● Resource (`color{violet}("food and space")`) availability is obviously essential for the `color{violet}("unimpeded growth of a population")`.

● Ideally, when resources in the habitat are unlimited, each species has the ability to realise fully its `color{violet}("innate potential to grow")` in number, as `color{violet}("Darwin observed")` while developing his `color{violet}("theory of natural selection.")`

● Then the `color{violet}("population grows")` in an exponential or `color{brown}("geometric fashion.")`

● If in a `color{violet}("population of size")` `N`, the `color{violet}("birth rates (not total number but per capita births)")` are represented as `b` and `color{violet}("death rates (again, per capita death rates)")` as `d`,

then the increase or decrease in `N` during a `color{violet}("unit time period")` `color{violet}("t (dN/dt)")` will be

`color{violet}("dN/dt = (b – d) × N")`

Let `color{violet}((b–d) = r),` then

`color{violet}("dN/dt = rN")`

● The `color{brown}(r)` in this equation is called the`color{brown}("intrinsic rate")` of `color{brown}("natural increase")` and is a very important parameter chosen for assessing impacts of any `color{violet}("biotic or abiotic factor")` on `color{violet}("population growth.")`

● To give you some idea about the `color{violet}("magnitude")` of `color{violet}(r)` values, for the `color{violet}("Norway rat")` the `color{violet}(r)` is 0.015, and for the `color{violet}("flour beetle")` it is 0.12.

● `color{violet}("In 1981")`, the `color{violet}(r)` value for `color{violet}("human population")` in India was `0.0205.`

● The above equation describes the `color{brown}("exponential or geometric growth pattern")` of a population and results in a `color{brown}("J-shaped curve")` when we plot `N` in relation to time.

● If you are familiar with basic calculus, you can derive the integral form of the `color{violet}("exponential growth equation")` as

`color{violet}(N_t = N_0 e^(rt))`
where
`color{violet}(N_t)` = Population density after time `color{violet}(t)`
`color{violet}(N_0)` = Population density at time zero
`color{violet}(r)` = intrinsic rate of natural increase
`color{violet}(e)` = the base of natural logarithms `color{violet}((2.71828))`

● Any species `color{violet}("growing exponentially")` under unlimited resource `color{violet}("conditions")` can reach `color{violet}("enormous population")` `color{violet}("densities")` in a short time.

● `color{violet}("Darwin showed")` how even a `color{violet}("slow growing animal")` like elephant could reach `color{violet}("enormous numbers")` in the `color{violet}("absence of checks")`.

● No population of any species in nature has its `color{violet}("disposal unlimited resources")` to permit `color{violet}("exponential growth")`.

● This leads to competition between individuals for `color{violet}("limited resources.")`

● Eventually, the `color{violet}("‘fittest’")` individual will `color{violet}("survive")` and `color{violet}("reproduce")`.

● The governments of many countries have also realised this fact and introduced various `color{violet}("restraints")` with a view to `color{violet}("limit human population growth.")`

● `color{violet}("In nature,")` a given habitat has enough resources to support a `color{violet}("maximum possible number")`, beyond which no further `color{violet}("growth is possible.")`

● This limit is called as `color{brown}("nature’s carrying capacity (K)")` for that species in that habitat.

● A `color{violet}("population growing")` in a habitat with limited resources show `color{violet}("initially a lag phase,")` followed by phases of `color{violet}("acceleration and deceleration")` and finally an `color{violet}("asymptote,")` when the`color{violet}(" population density")` reaches the `color{violet}("carrying capacity")`.

● A plot of `N` in relation to time `(t)` results in a`color{brown}("sigmoid curve")`.

● This type of `color{violet}("population growth")` is called `color{brown}("Verhulst-Pearl Logistic Growth")` and is described by
the `color{violet}("following equation")`:

`color{violet}(dN/dt = rN ((K-N)/K)`
Where `color{violet}(N)` = Population density at time `color{violet}(t)`
`color{violet}(r)` = Intrinsic rate of natural increase
`color{violet}(K)` = Carrying capacity

● Since resources for `color{violet}("growth")` for most `color{violet}("animal populations")` are `color{violet}("finite")` and become limiting sooner or later, the `color{violet}("logistic growth model")` is considered a more realistic one.