# II Population Growth Models

Divided
into three population models based on the method of organism reproduction

1) Populations with continuous reproduction
(Exponential growth in continuous time)

**N(****t + ****d****t) = N(t) + ****d****N**

** **

Colony size at time t+dt = Colony size at time t + Growth increment in time
(t, t+dt)

*Growth increment is directly proportional to the size
of the population*

Thus,

**d****N = r N ****d****t**

r = rate constant of growth ( reciprocal of time )

(relative growth rate –
specific growth rate – fractional growth rate)

r = dN /Ndt

dN / dt = rN

dN/dt = rN

By integrating both sides

ò dN/N = r ò dt

**Log N(t) = c + rt **

c = Log N(0)

If N(t) = N(0) at time t = 0

By taking exponentials of both sides

** **

**N(****t) = N(0) exp**^{rt}

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__Doubling Time (T__^{2})

The time taken to produce a doubling of the population
size

It is a property of rate constant of growth

** **

**T**^{2} = log 2 /r =
0.6931/r

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**Exponential decay**

Q(t) = quantity of material present at time t

dQ/dt = -kQ

k = rate constant of decay

** **

**Q(****t) = Q(0) exp **^{–kt}

__ __

__Half life __

The time taken to decay to half of the initial amount
present

** **

**T**_{1/2} = 0.6931 / k

2) Populations with discrete growth
(Exponential growth in discrete time)

The size of the population in one year is related to
its size in the previous year

Single annual breeding season and one year life span

Each female produces a fixed number R of females which
survive to breed

No. of females in year i+1 is related to the number of
females in year i

Ni+1 = RNi

N1 = RN0 , N2 = RN1 = R^{2}N0

** **

**Ni = R**^{i} N0

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## R = geometric growth rate or finite rate of increase

When R > 1 increase and when R < 1 decline

** **

**Log Ni = log N0
+ i log R**

Let R = exp ^{r}

r = log R

Thus,

**Ni = N0 exp**^{ir}

__ __

__Exponential
decline __

Log Ni = log N0
+ i log R

With slope R < 0

Then

** **

**Ni = N0 exp**^{i logR}

K = -log R

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### 3)
Populations with limited growth (Limited environmental growth factors)

The relative growth rate decreases with increasing
population size

Represented by the Monomolecular
model for limited growth in continuous time

R(t) is the number of the population at time t

Then

**dR/dt** **µ** - **R**

or

**dR/dt**** = -kR**

By integrating both sides

**R(****t) = A exp **^{–kt}

If C(t) is final population
number after long time t

Then

**R(****t) +C(t) = A **

** **

and

**C(****t) = A( 1 – exp **^{–kt})