AP Environmental Science Unit 3: Populations — Free Review Games.

Carrying capacity is the maximum population size that an environment can support indefinitely given available resources (food, water, space, etc.).

Density-dependent factors have a greater effect as population density increases. Disease spreads more easily in crowded populations. Natural disasters are density-independent.

Exponential growth produces a J-shaped curve where the population grows faster as it gets larger, with no environmental resistance. This is unsustainable long-term.

Type I curves (humans, elephants) show low mortality in early and middle life, with most deaths occurring in old age. These species typically have few offspring with high parental care.

The demographic transition model shows how nations shift from high birth and death rates (pre-industrial) to low birth and death rates (post-industrial) as they develop economically.

Logistic growth produces an S-shaped curve. Growth rate decreases as the population nears K due to density-dependent factors like competition for resources, predation, and disease.

A wide base means a large proportion of the population is in pre-reproductive age groups, indicating high birth rates and potential for rapid future growth.

r-selected species (insects, rodents, many fish) maximize reproductive rate: many offspring, little parental investment, early maturity, short lifespan. They thrive in unstable environments.

TFR estimates the average number of children born to a woman over her lifetime. A TFR of 2.1 is considered replacement level in developed countries.

Density-independent factors affect populations regardless of their size. Natural disasters like droughts, floods, and fires impact populations the same whether large or small.

Net change = (births - deaths) + (immigration - emigration) = (200-100) + (50-30) = 120. Growth rate = 120/10,000 = 0.012 = 1.2%.

Doubling time = 70 / growth rate (%). A population growing at 2% per year will double in 70/2 = 35 years. This applies to any exponentially growing quantity.

K-selected species (elephants, whales, large primates) invest heavily in few offspring with long maturation periods. When populations decline, recovery is extremely slow compared to r-selected species.

The IPAT equation (I = P x A x T) shows environmental impact as the product of population size, affluence (consumption per person), and technology (environmental impact per unit of consumption).

Replacement level exceeds 2.0 because not all children survive to reproductive age, and slightly more boys are born than girls. The extra 0.1 compensates for these factors.

A Type II survivorship curve plots as a straight diagonal line on a log scale, indicating a constant, age-independent mortality rate. Birds and squirrels are classic examples. Type III curves (choice B) show very high early mortality with few survivors reaching adulthood — typical of many fish and plants. Type I curves (choice C) describe humans and large mammals where most individuals survive to old age.

Biotic potential (often written \(r_{max}\)) is the highest intrinsic rate of natural increase a population could achieve if resources were unlimited and no environmental resistance existed. Choice B describes carrying capacity (\(K\)). Choice C describes survivorship. Choice D describes net primary productivity, which is an ecosystem-level measure.

Population density is a spatial measure — for example, 50 deer per square kilometer or 1,000 bacteria per milliliter. Choice B describes population size, which is an absolute count with no spatial component. Choice C describes a growth rate. Choice D describes a ratio used in demographic analysis, not density.

Emigration is the permanent departure of individuals from a population, which reduces population size. Immigration (choice B) is the opposite — arrival of new individuals — and increases population size. Choice C describes migration, which is reversible and seasonal. Choice D describes competitive displacement, an interspecific interaction.

K-selected species (e.g., elephants, whales, humans) channel energy into a small number of offspring, maximizing survival of each one. This contrasts with r-selected species (choices B, C, D), which maximize reproductive output, have rapid generation turnover, and excel at colonizing unstable or newly opened habitats where competition is low.

In logistic growth, the term \(\frac{K - N}{K}\) approaches zero as \(N\) approaches \(K\), progressively reducing the per capita growth rate and flattening the curve into the characteristic sigmoid shape. Choice C is incorrect — resources typically become more scarce, not more abundant, at higher densities. Choice D is partially true in direction but does not correctly describe the mechanism responsible for the S-curve shape.

