Population models form an essential tool to study the dynamics and characteristics of a population. They aim to represent the changes to the population structure over time in mathematical equations. All models, mathematical or otherwise are abstract representations of reality. According to Levins, modeling is a balancing act between three major parameters, generality, realism and precision.

Population models tend to characterize the general properties of the population and are idealized representations which ignore the existence of random variation among the members of the population.
These models are based on two patterns of growth of population namely exponential and geometric population growth patterns.

a. Exponential population growth assumes that populations grow in an exponential pattern. Individuals are added on a continuous scale to such populations. Bacteria and some kinds of insects fall into this group.
This is based on the Malthusian Principle which specifies the change in population density as
dN / dt = rN

The growth of population after t generations is given by the equation

N (t) = N (0) e rt

Where N (t) is the population density after t generations
N (0) is the initial population density
e is the base of natural logarithmic scale.
And r refers to the intrinsic rate of natural increase.

If r = 0 the population is stable. If r>0, then the population is found to be growing and if r<0, then the population will shrink and tend towards zero; i.e. they are facing extinction.
However, the second scenario where r>0 favoring unlimited growth of population is not true in nature. In natural ecosystems, the population growth is limited by other factors of the environment which clearly limits the population growth. In order to accommodate for these variations, a new model was proposed.

b. Geometric population growth models propose a geometric pattern of growth.
They describe the changes in populations across generations as a geometric increase in discrete intervals.
The population density after t+1 generation is defined by
N (t+1) = N (t) λ

Where 'λ' - geometric growth rate
This represents the proportional growth of population
If λ is 0.5 the population density gets reduced by 50 %. If λ is 1.5, the population density will increase by 50%. At λ=1, the population is considered to be stable.

The equation can also be represented as
N (t) = N (0)λ t

Where N (0) represents the initial population density.

When the equation is compared with that of exponential growth, we get the relation between geometric and exponential growth rates of population

As λ = er
Or ln λ = r

BIDE model

Geometric population growth rate is a derivative of this fundamental model and hence the two cannot be considered as separate.
There are four ways in which a population size can be altered, birth (B), death (D), immigration (I) and emigration (E). The change in population size is represented as

Δ N = B + I - D - E

Population size after t+1 generation is described as
N t+1 = Nt +Bt +It- Dt -Et

Closed populations does not have immigration or emigration, hence the equation is limited to rates of birth and death in such populations.
Here, It = 0 and Et = 0, and

N t+1 = Nt +Bt - Dt

When the birth and death rates are expressed as per capita rates the accuracy increases.
N t+1 = Nt +bt Nt - dt Nt

N t+1 = Nt + (bt- dt) Nt

Per capita rates are assumed to be constant over generations and this equation thus gives the geometric rate of growth. The difference b-d can be represented as a constant R, since b and d are assumed to be constants,

N t+1 = (1 +( bt- dt)) Nt
Hence, N t+1 = (1 +R) Nt

And N t+1 = λ Nt λ = 1+R

The difference between R, r and lambda is in the biological realm. It determines the type of reproduction in the population (either pulse or continuous), and survival rates (instantaneous or finite)

c. Logistic model

This model was developed by Verhulst in 1838 after the exponential growth models failed to accurately represent the characteristics of the original population, especially at lower values of r.
The model was further developed by Pearl, earning the title Verhulst-Pearl model.

The differential equation is given by

dN/dt = kN (1-N/K)
where K- carrying capacity of the population and N is the population size.
When N is smaller than K, this reaches the exponential growth rates.

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