Unit 3: Population Ecology

Population ecology is a subfield of ecology that deals with the dynamics of species populations and how these populations interact with the environment. Understanding populations requires analyzing their quantitative attributes, growth trajectories, regulatory factors, and interspecies interactions.

1. Unique and Group Attributes of Population

An individual organism represents the basic unit of life, but individuals do not exist in isolation. A population is defined as a group of individuals of the same species living in a specified geographical area at a given time, capable of interbreeding and sharing a common gene pool.

While individuals undergo birth, aging, and death, populations exhibit collective attributes that cannot be applied to individuals alone. These are divided into unique genetic attributes and group demographic attributes.

Unique Genetic Attributes

• Gene Pool: The sum total of all genes and their alleles present in a population at any given time.
• Evolutionary Trajectory: Populations, not individuals, evolve over time through changes in allele frequencies driven by natural selection, genetic drift, and mutation.

Group Demographic Attributes

Population Density

Population density refers to the number of individuals per unit area or volume. It is characterized in two ways:

• Crude Density: The number of individuals per unit of total space (e.g., total land area).
• Ecological (Specific) Density: The number of individuals per unit of habitable space (the actual area colonized by the population).

Natality (Birth Rate)

Natality is the production of new individuals in a population through reproduction (birth, hatching, or germination).

• Absolute (Physiological) Natality: The theoretical maximum production of new individuals under ideal, non-limiting environmental conditions. This is a constant value for a given species.
• Realized (Ecological) Natality: The actual increase in population size under specific, real-world environmental conditions. It is variable and depends on resource availability, competition, and predation.

Mortality (Death Rate)

Mortality refers to the loss of individuals from a population due to death.

• Minimum (Physiological) Mortality: The theoretical minimum death rate under ideal conditions, where individuals die only of old age or genetic senescence.
• Realized (Ecological) Mortality: The actual loss of individuals under specific environmental pressures (disease, predation, resource limitation). It fluctuates continuously.

Life Tables

A life table provides an age-specific summary of survival and mortality within a population. It serves as an actuarial tool to calculate life expectancy and reproductive potential. There are two primary types of life tables:

• Cohort (Horizontal) Life Table: Follows a single cohort of individuals born at the same time from birth until the death of the last surviving member.
• Static (Vertical) Life Table: Records the age-specific survival and mortality of different age classes within a population at one specific point in time. This is useful for long-lived species.

The standard components of a life table include:

Fecundity Tables

A fecundity table (or fertility table) combines age-specific survival data with age-specific birth rates to determine reproductive outputs. Because female population dynamics typically drive overall population growth, fecundity tables track female offspring produced per female.

• Age-Specific Fecundity (mx or bx): The mean number of female offspring produced per female during the age interval x.
• Net Reproductive Rate (R0): The average number of female offspring produced by a female during her entire lifetime.

Formula for Net Reproductive Rate: R0 = Sum of (lx * mx)

Evaluating R0 determines population trajectories:

• If R0 > 1, the population is growing.
• If R0 = 1, the population is stable (exact replacement).
• If R0 < 1, the population is in decline.

Using the fecundity table, the Generation Time (T) (the average time between the birth of parents and the birth of their offspring) can also be calculated:

Formula for Generation Time: T = [Sum of (x * lx * mx)] / R0

Survivorship Curves

A survivorship curve is a graphical representation plot of the number of individuals (or log of survivors, lx) surviving at each age interval. There are three generalized types of survivorship curves:

2. Exponential and Logistic Growth Models

Populations change size over time through births, deaths, immigration, and emigration. In closed systems (where immigration and emigration are negligible), population growth is a function of birth and death rates. Two primary mathematical models describe this growth: Exponential and Logistic Growth.

Exponential Growth Model

Exponential growth occurs in an idealized environment with unlimited resources (food, space, mates), no competition, and no predators. The rate of growth is directly proportional to the population size itself.

Differential Equation for Exponential Growth: dN / dt = rN

Where:

• N = Population size.
• t = Time.
• r = Intrinsic rate of natural increase (per capita rate of growth, calculated as per capita birth rate minus per capita death rate: r = b - d).
• dN / dt = The rate of change in population size over time.

To predict population size at any specific future point in time, we use the integrated form of the exponential equation:

Integrated Equation for Exponential Growth: Nt = N0 * e^(rt)

Where:

• Nt = Population size at time t.
• N0 = Initial population size at time zero.
• e = The base of natural logarithms (approximately 2.718).
• r = Intrinsic rate of natural increase.
• t = Time elapsed.

Graph Shape: When plotted as population size (N) over time (t), the curve is J-shaped. It starts with a slow lag phase and transitions into an accelerating, limitless upward trajectory.

Assumptions of Exponential Growth:

• Resources are unlimited.
• The value of r is constant.
• All individuals are identical (no age or size structure).
• The population is closed (no immigration or emigration).