The crude birth rate is expressed as live births per 1,000 people per year. It is called 'crude' because the denominator is the entire population rather than just women of reproductive age. Choice B defines the total fertility rate (TFR). Choice C would describe a birth-to-death ratio, not a standardized rate. Choice D describes a feature of age structure, not birth rate.

A narrow base signals a low birth rate — few individuals are entering the youngest age classes. As the larger middle cohort ages without sufficient replacement from below, overall population size will decrease. Countries such as Japan and Germany currently show this pattern. A pyramid with a wide base (choice B) indicates rapid growth, while a roughly equal-width column (choice C) indicates stability.

Overshoot happens when a population surpasses K faster than resource depletion becomes apparent, degrading the environment and potentially lowering the carrying capacity itself. The resulting sharp population decline is called a crash or dieback. Choice B describes unchecked exponential growth. Choice D describes standard logistic regulation — the opposite of overshoot — where growth is curtailed before K is exceeded.

In logistic growth, \(\frac{dN}{dt}\) is maximized at \(N = \frac{K}{2}\). Harvesting at this point removes individuals at the same rate the population naturally replaces them, yielding the largest sustainable harvest over time. Choice B ignores the replacement rate and would drive the population toward extinction. Choice C is incorrect because a population at \(K\) has near-zero growth and thus minimal surplus to harvest sustainably.

Stage 2 marks the early industrial or developing phase: advances in food production, clean water, and medicine reduce death rates significantly, but cultural norms favoring large families keep birth rates high. The resulting gap drives rapid natural increase. Choice B describes Stage 1, where both rates are high but roughly balanced, producing little net growth. Choice C reverses the actual pattern.

The Allee effect is a form of positive density dependence at low population sizes: individuals in very small populations may be unable to find mates, cooperate in predator defense, or sustain social group structures necessary for reproduction. This creates a minimum viable population threshold — fall below it, and per capita growth turns negative, accelerating extinction. Choice C describes a real density-dependent effect (disease transmission) but it operates at high density, not low.

Metapopulation theory describes a regional system of discrete subpopulations in separate habitat patches. Individual patches may experience local extinction, but dispersers from neighboring patches can recolonize them. This balance of local extinction and recolonization maintains regional persistence. Choice B describes a divided population but lacks the dynamic patch-network and recolonization process central to the metapopulation concept.

Density-dependent factors intensify as population density rises. As more deer compete for the same food supply, per capita food availability falls, weakening individuals and reducing reproduction and survival. Choices B, C, and D are density-independent — a cold snap, wildfire, or drought affects individuals regardless of how crowded the population is.

\(R_0\) is the average number of daughters born to one female that themselves survive to reproductive age. \(R_0 = 1\) means exact replacement; \(R_0 > 1\) means growth; \(R_0 < 1\) means each generation is smaller than the previous one — a declining population. An \(R_0\) of 0.7 means only 70 daughters survive to reproduce for every 100 mothers. Choice B incorrectly implies \(R_0\) is slightly above 1. Choice C would correspond to \(R_0 \approx 1\), not 0.7.

Fragmentation divides populations into smaller groups separated by inhospitable matrix habitat. Each fragment supports fewer individuals, increasing vulnerability to extinction from inbreeding depression and random demographic fluctuations. Reduced dispersal also weakens the rescue effect, by which immigration from neighboring patches can stabilize a declining subpopulation. Choice B is an oversimplification — edge effects benefit some species but harm interior specialists. Choice C states a precise fraction not supported by the general principle.

Lotka-Volterra dynamics produce out-of-phase oscillations: prey increase first (resources abundant), which fuels predator growth with a delay. Predators then over-consume prey, driving prey to a minimum. With food now scarce, predators decline to their minimum, allowing prey to recover — and the cycle repeats. Choice C is incorrect: when prey are near their trough, predator numbers are still relatively high and declining, so the two minima are offset, not simultaneous.

