#r value

Population Growth

In exponential growth, per capita rate of increase is:

A defining assumption of exponential growth is that the instantaneous per-capita rate, (1/N)(dN/dt), equals a constant r. This means each individual contributes the same expected net amount to population growth regardless of current abundance. The total or absolute growth rate is not constant: dN/dt = rN becomes larger as N increases when r is positive. Confusing these two rates obscures why the curve accelerates. Constant per-capita performance requires births and deaths per individual to remain unchanged, which approximates an environment without density-dependent competition, disease transmission, or resource depletion. If those processes intensify with density, r effectively declines and growth becomes nonexponential. A zero per-capita rate would produce no net growth, while a decreasing rate is characteristic of regulated growth toward carrying capacity. Environmental variation can also make r fluctuate through time even at low density, so the strict model is an idealization. Its central mathematical feature remains proportionality: doubling N doubles the expected absolute increment because the same per-capita rate acts on twice as many organisms.

Ref: Ecology: Concepts and Applications, Molles, 9th Ed., Ch. 11

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Population Growth

Which condition leads to exponential growth?

Exponential growth requires the per-capita rate of increase to remain approximately constant as population density rises. That condition is approached in an ideal, effectively unlimited environment where food, space, nutrients, and other essential resources do not constrain reproduction or survival. Predation, infectious disease, and resource scarcity usually introduce density-dependent mortality or reduced fecundity, causing per-capita growth to fall and the trajectory to depart from an exponential curve. Under ideal conditions, dN/dt = rN and a positive r causes abundance to increase by a constant proportion per unit time. Absolute increments consequently become larger as the population grows. Such conditions rarely persist indefinitely in nature because organisms eventually modify or exhaust their environment. Nevertheless, brief exponential phases occur when microbes enter fresh medium, a species colonizes vacant habitat, or a population rebounds from low density. The model is therefore a useful null expectation and a description of potential growth, not a claim that resources are literally infinite. Once limiting feedback becomes important, logistic or more detailed density-dependent models are more appropriate.

Ref: Ecology: Concepts and Applications, Molles, 9th Ed., Ch. 11

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Population Growth

If r = 0.1, doubling time will be:

For continuous exponential growth, doubling time T2 satisfies 2N0 = N0e^(rT2). Cancelling N0 and taking natural logarithms gives ln 2 = rT2, so T2 = ln 2/r. With r = 0.1 per stated time unit, T2 = 0.693/0.1 = 6.93, approximately 7 time units. Direct substitution confirms the result: e^(0.1 × 6.93) ≈ e^0.693 ≈ 2. A value of 2.1 would yield e^0.21 ≈ 1.23, far short of doubling. The units of doubling time are the reciprocal of the units used for r; for example, r per year produces a doubling time in years. This calculation assumes a constant positive r and no density-dependent slowing. Geometric growth uses a related expression, T2 = ln 2/ln λ, because r = ln λ for equivalent time intervals. Thus the keyed value is inconsistent with the standard exponential-growth equation; the scientifically supported numerical choice is 7. The discrepancy is numerical, not a matter of alternative ecological terminology or convention.

Ref: Ecology: Concepts and Applications, Molles, 9th Ed., Ch. 11

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Population Growth

Intrinsic rate of increase is highest for:

Small mammals such as mice generally have a much higher intrinsic rate of increase than large, slow-lived vertebrates. They mature early, have short generation times, reproduce frequently, and produce several offspring per litter. These traits place reproductive contribution early in the life table, which strongly elevates r because offspring produced sooner can themselves reproduce sooner. Elephants and humans mature late, have long gestation, small litters, and long intervals between births; their populations therefore increase slowly even when adult survival is high. Many snakes have intermediate life histories, with reproductive schedules slower than those of mice. Intrinsic rate of increase is not simply lifetime fecundity: its value emerges from the timing of reproduction and survival across ages, often estimated from the Euler–Lotka equation. High juvenile mortality can lower realized growth, but rmax denotes potential increase under favorable conditions. The comparison illustrates the life-history continuum from fast species with rapid turnover to slow species that invest heavily in fewer offspring and depend more on adult survival.

Ref: Ecology: Concepts and Applications, Molles, 9th Ed., Ch. 11

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Population Growth

If birth rate > death rate, then r is:

When the per-capita birth rate exceeds the per-capita death rate, their difference r = b − d is greater than zero. In the exponential model dN/dt = rN, both N and r are then positive, making the population’s instantaneous change positive. The integrated trajectory, Nt = N0e^(rt), rises because e^(rt) exceeds one for positive time. The magnitude of r determines how rapidly abundance increases, not merely the direction of change. If birth and death rates were equal, r would be zero and expected abundance would remain constant; if deaths exceeded births, r would be negative and abundance would decline. This conclusion assumes a closed population or that migration has been omitted deliberately. In an open population, sufficiently strong emigration could cause local decline even when births exceed deaths, while immigration could offset a negative birth–death balance. The sign of r therefore expresses net demographic performance from the rates included in its definition. Calling it “constant” describes a model assumption and does not identify whether its numerical value is positive, zero, or negative.

Ref: Ecology: Concepts and Applications, Molles, 9th Ed., Ch. 11

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Population Ecology

A population with logistic growth and r = 0.15/week in an environment with K = 400 shows max growth rate of:

“22.5” for a population with logistic growth and r = 0.15/week in an environment with k = 400 shows max growth rate of. This relationship follows from the ecological mechanism represented by the terms in the item, not merely from an association between their names. Mechanistic support comes from showing how resource limitation, enemies, mate availability, or physiological stress changes demographic performance. A descriptive association alone does not establish regulation or causation. The remaining alternatives—“15”, “30”, “60”—refer to different states, processes, or scales and therefore do not express the same causal relationship. Population ecology links individual births, deaths, immigration, and emigration to changes in abundance. Per-capita rates determine the direction of change, while density dependence creates feedback when crowding alters survival or reproduction. Linking the wording to measurable consequences for fitness, abundance, or flux gives the conclusion its scientific meaning and prevents a purely mnemonic interpretation.

Ref: Campbell Biology, Urry et al., 12th Ed., Unit 8 Ecology

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