#dN/dt

Population Growth

A population of 1000 with 100 births and 50 deaths/year shows dN/dt of:

Net population change from births and deaths is calculated by subtracting losses from gains. During one year, 100 births add individuals and 50 deaths remove them, giving ΔN = 100 − 50 = 50 individuals per year. If these events are treated as rates over that interval, dN/dt is approximated by +50 individuals per year. The initial population of 1,000 is not needed for the absolute change, but it permits calculation of the per-capita rate: r ≈ 50/1,000 = 0.05 per year. Likewise, the per-capita birth and death rates are 0.10 and 0.05 per year, whose difference is 0.05. Multiplying r by N recovers

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

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Population Growth

Plot of dN/dt against population density gives:

Plotting logistic dN/dt against N gives an inverted U-shaped, or dome-like, parabola. Growth is zero at N = 0, rises to a maximum at N = K/2, then falls to zero at N = K. Bell-shaped is an approximate visual description, although the mathematical curve is specifically parabolic rather than a normal distribution. The logistic model is a deliberately simplified, density-regulated model. It assumes a constant intrinsic rate and carrying capacity, no time delay, and no age or spatial structure. Its value lies in exposing the feedback mathematically; real populations can oscillate, overshoot, or tr

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

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Population Growth

If r = 0.15 and K = 400, what is maximum growth rate (dN/dt)?

For logistic growth, maximum dN/dt equals rK/4 and occurs at N = K/2. With r = 0.15 and K = 400, the calculation is 0.15 �� 400 �� 4 = 15 individuals per time unit. The keyed value 30 is therefore inconsistent with the standard logistic equation; 30 would not result from the supplied parameters. Graphically, logistic abundance through time is sigmoid, whereas its instantaneous increment plotted against abundance is a downward-opening parabola. Confusing these two plots leads to incorrect curve descriptions. The time trajectory approaches K, while the production curve peaks at K/2 and i

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

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Population Ecology

A plot of dN/dt as a function of population density shows:

“Bell-shaped curve” for a plot of dn/dt as a function of population density shows. This relationship follows from the ecological mechanism represented by the terms in the item, not merely from an association between their names. 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. The remaining alternatives—“Rectangular hyperbola”, “Negative exponential”, “Linear increase”—refer to different states, proc

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

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