A TFR below replacement level (~2.1 in developed nations) signals that each generation is smaller than the last, predicting eventual decline. A TFR of 3.8 predicts growth. Choice B may often hold empirically, but TFR data alone cannot establish per capita income. Choice C is incorrect: a high TFR is consistent with Stage 2 or other stages — Stage 1 is defined by high birth and death rates together, not TFR alone. Choice D is incorrect because population momentum from a large existing young cohort can sustain growth for decades even after TFR drops below replacement.

The rule of 70 states: doubling time \(\approx \frac{70}{r}\), where \(r\) is the growth rate expressed as a percentage. At \(r = 2.5\%\): doubling time \(= \frac{70}{2.5} = 28\) years. Choice B (35 years) corresponds to a growth rate of 2%. Choice D (14 years) would require a rate of 5%. Choice C (175 years) results from multiplying rather than dividing: \(70 \times 2.5 = 175\), a common algebraic error.

Net per capita rate of change \(= (40 - 15 + 10 - 5) = 30\) per 1,000 per year. Net annual change \(= \frac{30}{1{,}000} \times 5{,}000 = 150\) individuals. Choice B (175) results from omitting emigration. Choice C (200) results from incorrectly adding emigration instead of subtracting it. Choice D (125) results from counting only births minus deaths while ignoring migration entirely.

Rewriting the equation: \(\frac{dN}{dt} = \frac{r}{K}(KN - N^2)\). Differentiating with respect to \(N\) and setting equal to zero: \(\frac{d}{dN}\left(\frac{dN}{dt}\right) = \frac{r}{K}(K - 2N) = 0\), which gives \(N = \frac{K}{2}\). At \(N = K\), the term \(\frac{K - N}{K} = 0\), so growth rate is zero. At \(N = 0\), there are no individuals to reproduce. At \(N = 2K\), the term \(\frac{K - N}{K}\) is negative, meaning the population would be shrinking.

As a population shrinks, inbreeding reduces individual fitness, random variation in births and deaths (demographic stochasticity) becomes proportionally more impactful, and reduced genetic diversity limits adaptability to new threats — each effect further reducing population size in a downward spiral. Choice C is incorrect: most density-dependent factors (competition, disease) actually weaken at low densities, so they do not drive the vortex. The vortex is powered by genetic and stochastic forces that worsen as numbers fall.

Per capita footprint \(= \frac{\text{total footprint}}{\text{population}}\). Country X: \(\frac{300 \text{ million gha}}{50 \text{ million people}} = 6\) gha/person. Country Y: \(\frac{400 \text{ million gha}}{200 \text{ million people}} = 2\) gha/person. Country X's per capita impact is three times higher despite its smaller total footprint. Choice B confuses total and per capita impact — a larger country can have a bigger total footprint while each individual consumes far less. Choice D cannot be evaluated without knowing Earth's total biocapacity.

In the tragedy of the commons, each user captures the full private benefit of each additional unit harvested while the cost of depletion is shared among all users. This misalignment of individual and collective incentives makes overexploitation individually rational but collectively destructive. Choice B (competitive exclusion) is an ecological principle about interspecific competition, not an economic incentive problem. The Allee effect (choice D) may accelerate collapse once numbers are low, but it does not explain why rational actors overharvested in the first place.

Population momentum means a population with a young age structure continues to grow even if TFR drops to replacement level, because the oversized cohort now under 15 will soon enter reproductive age and generate a surge of births. Country P's 40% under-15 share signals strong future momentum. Country Q's small young cohort and large elderly cohort indicate declining birth rates and rising crude death rates — a trajectory toward stagnation or decline. Choice C is incorrect: current total population size does not determine future growth rate; age structure does. Choice D confuses economic dependency ratios with biological population dynamics.

The exponential growth model \(\frac{dN}{dt} = rN\) shows that population change is proportional to the current population size and the intrinsic growth rate, producing a J-shaped curve. The option \(\frac{dN}{dt} = rN\left(\frac{K-N}{K}\right)\) is wrong here because it is the logistic model that incorporates carrying capacity, not the unlimited-resource exponential model. Students should recognize that exponential growth applies only when resources are not limiting, such as in a population's early colonization phase.

