5.1 Population Dynamics: Exponential and Geometric Growth

Population Dynamics

Introduction to Population Growth Models

Population dynamics is the study of how and why the number of individuals in a population changes over time. Mathematical models help ecologists describe and predict these changes. Two primary models are used: exponential growth for continuous reproduction and geometric growth for discrete reproduction.

• Population: A group of individuals of the same species living in a defined area.

• Population size (N): The number of individuals in the population at a given time (t).

Factors Affecting Population Size

Births, Deaths, Immigration, and Emigration

Population size changes due to four main processes:

• Births (B): Add individuals to the population.

• Deaths (D): Remove individuals from the population.

• Immigration (I): Individuals entering from other populations.

• Emigration (E): Individuals leaving to other populations.

The general equation for population change over one time step is:

• $N_{t+1} = N_t + B + I - D - E$

• Rearranged: $\Delta N = (B + I) - (D + E)$

Closed populations (no immigration or emigration): $\Delta N = B - D$

Open populations (with immigration and emigration): $\Delta N = (B + I) - (D + E)$

Types of Populations: Discrete vs. Continuous

Discrete Populations

Populations with non-overlapping generations reproduce at specific intervals (e.g., annually). Growth is described by the geometric model.

• Examples: Salmon (breed once per year), migratory birds, desert annual plants.

Continuous Populations

Populations with overlapping generations reproduce continuously. Growth is described by the exponential model.

• Examples: Humans, tropical insects, bluehead wrasse (daily spawning).

Mathematical Models of Population Growth

Geometric (Discrete) Growth

For populations with discrete reproduction, each generation increases by a constant proportion, called the geometric growth rate ($\lambda$).

• Population size at time t: $N_t = N_0 \lambda^t$

• Geometric growth rate: $\lambda = \frac{N_{t+1}}{N_t}$

• Interpretation: $\lambda$ is the average number of offspring per individual per time interval.

Exponential (Continuous) Growth

For populations with continuous reproduction, growth is described by the exponential model. The intrinsic rate of increase ($r$) is the difference between per capita birth and death rates.

• Per capita birth rate (b): Number of births per individual per unit time.

• Per capita death rate (d): Number of deaths per individual per unit time.

• Intrinsic rate of increase: $r = b - d$

• Differential equation for population growth: $\frac{dN}{dt} = rN$

• Integrated form (predicting future population size): $N_t = N_0 e^{rt}$

Doubling time (time for population to double):

• $t_d = \frac{\ln 2}{r}$

Vital Rates and Their Role in Population Growth

Definitions and Calculations

• Total births (B): $B = bN$

• Total deaths (D): $D = dN$

• Population growth rate: $\frac{dN}{dt} = (b - d)N = rN$

Comparing Geometric and Exponential Growth

Key Differences

• Geometric growth: Population increases at discrete intervals (non-overlapping generations).

• Exponential growth: Population increases continuously (overlapping generations).

Graphically, geometric growth appears as steps at each time interval, while exponential growth is a smooth curve.

Behavior of Growth Parameters ($\lambda$ and $r$)

Parameter	Value	Population Trend
$\lambda < 1$	or $r < 0$	Population decreases
$\lambda = 1$	or $r = 0$	Population stable (no change)
$\lambda > 1$	or $r > 0$	Population increases

Relationship Between $\lambda$ and $r$

• $\lambda = e^r$

• $r = \ln \lambda$

This relationship allows conversion between geometric and exponential growth rates.

Assumptions of the Exponential Growth Model

• Population is closed (no immigration or emigration).

• Birth and death rates are constant.

• Unlimited resources (no environmental resistance).

• No genetic, age, or size structure in the population.

• Continuous growth with no time lags.

Example: Reindeer on St. Matthew Island

In 1944, 29 reindeer were introduced to St. Matthew Island, Alaska. With abundant resources and no predators, the population grew exponentially, reaching 6,000 by 1963. This real-world example illustrates exponential growth under ideal conditions.

Summary Table: Discrete vs. Continuous Growth

Growth Type	Equation	Parameter	Population Type
Geometric (Discrete)	$N_t = N_0 \lambda^t$	$\lambda$	Non-overlapping generations
Exponential (Continuous)	$N_t = N_0 e^{rt}$	$r$	Overlapping generations

Key Equations

• Geometric growth: $N_t = N_0 \lambda^t$

• Exponential growth: $N_t = N_0 e^{rt}$

• Relationship: $\lambda = e^r$, $r = \ln \lambda$

• Doubling time: $t_d = \frac{\ln 2}{r}$

Conclusion

Understanding population growth models is essential for predicting changes in population size and for managing wildlife, conservation, and resource use. The choice between geometric and exponential models depends on the reproductive biology of the species and the time scale of interest.