(I) When resources are limiting, the population growth curve is sigmoid and it is represented by the equation:
\(dN/dt=rN\left( \frac { K-N }{ K } \right) \), where
N = population density at time t
r = intrinsic rate of natural increase
K = carrying capacity
When resources are not limiting, the growth curve is J shaped and is represented by the equation: dN/dt = rN 
\(N_{ t }={ N }_{ 0 }^{ e^{ rt } },where\)
\(N_{ t }\) =population density after time t
\(N_{ 0 }\)= population density at time zero
r = intrinsic rate of natural increase
e = the base .of  natural logarithm (2.71828)

The different ways to cope with abiotic stresses are:
(i) Regulation
(ii) Conformation
(iii) Migration and
(iv) Suspending.
(i) Regulation: Some animals maintain homeostasis by physiological means that ensure a constant body temperature, constant osmolarity of body fluids, etc; mammals maintain a constant body temperature.
(ii) Conformation: Those animals called conformers, which cannot maintain homeostasis, change their body temperature or osmolarity of body fluids according to the ambient environment.
(iii) Migration: If the stressful condition is localised, animals move away temporarily from the stressful environment to more hospitable area and return when the stressful conditions is over.
(iv) Suspending: Some animals avoid the stress by escaping in time; bears go into hibernation during winter and snails go into aestivation in summer. (any three)

Three important characteristics of a population are:
(i) Population size and population density
(ii) Birth or natality rate
(iii) Death or mortality rate.
(i) Population size.  The size of population keeps changing with time depending on the factors like:
(a) Food availability
(b) Weather
(c) Predation pressure and
(d) Competition.
The population density in a given habitat during a given period, changes due to four basic processes namely
(a) Natality,
(b) Mortality,
(c) Immigration and
(d) Emigration.
While natality and immigration contribute to an increase in the size and density of a population, mortality and emigration contribute to a decrease in them.
So the equation for population growth is:
\(N_{ t+1 }=N_{ t }+[(B+I)-(D+E)],\)
where, N = population density at time t.
B = Birth rate.
I = Immigration.
D = Death rate.
E  = Emigration.
If B + I is more than D + E, the population density increases.
If B + I is less than D + E, the population density decreases.
(ii) Birth or Natality rate. It is generally expressed as number or births per 1,000 individuals of a population per year. It increases the population size (total number of individuals of a population) and population density.
Natal or Birth Rate                  
If B + I is more than D + E, the population density increases.
\(=\frac { Total\quad Number\quad of\quad Live\quad Births\quad in\quad a\quad year }{ Total\quad Population\quad in\quad the\quad Middle\quad of\quad the\quad year } X1000\)
(iii) Death or Mortality rate. It is opposite of Mortality (Death) Rate natality rate. It is commonly expressed as the number of deaths per 1,000 individuals of a population per year.
\(=\frac { No.of\quad Deaths\quad in\quad a\quad year\qquad }{ Total\quad Population\quad in\quad the\quad Middle\quad of\quad the\quad year } X1000\\ \)

Commensalism:
Commensalism is defined as an interspecific interaction where one species is benefitted, while the other species is neither benefitted nor harmed, e.g.
(i) Orchids grow as epiphytes on mango or other fruit trees.
Orchids are benefitted by getting a shelter, while the tree is neither benefitted nor damaged.
(ii) Barnacles growing on the whale are benefitted to move where food is available.
(iii) The cattle egrets always forage near to where the cattle animals graze; the cattle animals stir up the ground and the insects are flushed' out from the vegetation and make it easy for the egrets to catch them.
(iv) The clown fish living among sea anemones get protection from their predators, which stay away from the stinging tentacles of the sea anemone

Mutualism:
Mutualism is defined as the interspecific interaction in which both the interacting species are benefitted, e.g.
(i) Lichens show an intimate mutualistic relationship between a fungus and an alga, where the fungus provides protection and helps the alga to absorb nutrients.
The alga prepares the food for the fungus too.
(ii) Mycorrhizae are mutualistic association between certain fungi and the roots of higher plants.
The fungus helps the plant by making the nutrients available in the vicinity of roots, and stores certain minerals like phosphorus in the mantle of hyphae
The fungus gets food and shelter from the plant.
(iii) Plants depend on insects for pollination; the insects are rewarded in the form of nectar or edible pollen or sometimes place for oviposition.
(iv) Rhizobium in the root nodules of legumes helps in nitrogen fixation and nitrogen nutrition to the plants and in turn gets shelter and food from the plant.

Adaptation is the quality of the organism (morphological, physiological, behavioural) that enables the organism to survive and reproduce in its habitat. It adopts organism to live in different types of habitats.
Adaptation of desert plant are as follows:
(i) Desert plants have cuticle to minimise transpiration.
(ii) In some desert plant, leaves modify into spines to minimise loss of water.
(iii) They have long roots and adaptations to reduce transpiration, e.g., Acacia.
Adaptation of desert animal are as follows:
(i) Desert animals concentrate their urine for minimum loss of water, e.g., Kangaroo rat.
(ii) Desert animals absorb heat from the sun when the body temperature drop below the comfort zone.
(iii) They live in burrows during hot season and have little water requirement.

(i) Predation (Tiger-predator and deer-prey)
(ii) The role of predators is
(a) To keep prey population under control
(b) To help in maintaining species diversity in a community by reducing the intensity of competition among prey species.
(iii) Population interaction seen in nature are:
(a) Predation
(b) Parasitism
(c) Commensalism
(d) Mutualism
(iv) Animals lover and sympathy towards animals

(a)The two types of growth models are as follows:
(i): Exponential growth:When the resources are unlimited, population tends to grow in an exponential pattern
If the population size is N and the birth and death rates (not per capita) are band d respectively, then increase or decrease of N at t (time period) is given by
dN/dt = (b-d) * N
If (b - d) = r, then
dN/dt = rN
where r = Intrinsic rate of natural increase
or  N₁=N₀e^N

where
N, = Population density at time t
N_o = Population density at time 0
r = Intrinsic rate of natural increase
e = Base of natural algorithm (2.71828)
(ii) Logistic Growth : When the resources are limited leading to competition between individuals and survival of fittest, the population tends to grow in a logistic manner
A population with limited resources shows initially a lag phase, followed by phases of acceleration and deceleration and finally an asymptote, when the population density reaches the carrying capacity. This type of population growth is called Verhulst-Pearl Logistic Growth
\(\frac { dN }{ dt } =rN\left( \frac { K-N }{ K } \right) \)
where,
N = Population density at time t
r = Intrinsic rate of natural increase
K = Carrying capacity

(b) The difference in the shape of these curves is the amount of resources available for the given population. For exponential growth, the amount of resources is considered infinite, while, logistic growth happens when there are
limited resources.
(c) The human population represents the logistic growth curve as the numbers of human beings are increasing rapidly but available resources are not increasing at the same pace. Such a growth pattern is not sustainable because at one point the human population would reach a place where there would not be enough resources for everyone. For sustainable growth, we must find out ways to develop and use already present resources more intelligently.

The cactus plants grow in desert regions where they adapt to store water in the roots, stems or fruits. So, they need very less water to survive.If she waters it daily, it will gradually decay due to excess of water.
Kangaroo rats, camels and desert lizards are animals that survive without water. This is because, the kangaroo rat in North American deserts is capable of meeting all its water requirements through its internal fat oxidation (in which water is a by product). It also has the ability to concentrate its urine so that minimal volume of water is used to remove excretory products.