Carrying capacity is the population size that a given environment can support long-term without degrading the resources the population depends on, such as food, water, and habitat space. The choice describing 'the theoretical maximum growth rate of a species under ideal conditions' instead describes biotic potential, a different concept entirely from carrying capacity. Students should remember that \(K\) is set by environmental limits, not by a species' inherent reproductive ability.

Type I survivorship curves, seen in species like humans and large mammals, reflect high parental investment and low early mortality, with most deaths occurring near the maximum lifespan. The option describing 'relatively constant mortality rates throughout life' instead describes a Type II curve, common in species like some birds and squirrels. Students should link survivorship curve shape to reproductive strategy: high parental care correlates with Type I, while minimal care correlates with Type III.

r-selected species prioritize rapid reproduction and high offspring numbers to exploit unstable or unpredictable environments, trading off parental investment for quantity, which matches 'small body size, short lifespan, large number of offspring with little parental care.' The option describing 'large body size, long lifespan, few offspring with extensive parental care' instead describes K-selected species adapted to stable environments near carrying capacity. Students should associate r-selection with opportunistic colonizers like insects and weeds, and K-selection with species like elephants and whales.

Stage 1 reflects pre-industrial conditions where both birth rates and death rates are high due to limited healthcare and sanitation, so the population grows slowly despite high fertility because mortality offsets it. The option describing 'high birth rates and rapidly falling death rates' instead characterizes Stage 2, when improvements in medicine and sanitation reduce death rates first. Students should remember that Stage 1 has virtually no modern industrialized examples remaining today, making it primarily a historical reference point.

The rule of 70 estimates doubling time as \(\frac{70}{r}\), so dividing 70 by a growth rate of 2 gives \(\frac{70}{2} = 35\) years. The choice of '70 years' incorrectly uses the constant itself as the answer without dividing by the growth rate. Students should memorize this shortcut formula since AP exam questions frequently test quick doubling-time estimates from a given percentage growth rate.

A wide-based, rapidly narrowing pyramid indicates a large proportion of young people and high birth rates combined with falling death rates, the hallmark of Stage 2 rapid population growth. The option describing 'Stage 4, with low birth and death rates' instead produces a more column-shaped pyramid reflecting an even age distribution, not a wide youthful base. Students should use pyramid shape as a visual shortcut for identifying a country's stage: wide base means rapid growth, while a rectangular shape signals stabilization.

Density-dependent factors intensify in effect as population density increases, and disease transmission is a classic example because crowded populations facilitate faster spread of pathogens. The option describing 'a hurricane that destroys habitat regardless of population size' instead illustrates a density-independent factor, since its impact does not scale with how crowded the population is. Students should sort limiting factors into these two categories, since AP questions often ask you to classify a scenario as one or the other.

Density-independent factors affect populations with similar severity no matter how many individuals are present, and a wildfire destroying habitat is a prime example since it does not depend on population crowding. The option describing 'increased competition for nesting sites as a bird population grows denser' instead depends directly on population size, making it density-dependent rather than independent. Students should remember that natural disasters, extreme weather, and pollution events are typically density-independent, while competition, predation, and disease are typically density-dependent.

K-selected species invest heavily in fewer offspring with extended parental care, a strategy favored in stable environments where competition for limited resources near carrying capacity determines survival. The option describing 'newly colonized habitats with abundant unused resources' instead favors r-selected species, which reproduce quickly to exploit open niches. Students should link environmental stability to K-selection and environmental unpredictability or disturbance to r-selection when analyzing life-history strategies.

Type III curves show extremely high mortality among young offspring because parental investment per offspring is minimal, but the few survivors that reach maturity tend to live relatively long. The option describing 'high survival through youth and middle age with mortality concentrated in old age' instead describes a Type I curve seen in species with high parental care. Students should connect Type III curves to high-fecundity, low-investment r-selected species that produce huge numbers of offspring to offset early losses.

