CBSE · Class 12 · Biology · Chapter 11

Organisms and Populations — Formula Sheet

Board Formulas

14 formulas

1.Population Density
$N = \frac{number of individuals}{unit area or volume}$
$N$ : Population density (individuals per unit area or volume)
Total number is not always meaningful: one huge banyan tree versus 200 Parthenium plants. Then per cent cover or biomass is a better measure. Relative density (fish caught per trap) or indirect signs (tiger pug marks and fecal pellets) are also used.
2.Birth Rate (Natality Rate)
$b = \frac{number of births in a period}{population at the start}$
$b$ : Per capita birth rate (births per individual per unit time)
A per capita rate. Example from NCERT: 8 new lotus plants added to 20 gives 8/20 = 0.4 offspring per lotus per year.
3.Death Rate (Mortality Rate)
$d = \frac{number of deaths in a period}{population at the start}$
$d$ : Per capita death rate (deaths per individual per unit time)
A per capita rate. Example from NCERT: 4 of 40 fruit flies die in a week gives 4/40 = 0.1 individuals per fruit fly per week.
4.Age Pyramids
$Expanding (growing); Stable; Declining$
Age groups are pre-reproductive, reproductive and post-reproductive. An expanding pyramid has a broad base (many young); a declining one has a narrow base. Sex ratio is another attribute, e.g. 60% females and 40% males.
5.Population Growth Equation★
$N_{t + 1} = N_{t} + [(B + I) - (D + E)]$
$N_{t}$ : Population density at time t · $N_{t + 1}$ : Population density one time step later · $B$ : Number of births (natality) · $I$ : Number of immigrants (same species coming in) · $D$ : Number of deaths (mortality) · $E$ : Number of emigrants (leaving the habitat)
Population increases if (B + I) > (D + E) and decreases if (B + I) < (D + E). Births and deaths matter most normally; immigration can dominate when a new habitat is being colonised.
6.Exponential Growth (Differential Form)★
$\frac{d N}{d t} = (b - d) \times N = r N$
$N$ : Population size (density) · $r$ : Intrinsic rate of natural increase, r = b − d (per unit time) · $b, d$ : Per capita birth and death rates
Applies when resources are unlimited. Each species then realises its full innate potential to grow, and the curve is J-shaped.
7.Exponential Growth (Integral Form)
$N_{t} = N_{0} e^{r t}$
$N_{t}$ : Population density after time t · $N_{0}$ : Population density at time zero · $e$ : Base of natural logarithms (2.71828) · $t$ : Time elapsed
Use for the population after time t under unlimited resources. Keep r and t in matching time units.
8.NCERT Values of r
$r_{Norway rat} = 0.015; r_{flour beetle} = 0.12; r_{India, 1981} = 0.0205$
r measures how quickly a population can grow under ideal conditions. These are the values quoted in NCERT; they are useful for comparison questions.
9.Logistic Growth (Verhulst–Pearl)★
$\frac{d N}{d t} = r N (\frac{K - N}{K})$
$N$ : Population density at time t · $r$ : Intrinsic rate of natural increase · $K$ : Carrying capacity — the maximum population the habitat can support
Applies when resources are limited. Growth slows as N approaches the carrying capacity K, giving an S-shaped (sigmoid) curve. NCERT calls this the more realistic model.
10.Limits of the Logistic Equation
$N ≪ K \Rightarrow \frac{d N}{d t} \approx r N; N = K \Rightarrow \frac{d N}{d t} = 0$
When the population is small the logistic curve looks exponential; at N = K growth stops and the curve reaches its asymptote.
11.Growth Curve Shapes
$Exponential \Rightarrow J-shaped; Logistic \Rightarrow S-shaped (lag, acceleration, deceleration, asymptote)$
Plot population density N on the y-axis and time t on the x-axis. Mark K as a horizontal dashed line on the logistic curve.
12.Life History Variation
$Breed once: Pacific salmon, bamboo; Breed many times: most birds and mammals$
Some organisms produce many small offspring (oysters, pelagic fishes); others produce a few large ones (birds, mammals). Populations evolve to maximise reproductive fitness in their habitat.
13.Population Interactions★
$Interaction Mutualism Competition Predation Parasitism Commensalism Amensalism Species A + - + + + - Species B + - - - 00$
+ means benefit, − means harm, 0 means unaffected. Mutualism: lichens, mycorrhizae, fig and wasp. Commensalism: orchid on a mango branch, barnacles on a whale, cattle egret and cattle, clown fish and sea anemone. Brood parasitism: cuckoo (koel) laying eggs in a crow's nest.
14.Gause's Competitive Exclusion Principle
$Two closely related species competing for the same resources cannot co-exist indefinitely$
The competitively inferior species is eliminated eventually. Species can co-exist by resource partitioning, as MacArthur showed for five warbler species feeding on the same tree.