

  • Soybeans: Plotting initial seed planting density (seeds per $\text{m}^2$) against surviving plants per $\text{m}^2$ demonstrates that higher initial densities deviate downward from a $1:1$ survival line due to density-dependent mortality.
  • Beetles: Laboratory trials with beetles demonstrate a direct positive density-dependent relationship where beetle mortality rate increases as egg density inside culture tubes increases.
  • Grasses: Measurement of finite rate of increase ($\lambda$) in grass species displays a negative slope as population density increases.
  • Aquatic Organisms: Per capita growth rate ($r$) declines monotonically with increasing population density.

Mathematical Formulation of Logistic Growth

• Modifying the Exponential Model:

  • To account for density dependence, the exponential growth equation $\frac{dN}{dt} = rN$ is modified by incorporating a density-dependent feedback term: \frac{dN}{dt} = rN \times \begin{pmatrix} 1 - \frac{N}{K} \frac{}{}\text{} \right)
  • Variables and Parameters:
  • $N$: Current population size (density).
  • $r$: Intrinsic per capita rate of population growth.
  • $K$: Carrying capacity, defined as the maximum sustainable population size in a given environment.
  • \begin{pmatrix} 1 - \frac{N}{K} \frac{}{}\text{} \right): Represents the unused fraction of the environmental carrying capacity.

• Behavior at Density Extremes:

  • When $N \to 0$ ($N \boldsymbol{\nu} K$): The term $\frac{N}{K} \to 0$, making \begin{pmatrix} 1 - \frac{N}{K} \frac{}{}\text{} \right) \to 1. The logistic model reduces to exponential growth $\frac{dN}{dt} \to rN$. At very small population sizes, density-dependent growth is mathematically indistinguishable from density-independent growth.
  • When $N = K$: The term $\frac{N}{K} = 1$, making \begin{pmatrix} 1 - 1 \right) = 0. Thus, $\frac{dN}{dt} = 0$, and population growth ceases entirely.
  • When $N > K$: The term $\frac{N}{K} > 1$, resulting in a negative multiplier \begin{pmatrix} 1 - \frac{N}{K} \right) < 0, yielding negative population growth ($\frac{dN}{dt} < 0$).

Population Dynamics and Maximum Growth Rate

• Inflection Point of Maximum Population Growth:

  • Absolute population growth rate ($\frac{dN}{dt}$) follows a non-linear trajectory relative to population density ($N$).
  • Growth is slow at very low density ($N$ is small despite high per capita rate) and approaches zero at carrying capacity ($N \to K$).
  • Absolute growth rate $\frac{dN}{dt}$ is strictly maximized at an inflection point where population density is exactly half of the carrying capacity: $N = \frac{K}{2}$

• Numerical Comparison Example:

  • Assume carrying capacity $K = 500$ and intrinsic growth rate $r = 0.1$:
  • At $N = 15$: \frac{dN}{dt} = 0.1 \times 15 \times \begin{pmatrix} 1 - \frac{15}{500} \right) = 1.5 \times 0.97 = 1.455
  • At $N = 250$ ($N = \frac{K}{2}$): \frac{dN}{dt} = 0.1 \times 250 \times \begin{pmatrix} 1 - \frac{250}{500} \right) = 25 \times 0.5 = 12.5
  • At $N = 450$: \frac{dN}{dt} = 0.1 \times 450 \times \begin{pmatrix} 1 - \frac{450}{500} \right) = 45 \times 0.1 = 4.5
  • Absolute population growth is substantially higher at $N = 250$ than at either $N = 15$ or $N = 450$

Derivation of Carrying Capacity from Linear Birth and Death Rates

• Linear Equations for Birth and Death Rates:

  • Baseline per capita birth rate at zero density ($N = 0$): $b_0$
  • Baseline per capita death rate at zero density ($N = 0$): $d_0$
  • Density-dependent death rate function: $d = d_0 + cN$
  • Slope $c$: Represents the strength/magnitude of density dependence on mortality (increase in death rate per added individual).
  • Density-dependent birth rate function: $b = b_0 - aN$
  • Slope $a$: Represents the strength/magnitude of density dependence on reproduction (decrease in birth rate per added individual).