Replacement-level fertility is approximately 2.1 in developed nations because two children replace the parents, with the extra 0.1 accounting for child mortality before reproductive age. The option of '1.0 children per woman' is far too low to sustain a stable population since it would only replace one parent, leading to long-term decline. Students should know that TFR below 2.1 in a country signals eventual population decline absent immigration, a concept tested alongside demographic transition stages.

Rate of natural increase is calculated as \(\frac{\text{birth rate} - \text{death rate}}{10}\) expressed as a percentage, so \(\frac{18-7}{10} = 1.1\%\). The distractor of '2.5%' incorrectly adds rather than subtracts the rates or misapplies the conversion factor. Students should practice this quick conversion since AP free-response and multiple-choice questions frequently require deriving percentage growth from per-1,000 birth and death rate data.

Stage 4 reflects a post-industrial society where both birth rates and death rates have stabilized at low levels, resulting in minimal or zero natural population growth, as seen in many developed nations. The option describing 'birth rates falling below death rates, causing population decline' instead characterizes the proposed Stage 5, an extension some demographers add for countries with sub-replacement fertility. Students should distinguish Stage 4's stability from Stage 5's decline, since both feature low death rates but differ in the birth rate relative to it.

In boom-bust dynamics, a population grows rapidly, temporarily overshoots the environment's carrying capacity due to a lag in resource depletion effects, then crashes sharply as resources become scarce before potentially stabilizing at a lower level. The option claiming the population 'remains constant because predators regulate it precisely at carrying capacity' ignores the lag effects and overshoot that define boom-bust cycles, describing instead an idealized logistic equilibrium. Students should recognize that real populations often overshoot \(K\) due to reproductive lag time, producing oscillations rather than smooth logistic curves.

A wide base indicates a large cohort of children who will soon enter reproductive age, meaning even if fertility per woman declines, the sheer number of potential parents sustains high population growth through demographic momentum. The option describing 'an aging population with a shrinking workforce and rising dependency ratio' instead applies to countries with narrow bases and wide tops, the opposite pyramid shape. Students should understand demographic momentum as a key reason population growth continues for decades even after fertility rates begin falling.

Applying the IPAT framework, impact scales as \(P \times A \times T = 2 \times 3 \times 0.5 = 3\), so total environmental impact triples despite technology improvements per unit reducing impact. The option stating impact 'stays the same' incorrectly assumes the technology factor fully offsets population and affluence growth, but the math shows a net threefold increase. Students should practice multiplying these three factors together rather than assuming efficiency gains automatically cancel out growth in population and consumption.

Stage 3 birth rate declines are driven by factors like expanded education (especially for women), greater availability of contraception, and urbanization, which shifts children from economic assets in agrarian settings to economic costs in urban settings. The option citing 'declining access to healthcare and contraception' contradicts the actual trend, since Stage 3 is marked by increasing, not decreasing, access to these services. Students should connect socioeconomic development factors to fertility decline, since this causal chain is a frequently tested AP concept.

Total growth combines natural increase and migration: births minus deaths from per capita rates gives \((0.04-0.02)\times10{,}000 = 200\), and adding net migration of \(200-50=150\) yields a total of \(200+150=350\) individuals per year. The distractor of '200' only accounts for the birth-death difference and ignores net migration, an incomplete calculation. Students should remember that total population change equals births minus deaths plus immigration minus emigration (BIDE model), not birth and death rates alone.

Carrying capacity is not fixed but depends on environmental resource availability, so a drought that reduces water and forage lowers \(K\), and if the population was previously near the old higher \(K\), it may temporarily exceed the new lower limit and experience a die-off. The option claiming carrying capacity 'remains unchanged because \(K\) is a fixed biological constant' misunderstands that \(K\) is dynamic and environment-dependent, not an intrinsic species trait. Students should treat carrying capacity as a variable shaped by current environmental conditions, which is essential for understanding how climate events or habitat degradation can trigger population crashes.