Logistic Growth Model

In the real world, resources are finite. As population density increases, resource depletion, waste accumulation, and crowding occur, reducing per capita birth rates and increasing per capita death rates. The population growth rate slows down and eventually halts at a maximum sustainable population size called the carrying capacity (K).

Differential Equation for Logistic Growth (Verhulst-Pearl Equation): dN / dt = rN * [(K - N) / K]

Where:

• N = Population size.
• r = Intrinsic rate of natural increase.
• K = Carrying capacity of the environment.
• (K - N) / K = The environmental resistance factor. This term represents the fraction of carrying capacity still available for population growth.

Dynamics of the Environmental Resistance Factor:

• When N is very small (close to 0), the factor `(K - N) / K` is close to 1, meaning the population grows almost exponentially (`dN/dt ~ rN`).
• When N approaches K, the factor `(K - N) / K` approaches 0, causing the growth rate `dN/dt` to slow to zero.
• If N exceeds K, the factor becomes negative, and the population declines back toward K.

Graph Shape: When plotted, the logistic curve is S-shaped (sigmoidal). It exhibits four distinct phases:

1. Lag Phase: Initial slow growth due to small population size.
2. Log (Exponential/Acceleration) Phase: Rapid growth where resources are highly available and population density is moderate.
3. Deceleration Phase: Growth slows down as resource limitations and density-dependent factors begin to exert pressure.
4. Stationary (Equilibrium) Phase: The population size stabilizes around carrying capacity (K), where birth rates equal death rates.

Inflection Point: The maximum rate of population growth (`dN/dt`) occurs exactly at the inflection point of the S-shaped curve, which is located at half the carrying capacity: N = K / 2.

3. Population Regulation: Density-Dependent and Independent Factors

Population sizes do not grow indefinitely; they are regulated and kept within sustainable bounds by biotic and abiotic environmental factors. These factors are classified based on whether their intensity changes with population density.

Density-Dependent Factors

Density-dependent factors have an effect that varies in intensity as a function of population density. They typically act as negative feedback mechanisms, stabilizing population sizes around the carrying capacity.

• Mechanism: As population density increases, the intensity of these factors increases, which in turn reduces the birth rate and/or increases the death rate.
• Biotic Factors: Most density-dependent factors are biological interactions.
• Key Examples:
  • Intraspecific Competition: Competition among members of the same species for limited resources (food, space, nesting sites).
  • Predation: Predators may concentrate their hunting efforts on highly abundant prey populations (numerical and functional responses).
  • Disease and Parasitism: High-density populations allow pathogens and parasites to transmit more quickly and easily among hosts.
  • Toxic Waste Accumulation: In closed environments, metabolic byproducts can accumulate to lethal levels (e.g., ethanol buildup by yeast during fermentation).
  • Intrinsic Factors: Physiological or behavioral changes, such as stress-induced hormonal changes that lower fertility in overcrowded conditions.

Density-Independent Factors

Density-independent factors affect a population's size regardless of how dense the population is. They lack any feedback loops linked to population density.

• Mechanism: These factors cause mortality or alter reproductive rates by the same proportion, whether the population consists of 10 individuals or 10,000 individuals.
• Abiotic Factors: Most density-independent factors are physical or chemical environmental influences.
• Key Examples:
  • Natural Disasters: Wildfires, floods, volcanic eruptions, earthquakes, and hurricanes.
  • Weather and Climate: Sudden freezes, severe heatwaves, prolonged droughts, or seasonal changes.
  • Anthropogenic Disturbances: Habitat destruction, pesticide application, and chemical pollution.

Comparison of Regulation Factors

4. Prey vs Predator, Gause's Principle, and Lotka-Volterra Models

Populations do not exist in isolation. They interact with other species, which influences their demographic trajectories. This section details prey-predator dynamics, competitive outcomes, and their quantitative representations.

Prey vs Predator Dynamics (p v p v)

Predation is an antagonistic species interaction where one organism (the predator) kills and consumes another organism (the prey). These interactions generate linked, cyclic oscillations in the populations of both species over time.

• The Lagged Oscillation Cycle: An increase in prey density provides more food for predators, leading to an increase in the predator population. Eventually, the growing predator population overconsumes the prey, causing the prey population to crash. Deprived of food, the predator population subsequently crashes, which removes predation pressure and allows the prey population to recover, starting the cycle anew.
• Classic Example: The historical population cycles of the snowshoe hare (prey) and the Canadian lynx (predator), tracked using trapping records from the Hudson's Bay Company over a century.

Gause's Principle (G ' p)

Gause's Principle, also known as the Competitive Exclusion Principle, was formulated by Soviet biologist Georgy Gause based on his classic laboratory experiments with protozoans.

Gause's Competitive Exclusion Principle: Two species competing for the exact same limiting resource cannot stable-coexist if all other environmental factors remain constant. The species that has even the slightest ecological advantage will outcompete and eventually exclude the other.

