Population Ecology: Logistic Growth, Life Histories, and r-/K-Selection

Logistic Growth and Overshoot

Logistic growth equation: \frac{dN}{dt} = rN\left(1-\frac{N}{K}\right)
As population size N approaches carrying capacity K, the per-capita growth rate approaches zero; overall growth slows
Overshoot: under some conditions (e.g., delays in reproduction when resources become limiting), N can exceed K temporarily
If N > K, 1-\frac{N}{K} < 0 and \frac{dN}{dt} < 0; population declines toward K
Exercise snapshot (illustrative): with r = 1.0, K = 1500, evaluate dN/dt for N = 1510, 1600, 1750, 2000
- N = 1510: \frac{dN}{dt} = 1.0 \cdot 1510 \left(1-\frac{1510}{1500}\right) \approx -10.07
- N = 1600: \frac{dN}{dt} \approx -106.67
- N = 1750: \frac{dN}{dt} \approx -291.67
- N = 2000: \frac{dN}{dt} \approx -666.67
Highest growth rate among these (least negative) is at N = 1510
If r is doubled (r = 2) with K = 1500, growth rates double in magnitude:
- N = 1510: \frac{dN}{dt} \approx -20.13
- N = 1600: \frac{dN}{dt} \approx -213.33
- N = 1750: \frac{dN}{dt} \approx -583.33
- N = 2000: \frac{dN}{dt} \approx -1333.33

Life history traits define an organism's schedule of reproduction and survival; three key components:
- Age at first reproduction (maturity)
- How often reproduction occurs (reproduction frequency)
- How many offspring per reproductive event (offspring number)
Variation across species demonstrates broad life-history strategies driven by environment and trade-offs
Examples of maturity timing:
- Loggerhead turtle: first reproduction ~30 years
- Coho salmon: first reproduction ~3–4 years
Reproduction strategies:
- Semelparity (one-shot reproduction): e.g., coho salmon, agave
- Iteroparity (repeated reproduction): e.g., loggerhead turtle, oaks, horses
Trade-offs: finite resources force compromises between reproduction and survival; higher investment in offspring often reduces parental survival or future fecundity
Examples illustrating trade-offs:
- Larger brood sizes can reduce parental survival (kestrel brood-size manipulation study)
- In some species, more offspring necessitates less parental care per offspring, affecting survival and growth

r-selection: selection for traits that maximize reproduction in low-density or disturbed environments; high intrinsic rate of increase (r) and colonization ability
- Common in disturbed or newly recolonizing habitats
- Examples: weeds in abandoned fields; many small-seeded species (e.g., dandelions)
K-selection: selection for traits that maximize efficiency near carrying capacity; competitive in crowded environments
- Traits favor survival and competitive ability with fewer offspring but greater parental investment
- Examples: mature trees in old-growth forests; large mammals; many birds with parental care
The r/K framework describes a continuum of life-history strategies rather than strict categories; relates to how populations approach carrying capacity and respond to competition

Trade-off between seed size and seed number
- r-selected plants tend to produce many small seeds to spread risk and colonize new habitats
- Large-seeded species invest more resources per offspring, increasing per-offspring survival probability
Examples:
- Dandelions: many tiny seeds to maximize colonization
- Brazil nut trees and walnuts: few large seeds rich in nutrients, aiding seedling establishment
Reproductive strategies reflect environments: high predation or disturbance favors high offspring numbers; resource-rich, stable environments favor larger investment in fewer offspring

Organisms balance offspring number with parental survival after reproduction
Experimental insight: increasing brood size can reduce parental survival in the following season, illustrating a cost of parental care
Implication: life-history diversity arises from trade-offs between offspring quantity and offspring quality/parental investment

Identify three key life-history traits and give examples across species
Explain the reproductive trade-offs illustrated by the peacock wrasse strategy (egg dispersal vs. parental care)
What if stress (e.g., food shortage) leads to parental abandonment? How could this evolve in the context of reproductive trade-offs?

Disturbance and colonization favor r-selection with many small seeds
Stable, competitive environments favor K-selection and larger seed/offspring investment
Life-history variation is an evolutionary outcome shaped by resource limits, density dependence, and environmental pressures