Why No Population Can Keep Growing Forever

The fish in a village pond, the deer in a forest and the bacteria on a laboratory plate each form a population — individuals of one species living in one area. Populations have their own measurable features, they grow and shrink, and they interact with other species all the time.

This lesson covers population attributes and density, population growth models, and population interactions.

A population has attributes that single individuals do not — birth rate, death rate, sex ratio, age distribution and density — and density is measured by direct counts or estimated indirectly from cover, catches or signs such as pugmarks and faecal pellets.

Population attributes:

- Birth rate (natality) — births per member of the population in a given period
- Death rate (mortality) — deaths per member in a given period
- Sex ratio — the proportion of females to males
- Age distribution — shown as an age pyramid of pre-reproductive, reproductive and post-reproductive groups; an expanding pyramid has a broad base, a stable one has similar layers, and a declining one has a narrow base

Worked example. If 8 new lotus plants appear in a pond of 20 lotus plants over a season, the birth rate is

$$\frac{8}{20}=0.4\text{ offspring per lotus per season}$$

If 4 of 40 fruit flies in a jar die in a week, the death rate is $\frac{4}{40}=0.1$ per fruit fly per week.

Measuring density:

- Direct counts — for large animals or small areas
- Percentage cover or biomass — when individuals differ greatly in size
- Relative density — such as the number of fish caught per trap
- Indirect estimates — tiger numbers from pugmarks and faecal pellets

An everyday example. Tiger surveys in Indian reserves rely on pugmarks, faecal pellets and camera traps instead of trying to count every tiger directly.

The substance. Counting individuals can mislead — 200 carrot grass plants may matter far less in an area than a single huge banyan tree, so cover or biomass is used.

Exponential growth happens when resources are unlimited, so the population grows at a constant rate into a J-shaped curve, while logistic growth happens when resources are limited, so growth slows near the carrying capacity and gives an S-shaped curve.

Change in population size:

$$N_{t+1}=N_t+[(B+I)-(D+E)]$$

where B is births, I immigration, D deaths and E emigration.

Exponential growth:

$$\frac{dN}{dt}=rN,N_t=N_0{e}^{rt}$$

- $r$ is the intrinsic rate of natural increase
- Growth becomes explosive, forming a J-shaped curve

Worked example. A bacterial culture starts with $N_0=100$ cells and grows exponentially with $r=0.1$ per hour. After 10 hours,

$$N_{10}=100\times e^{0.1\times 10}=100\times e\approx 272\text{ cells}$$

Logistic growth:

$$\frac{dN}{dt}=rN\left(\frac{K-N}{K}\right)$$

- $K$ is the carrying capacity, the largest population the environment can support
- Growth passes through lag, acceleration and deceleration phases to an asymptote at $K$, forming an S-shaped curve

Worked example. With $r=0.1$ per hour, $K=1000$ and $N=500$,

$$\frac{dN}{dt}=0.1\times 500\times \frac{1000-500}{1000}=25\text{ individuals per hour}$$

An everyday example. Water hyacinth spreading across a nutrient-rich lake grows almost exponentially until space and nutrients begin to run out.

The substance. Logistic growth is the more realistic model — no natural population has unlimited resources for long.

Population interactions are classified by their effect on each species: mutualism benefits both, competition harms both, predation and parasitism benefit one and harm the other, commensalism benefits one without affecting the other, and amensalism harms one without affecting the other.

Interactions, with the effect on each species:

- Mutualism (+, +) — lichens, a partnership of fungi and algae; mycorrhiza between fungi and plant roots; fig trees and their pollinating wasps
- Competition (-, -) — flamingos and fish in South American lakes competing for zooplankton
- Predation (+, -) — tigers eating deer; prey defend themselves with camouflage or toxic chemicals, as in the distasteful monarch butterfly
- Parasitism (+, -) — Cuscuta, a parasitic plant on hedge plants; lice on humans; the cuckoo laying eggs in a crow's nest, called brood parasitism
- Commensalism (+, 0) — an orchid growing on a mango branch; cattle egrets feeding on insects stirred up by grazing cattle; clownfish living among sea anemone tentacles
- Amensalism (-, 0) — the fungus Penicillium releasing substances that stop nearby bacteria from growing

An everyday example. Cattle egrets following buffaloes in paddy fields catch insects disturbed by the grazing animals without affecting the buffaloes — commensalism.

The substance. Predators help maintain species diversity — by stopping any one prey species from becoming too numerous, they reduce competition among prey species.