Population Dynamics: Growth Rates, Carrying Capacity, and Logistic Growth

Population Growth Rate vs. Per Capita Growth Rate: Population growth rate ($\frac{\text{Change in N}}{\text{Change in T}}$) is units per time, while per capita growth rate ($\frac{\text{Change in N}}{\text{Change in T} \times \text{Initial N}}$) is units per time per individual.

Intrinsic Rate of Change ($\lambda$ and $r$):

• $\lambda$ (lambda) is the per capita growth rate (ratio of population at time $t+1$ to time $t$ ($\frac{N<em>{t+1}}{N</em>t}$)).

• $r$ is the instantaneous per capita growth rate, exponential population constant, where $\lambda = e^r$ .

Exponential Growth: Occurs under ideal, unrestricted conditions with no limiting factors. The rate of change is described by $\frac{dN}{dt} = rN$.

• If $r > 0$, population increases.

• If $r = 0$, population is stable (flat line).

• If $r < 0$, population declines (crash).

Three Meanings of $r$ (Ecological Context):

• Empirically Determined $r$: Based on real data, calculated without assumptions about age structure.

• Malthusian Constant ($r_m$): Represents a population with stable age distribution and reproductive rates, typically at carrying capacity (stable, replacement-level growth).

• $r_{max}$: The theoretical maximum potential for growth and reproduction under ideal conditions, often leads to population explosion and subsequent collapse.

Limiting Factors: Competition (intraspecific and interspecific), resources (food, habitat), predation, pathogens, and stochastic (random) events. Competition is a lose-lose situation, increasing with population density.

Carrying Capacity ($K$): The maximum population size that a specific environment can support at a given time.

• As population ($N$) approaches $K$, competition for critical resources increases.

• When $N = K$ , the environment is saturated, and competition is extremely high.

Logistic Growth: Models population growth when limiting factors and carrying capacity are present. The logistic equation is $\frac{dN}{dt} = r_{max}N \left(1 - \frac{N}{K}\right)$.

S-curve (Logistic Growth Curve): Represents logistic growth, showing an initial exponential phase, followed by a slowing growth rate as $N$ approaches $K$, eventually stabilizing at $K$ (where births approximate deaths).