Table of Contents
Population Size Over Time: Basic Patterns
When ecologists talk about growth and development of a population, they mean how the number of individuals changes over time and how the population’s structure and dynamics shift as it grows, stabilizes, or declines. Several typical patterns and concepts are used to describe this.
Exponential Growth: Unlimited Growth in Theory
If each individual in a population reproduces at a constant average rate and there are no important limits (enough food, space, no crowding effects), the population can grow exponentially.
- Let $N$ = number of individuals in the population
- Let $r$ = intrinsic rate of increase (per capita growth rate)
Then, in a simple continuous model:
$$
\frac{dN}{dt} = rN
$$
The solution is:
$$
N(t) = N_0 e^{rt}
$$
Where:
- $N_0$ = initial population size
- $e$ = base of the natural logarithm
- $t$ = time
Key features of exponential growth:
- The larger the population, the more individuals are added per time unit (growth accelerates).
- A graph of $N$ vs. time is J-shaped.
- It typically occurs:
- In newly colonized habitats with abundant resources.
- In populations recovering after a crash or after being introduced to a new environment (e.g., invasive species at first).
In nature, pure exponential growth is usually temporary, because resources and environmental conditions are not limitless.
Logistic Growth: Growth With Limits (Carrying Capacity)
As a population grows, individuals begin to compete for limited resources (food, space, nesting sites), and negative effects of crowding appear (disease spread, stress, etc.). Growth slows and may eventually stabilize. This is often described by the logistic growth model.
Let $K$ = carrying capacity of the environment (maximum population size that can be sustained long term, under given conditions).
The logistic equation:
$$
\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)
$$
Interpretation:
- When $N$ is very small, $\left(1 - \frac{N}{K}\right) \approx 1$ and growth is nearly exponential.
- When $N$ approaches $K$, $\left(1 - \frac{N}{K}\right) \rightarrow 0$ and growth rate falls toward zero.
- If $N$ > $K$, the term becomes negative and the population tends to decrease.
Graphically:
- S-shaped (sigmoid) curve.
- Characteristic phases:
- Lag phase: slow increase; few individuals; growth still exponential but starting from low numbers.
- Exponential phase: steep, rapid increase; resources still plentiful.
- Deceleration phase: growth slows as competition and crowding increase.
- Stationary phase: population fluctuates around $K$.
Carrying capacity is not fixed forever: it can change with seasons, climate, resource availability, and human impacts (e.g., habitat degradation, supplemental feeding).
Life Histories and Growth Patterns: r- and K-Selection (Conceptual)
Many species show characteristic patterns related to how their populations grow and develop in relation to environmental limits. A classical conceptual distinction (though simplified) is between r-selected and K-selected strategies.
r-Selected Strategies
- Often associated with environments that are unpredictable or frequently disturbed.
- Typical traits:
- Early reproduction.
- High number of offspring.
- Little or no parental care.
- Short lifespan.
- High mortality in early life stages.
- Population dynamics:
- Rapid, often exponential growth when conditions are good.
- Large fluctuations: booms and crashes.
- Examples (in general terms): many annual plants, many insects, many small, short-lived organisms.
K-Selected Strategies
- Often associated with relatively stable environments where populations frequently exist near carrying capacity.
- Typical traits:
- Later reproduction.
- Few, larger offspring.
- Intensive parental care.
- Longer lifespan.
- Lower mortality in early life stages.
- Population dynamics:
- Slower growth.
- Densities often near $K$, with smaller fluctuations around it.
- Examples (in general terms): many large mammals, some large birds, long-lived trees.
Real species often show mixed or intermediate strategies, but the r/K framework helps to think about how life-history traits relate to population growth patterns.
Density-Dependent and Density-Independent Influences on Growth
Changes in population size result from births, deaths, immigration, and emigration. How strongly these processes respond to population density is central for understanding growth.
Density-Dependent Factors
These are factors whose effect on birth or death rates changes with population density. They often stabilize populations.
- Examples:
- Competition for food, space, nesting sites.
- Predation: predators may focus on more abundant prey.
- Parasitism and disease: spread more easily when individuals are crowded.
- Social stress: aggression or reduced reproduction when many individuals are packed together.
Typical effects:
- As density increases:
- Birth rates tend to decrease (reduced fertility, fewer offspring).
- Death rates tend to increase (starvation, more disease, more predation).
- This negative feedback can slow population growth and lead to an equilibrium near $K$.
Graphically, you can imagine:
- Birth rate curve declining with increasing density.
- Death rate curve increasing with increasing density.
- They intersect at the density where birth and death balance (zero net growth).
Density-Independent Factors
These affect populations regardless of their density.
- Examples:
- Extreme weather (storms, heat waves, frost, drought).
- Natural disasters (fires, volcanic eruptions, floods).
- Certain human impacts (pollution, habitat destruction, pesticide application).
Typical effects:
- They can sharply reduce populations at any density.
