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 + dt) = N(t) + dN

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,

dN = r N dt

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

Doubling Time (T²)

The time taken to produce a doubling of the population size

It is a property of rate constant of growth

T² = log 2 /r = 0.6931/r

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 = R²N0

Ni = Rⁱ N0

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

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)