Exponential or J-shaped growth curve and Sigmoid Growth curve

In a new environment population follow a J-shaped curve when the density increases exponential in a logarithmic form and then due to environment resistance halt abruptly. It is summarised as: dN/dt= r (r=constant for organism’s biotic potential)

Number of population show characteristic great fluctuations. It’s a density independent population growth curve i.e. the population growth is independent of density until the final crash.  

Figure 20: J-shaped or exponential growth curve.

Sigmoid Growth curve

Sigmoid or S-shaped curve is a population growth pattern where initially density of population accelerated slowly and then rapidly at an exponential growth rate. Further, the population stabilizes at zero growth rates due to environmental resistance at higher population density. Hence, it’s a density dependent population growth. The stabilized zero population growth rates are called as “carrying capacity” (K) or saturation value. It’s a graph plotted against time and change in number of population.

When the biotic potential interacts with environment resources the population shows an S-shaped growth curve and summarised as:

dN/dt= rN (K-N)/K

Table 6: Difference between exponential and Logistic Growth curve

S.No.	Exponential or J-shaped growth	Sigmoid or Logistic or S-shaped growth curve
1.	Abundance of resources	Limited resources (strong intraspecific competition)
2.	Population exceeds carrying capacity	Population never reaches carrying capacity
3.	Seldom reaches steady phase	Reaches a stationary phase
4.	Mass mortality leads to population decline	Seldom decline in population
5.	Two phase curve: lag and log phase	Four phase curve: lag phase, log phase, deceleration and steady phase.
6.	Eg. algal bloom. Lemming and other few organisms	Common curve eg. members of wildlife

Living organism differ from non living things by means of reproduction. Thus, population dynamics is majorly discussed under two models of population growth; (a) exponential and (b) logistic model, to answer some basic questions as how long a population will take to reach certain density and the expected duration in non favourable environment the population can withdraw and also what will be the generation time of population.

a.    Exponential Model

Exponential model of population growth is applicable in fishery to predict the fish population dynamics, for predicting the yield of insect rearing, conservation biology, in predicting the population growth of introduced species i.e. insect quarantine and in microbiology for predicting the growth kinetics of multiplying bacteria.

Figure 21: Exponential growth curve of Human population

Thomas was the first who suggested that the size of population increases in a geometric series. For example, in an annual plant, suppose each species produces R offspring, the population size a number after generation 0 (zero) will be:

N₀ = N₀R₀

N₁ = N₁R₁

After time “t”;                                 N_t = N₀R

But when the generation time is large, the population size became an exponential function.

N_t = N₀ exp (r*t) = N₀ e^rt

Figure 22: Three outcomes of Population growth rate

1.     Population declines exponentially (r<0)

2.     Increases exponentially (r>0)

3.     No change (r=0)

This “r” is called “population growth rate”, sometimes called as “Intrinsic rate of increase” or Malthusian parameter or instantaneous rate of natural increase.

Assumptions of Exponential model

The exponential growth curve is based on the assumptions that the fecundity rate is constant i.e. continuous reproduction and the age structure of all individuals is identical and constant environment (no environmental resistance, unlimited resource availability).

However, even if the following assumptions are unmet (such as age, mortality and survival rate are different among organisms) due to large population size, average birth and death rate gives a reasonable precision.

In the exponential model of population dynamics, the fecundity rate and death rate can be differentiated by parameter “r” analysis.

r = (b-m)

dN/dt = (b-m)N = rN

b= birth rate, one organism producing number of offsprings per unit time in a population.

m= death rate, probable number of organisms dying in a population.

When the death rate (m) is subtracted from the birth rate (b) it gives the population growth rate (r).

b.     Logistic Model

In 1838, Pierre Verhulst developed logistic method and depicted the population density dependent population growth i.e. the increase in population growth rate is dependent on the density of population.

Figure 23: Population growth rate curve

r = r₀ (1- N/K)

In this model the resources are limited; hence, a maximum population size is set which can be supported by the environment.

According to this model, the rate of population growth at low density of population (N<<K) and at this point it is equal to “r₀” i.e. the rate of population growth when the intraspecific competition is no more.

The growth rate of population is directly proportional to N but become “0” when N=K. so as the “N” i.e. population size declines the growth rate also decreases. “K” is a parameter called as “carrying capacity” which is the upper limit of population growth. The amount of resources available to support certain number of organisms can be expressed in terms of carrying capacity. In simple terms it is the maximum number of organism an environment can support.

A population growth declines and become negative if exceeds K. the differential equation for population dynamics.

dN/dt= rN=r₀N (1- N/K)

Nt= N₀K / N₀ + (K+N₀) exp (-r₀t)

Figure 24: Three outcomes of population growth curves

Outcomes of model

1.     N₀ < K; increase in population

2.     N₀ > K; declining population

3.     N₀ = 0 = K; no change in population

The logistic model for population growth resulted into two equilibrium states, one stable and one unstable. The stable equilibrium is when N=K, any small fluctuations in population is buffered and the population reaches the equilibrium state.

The unstable equilibrium is when N=0, where even a small fluctuation will leads to population growth.

The density dependent factors, reproduction and competition in combination are the ecological process affecting logistic model.

Parameters

a.     “K”: carrying capacity is a intrinsic biological which regulate reproduction and population size. It is the maximum number of population size supported by the environment.

Intraspecific competition becomes intense when the resources become limited as the population size increases.

b.     “r₀” is a maximum population growth rate and is directly proportional to the fecundity rate and is a combined result of both mortality and fecundity. For example pest insects have high r₀ as they are rapidly reproducing while low r₀ is depicted in slow reproducing Elephants.

r₀ regulate both population decline (N>K) as well as population growth rate.

Logistic model does not work when the mortality is high and reproductively is slow.