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




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) exprt




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



T1/2 = 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





Ni = N0 expir














Exponential decline



Log Ni = log N0 + i log R



With slope R < 0





Ni = N0 expi 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









dR/dt µ - R




dR/dt = -kR


By integrating both sides



R(t) = A exp –kt



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




R(t) +C(t) = A




C(t) = A( 1 – exp –kt)