• Algebraic Derivation of Carrying Capacity ($K$):

  • Environmental carrying capacity ($K$) occurs at equilibrium where per capita birth rate equals per capita death rate ($b = d$):     $b_0 - aK = d_0 + cK$
  • Rearranging terms to solve for $K$:     $b_0 - d_0 = aK + cK$$b_0 - d_0 = (a + c)K$$K = \frac{b_0 - d_0}{a + c}$

• Effects of Parameters on Carrying Capacity:

  • Strength of Density Dependence ($a$ and $c$): As the density-dependent constants $a$ or $c$ increase (steeper linear slopes), the denominator $(a + c)$ increases, causing carrying capacity $K$ to decrease.
  • Single Parameter Density Dependence: Logistic growth occurs even if only one rate is density-dependent. If $c = 0$ (death rate independent of density) and $a > 0$, $K = \frac{b_0 - d_0}{a}$. The birth and death lines will still cross as long as they are not parallel.
  • Density Independence Condition: If both $a = 0$ and $c = 0$, the denominator becomes zero, $K \to \text{∞}$, birth and death rate lines remain parallel, and the system exhibits unbounded exponential growth.

Equilibria and Dynamics Above Carrying Capacity

• System Equilibria:

  • Solving $\frac{dN}{dt} = 0$ yields two distinct equilibria:
  1. $N^* = 0$: Trivial/unstable equilibrium (no population present).
  2. $N^* = K$: Stable equilibrium (carrying capacity).

• Perturbations Above Carrying Capacity ($N > K$):

  • If external individuals are introduced into a population already at carrying capacity (e.g., adding $50$ additional laboratory beetles into a stable culture at capacity), $N$ exceeds $K$.
  • When $N > K$, death rate exceeds birth rate ($d > b$).
  • The evaluating term \begin{pmatrix} 1 - \frac{N}{K} \right) yields a negative value, resulting in $\frac{dN}{dt} < 0$.
  • The population experiences negative growth and declines monotonically until returning to equilibrium $N = K$

• Relationships Between Population Trajectories and $\frac{dN}{dt}$ Graphs:

  • Fluctuating Population Trajectory: The growth rate $\frac{dN}{dt}$ is zero at every local peak and trough of the population size curve ($N$ vs $t$). $\frac{dN}{dt}$ reaches its positive maximum at the steepest upward slope of $N$ and its negative minimum at the steepest downward slope of $N$
  • Exponential Growth Trajectory: The plot of $\frac{dN}{dt}$ vs $t$ increases continuously and exponentially without bound.
  • Logistic Growth Trajectory: The plot of $\frac{dN}{dt}$ vs $N$ forms a symmetric downward parabola, starting at zero when $N = 0$, reaching its peak at $N = \frac{K}{2}$, and returning to zero when $N = K$

Positive Density Dependence and the Allee Effect

• Concept of Non-Monotonic Density Dependence:

  • Density dependence is not exclusively negative. At low population densities, per capita birth rates may increase or survival rates may improve as population density increases.

• Biological Mechanisms for Positive Density Dependence at Low Density:

  • Pollen Limitation in Plants: At extremely low plant densities, individuals are sparsely distributed, leading to low pollination efficiency by vectors and reduced birth rates. Increasing plant density enhances pollination success and birth rate.
  • Obligate Social Breeders: Animals that rely on group behaviors (e.g., cooperative breeding, group vigilance, group defense against predators) suffer reduced per capita survival or reproduction at low densities due to insufficient group size.

• The Allee Effect Threshold Dynamics:

  • Definition: An Allee effect is a biological phenomenon characterized by positive density dependence at low population sizes, resulting in a critical threshold density below which population growth rate becomes negative.
  • Mathematical Dynamics:
  • Below Allee Threshold ($N < N_{\text{threshold}}$): Per capita death rate exceeds birth rate ($d > b$), causing negative population growth ($\frac{dN}{dt} < 0$). Populations falling below this critical density threshold decline toward extinction ($N = 0$).
  • Intermediate Density ($N_{\text{threshold}} < N < K$): Per capita birth rate exceeds death rate ($b > d$), yielding positive population growth ($\frac{dN}{dt} > 0$).
  • High Density ($N > K$): Negative density dependence takes over due to resource limitation ($d > b$), driving population growth back down to zero at carrying capacity $K$