EE 3- Carrying Capacity and Density Dependence Study Notes

Exponential growth assumes the per capita growth rate ( $r$ or $\lambda$ ) is constant and resources are unlimited.
Standard models include $N = N_0e^{rt}$ (continuous) and $N_t = \lambda^t N_0$ (discrete).
In reality, populations encounter limits such as nutrient/water availability, physical space, predation, and disease.

Density-Independent Limitation: Factors like weather or natural disasters affect mortality regardless of population density (e.g., drought impacts on Galapagos finches).
Density-Dependent Regulation: Factors like disease or resource competition increase their negative effects as density rises, leading to population stabilization.
Negative Density Dependence: Results in a negative growth rate when the population is large and a positive growth rate when it is small.

The logistic equation accounts for environmental limitation: $\frac{dN}{dt} = rN(1 - \frac{N}{K})$ .
Carrying Capacity ( $K$ ): The maximum population density the environment can support; at this point, birth rates balance death rates, and $\frac{dN}{dt} = 0$ .
The per-capita growth rate, $(\frac{dN}{dt})/N$ , declines linearly as $N$ approaches $K$ .

Discrete models, such as those visualized in Ricker diagrams, predict population size for the next generation ( $N_{t+1}$ ).
Dynamics depend on the value of $R$ : * R < 1: Movement to a single equilibrium point without oscillation. * 1 < R < 2: Damped oscillations toward $K$ . * R > 2: Limit cycles (e.g., 2-point or 4-point cycles) and the onset of chaos.

The Allee Effect: A decline in individual fitness at low population densities caused by mate limitation, inbreeding, or loss of cooperative strategies (e.g., group hunting/defense named after W.C. Allee).
High Density Competition: * Exploitation (Scramble): Indirect competition for shared resources. High density can lead to a population crash if no individual gets enough resources (e.g., Gypsy moth boom/bust cycles). * Interference (Contest): Direct interaction, such as territoriality in great tits at Wytham Woods. It provides more stable regulation through a constant number of survivors.

Undercompensation: Increased mortality at high density doesn't prevent the population from growing larger than a low-density population.
Exact Compensation: Often seen in contest competition; higher numbers are perfectly balanced by mortality to maintain the same final size.
Overcompensation: Mortality is so high at high densities that the resulting population is smaller than one starting at a lower density; extreme overcompensation is characteristic of pure scramble competition.