Gause's Classic Experiment

Gause cultured two species of ciliated protozoans, Paramecium aurelia and Paramecium caudatum, under controlled conditions with a fixed supply of food (bacteria).

• Grown Separately: Both species grew logistically, stabilizing at carrying capacity.
• Grown Together: Paramecium aurelia outcompeted Paramecium caudatum for the bacterial food. P. caudatum was driven to extinction because P. aurelia had a higher rate of population growth and could forage more efficiently at lower food densities.

How Competitors Coexist in Nature

In natural ecosystems, complete competitive exclusion is often avoided through evolutionary and behavioral adaptations:

• Resource Partitioning: Competitors evolve to use different resources, occupy different microhabitats, or feed at different times of the day (e.g., Robert MacArthur's study of five warbler species partitioning different zones of spruce trees).
• Character Displacement: Evolutionary divergence of physical traits in competing species when they occur in the same geographical area (sympatry), compared to when they occur separately (allopatry). A classic example is the divergence in beak sizes among Darwin's finches on the Galápagos Islands.

Lotka-Volterra Equation for Competition

The Lotka-Volterra competition equations are a mathematical extension of the logistic growth model. They describe the dynamics of two species competing for shared, limiting resources. They incorporate competition coefficients to convert the individuals of one species into equivalent numbers of the other.

Equations for Competing Species: Species 1: dN1 / dt = r1 * N1 * [(K1 - N1 - alpha * N2) / K1] Species 2: dN2 / dt = r2 * N2 * [(K2 - N2 - beta * N1) / K2]

Where:

• N1, N2: Population sizes of Species 1 and Species 2.
• r1, r2: Intrinsic growth rates of Species 1 and Species 2.
• K1, K2: Carrying capacities of the environment for Species 1 and Species 2 when grown in isolation.
• alpha (α): The effect of Species 2 on Species 1 (competition coefficient). It represents the per capita competitive effect of an individual of Species 2 on the growth of Species 1, relative to an individual of Species 1. For example, if alpha = 2, one individual of Species 2 uses twice as many resources as one individual of Species 1.
• beta (β): The effect of Species 1 on Species 2 (competition coefficient). It represents the per capita competitive effect of an individual of Species 1 on the growth of Species 2, relative to an individual of Species 2.

Outcomes of the Competition Model

By plotting the zero-force growth isoclines (lines where population growth `dN/dt = 0` for each species) on a state-space graph, the model yields four possible outcomes:

1. Species 1 Wins: The isocline of Species 1 lies completely outside the isocline of Species 2. Species 1 drives Species 2 to extinction.
2. Species 2 Wins: The isocline of Species 2 lies completely outside the isocline of Species 1. Species 2 drives Species 1 to extinction.
3. Unstable Equilibrium: The isoclines cross, and the equilibrium point is unstable. Whichever species starts with a higher initial density will outcompete and exclude the other. This occurs when interspecific competition is stronger than intraspecific competition.
4. Stable Coexistence: The isoclines cross, and the equilibrium point is stable. Both species coexist at a stable population density below their respective carrying capacities. This occurs when intraspecific competition is stronger than interspecific competition (i.e., each species limits its own growth more than it limits the competitor's growth; alpha < K1/K2 and beta < K2/K1).

Lotka-Volterra Equation for Predation

The Lotka-Volterra predator-prey equations are a pair of first-order, non-linear differential equations that describe the feedback loop between a predator population and its prey.

Prey Population Equation (V): dV / dt = r * V - c * V * P Predator Population Equation (P): dP / dt = g * c * V * P - d * P

Where:

• V: Density of the prey (or host) population.
• P: Density of the predator population.
• r: Intrinsic per capita growth rate of the prey in the absence of predators.
• c: Capture efficiency of the predator (the probability that a predator-prey encounter results in a kill).
• V * P: Encounter rate between predators and prey.
• c * V * P: Rate of prey consumption by predators (predation rate).
• g: Conversion efficiency (the efficiency with which consumed prey is converted into new predator offspring).
• d: Per capita death rate of the predator in the absence of prey.

Key Assumptions of the Predator-Prey Model

• Prey populations have access to unlimited food resources and grow exponentially in the absence of predators.
• The predator is a specialist and will starve to death if prey is absent.
• Predators can consume an infinite amount of prey (no handling time or satiation; linear functional response).
• Encounter rates between predators and prey are random and proportional to their densities.
• The environment is homogeneous.

Model Behavior and Oscillations

Analyzing these equations mathematically shows that they do not reach a single stable equilibrium point. Instead, they produce neutral stable cycles. When plotted over time, the predator and prey populations show continuous sinusoidal oscillations that are 90 degrees (or a quarter of a cycle) out of phase. The prey population peaks first, followed shortly after by the predator population.