- They tend to cause fluctuations but do not create a self-correcting feedback based on density.
In reality, both density-dependent and density-independent influences interact to shape actual growth curves.
Age Structure and Population Development
Besides total size, the age structure (how many individuals are in different age classes) is crucial for understanding how a population will develop.
Age Classes and Reproductive Value
Populations are often divided into age groups:
- Pre-reproductive (juveniles)
- Reproductive (adults producing offspring)
- Post-reproductive (older individuals, usually no longer reproducing)
Key points:
- A population with many pre-reproductive individuals will likely increase, if they survive to reproductive age.
- A population dominated by post-reproductive individuals is often on a path to decline, unless many younger individuals are added.
Related ideas:
- Different ages have different probabilities of survival and reproduction.
- The average number of offspring produced by an individual of a given age, combined with the probability of reaching that age, determines that age class’s contribution to future population growth.
Age Pyramids and Growth Trends
Age structure is often visualized as an age pyramid (or age distribution diagram):
- Expanding populations:
- Broad base (many young), narrow top.
- Indicates high birth rates and/or high recent reproduction.
- Stable populations:
- More column-shaped.
- Similar width in pre-reproductive and reproductive classes.
- Declining populations:
- Narrow base (few young) and sometimes a relatively wider middle or top.
- Indicates low birth rates or strong recent reductions in survival of young.
Such diagrams are especially used for human populations, but the concept applies to other species as well.
Survivorship Curves and Mortality Patterns
Another way to describe population development is to look at survivorship: how many individuals of a cohort (group born at the same time) remain alive at different ages.
Three idealized survivorship curves are often distinguished:
- Type I:
- High survival through early and middle life.
- Most deaths occur at older ages.
- Typical of many large mammals with parental care.
- Type II:
- Constant probability of death at all ages.
- Straight line decline on a plot of survivors vs. age.
- Seen in some birds, some reptiles, some small mammals.
- Type III:
- Very high mortality at early life stages.
- Individuals that survive early hazards can live relatively long.
- Many plants, marine invertebrates, and fish that produce many small offspring fall into this pattern.
These curves reveal how mortality is distributed across the life span and help link life-history traits (e.g., many small offspring vs. few large ones) to population dynamics.
Discrete Generations and Geometric Growth
Not all populations reproduce continuously. Some species have discrete generations:
- All individuals reproduce in a particular season, then die (e.g., many annual plants, some insects).
- Population size is measured from generation to generation.
In this case, a simple geometric growth model is often used:
Let:
- $N_t$ = population size in generation $t$
- $\lambda$ = finite rate of increase (ratio of population sizes between generations)
Then:
$$
N_{t+1} = \lambda N_t
$$
- If $\lambda > 1$, the population grows.
- If $\lambda = 1$, size stays constant.
- If $\lambda < 1$, the population declines.
Over multiple generations:
$$
N_t = N_0 \lambda^t
$$
This is similar to exponential growth, but in discrete time steps, which fits species with non-overlapping generations.
Fluctuations, Cycles, and Irregular Dynamics
Real populations seldom follow a perfect logistic curve. They often fluctuate around the carrying capacity or show more complex patterns.
Oscillations Around Carrying Capacity
- Time delays in reproduction (e.g., taking time to mature) or in how density-dependent factors act can lead to overshoots and undershoots of $K$.
- Population size may oscillate around $K$ rather than smoothly approach it.
Predator–Prey and Other Cycles
- Interactions with other species (predators, parasites, hosts, competitors) can create regular cycles.
- Classic examples include multi-year cycles where prey numbers rise, then predator numbers increase and reduce prey, then predator numbers fall, and so on.
- Such feedbacks arise from linked birth and death rates between interacting populations.
Outbreaks, Crashes, and Chaos
Under some conditions:
- Populations can experience outbreaks (rapid booms) followed by crashes (sharp declines), for example in some pest insects.
- Nonlinear feedbacks and time delays can theoretically produce irregular, seemingly chaotic population dynamics, even in simple models.
Recognizing that populations often fluctuate rather than sit quietly at an equilibrium is essential for understanding real-world population development.
Human Influence on Population Growth and Development
Human activities strongly alter how populations grow and develop:
- Habitat modification (deforestation, urbanization, agriculture) changes carrying capacities and often fragments populations.
- Harvest and exploitation (fishing, hunting, logging) add an extra mortality factor that can:
- Drive populations below $K$.
- Select for different life-history traits (earlier reproduction, smaller body size).
- Introductions and invasions:
- Introduced species may experience rapid exponential growth in new environments, sometimes becoming invasive when natural enemies are lacking.
- Conservation measures:
- Protection, habitat restoration, and captive breeding can help small populations recover, altering their growth trajectory.
- Climate change:
- Alters resource availability, survival, reproduction, and thus carrying capacities and growth patterns.
Understanding growth and development of populations is therefore essential not only for basic ecology, but also for population management, conservation biology, and sustainable use of natural resources.