Deterministic Models of Growth


I Growth of individual organisms


Von Bertalanffy’s Growth Model




·        Growth is expressed as the net result of material gained through anabolic processes less material lost through catabolism

·        The gain in weight is proportional to the surface area S(t) of the organism

·        A rate constant H is used to measure the average effect of the anabolic processes, thus

At small time interval dt


Weight gain = HS(t) dt


·        Loss in weight is proportional to the weight of the organism W(t)

·        A rate constant C measures the average effect of the catabolic processes




Weight loss = CW(t) dt


The resulting growth increment dW(t) is then


dW(t) = HS(t)dt – CW(t)dt


The rate of growth in weight can be described by the differential equation



dW/dt = HS – CW




·        The relationship between surface area and weight was assumed based on organisms that maintain the same shape while growing

·        The surface area and the weight and volume and weight are related to the square and cube of linear dimension

·        Example: a sphere of radius r has surface area 4pr2 and the volume 4pr3/3

·        The relationship in general between surface area and weight to length is:



S = a2 L2


W = a3 L3


a2 and a3 are model parameters





d(a3L3)/dt = H a2 L2 – Ca3 L3



Differentiating L3 with respect to time


dL3/dt = 3L2 dL/dt


By dividing both sides by 3 a3 L2


dL/dt = Ha2/3a3 – CL/3



When dL/dt = 0    that is when growth has reached the maximum


L max = Ha2/Ca3





dL/dt = k ( L max – L )


k = C/3







By integrating


ò dL / L max – L  = ò k dt


-         log ( L max – L ) = kt + c


at t = 0    L = L (0)


-         log ( L max – L ) = kt – log ( L max – L (0))


by taking exponentials of both sides


L = L max – [ L max – L (0)] exp (-kt)





L(t) = L max ( 1 – A exp(-kt))



A = 1 – (L(0) / L max )



Since W = a3L3



W(t) = W max ( 1 – A exp(-kt))3













Richards’ Growth Models


·              A flexible empirical model for describing different patterns of organic growth

·              Rate of growth in weight is given by the differential equation



  dW/dt = kW/ (1-m) [ (Wmax /W)1-m   -1 ]



Wmax = maximum attainable weight


K = rate constant





W(t) = Wmax ( 1- A* exp(-kt)) 1/(1-m)



Flexibility is achieved by varying the parameter m












Models for Special cases


Growth Model


Growth Equation




W(t) = Wmax (1- A* exp(-kt))

Von Bertalanffy



W(t) = Wmax (1 – A * exp(-kt))3




W(t) = Wmax exp(-A * exp(-kt))




W(t) = Wmax/(1+A* exp(-kt))





Allometric Growth


·              Allometry is the study of the relative size of different parts of an organism and how the parts grow in relation to each other

·              Huxley proposed a model in which the ratio of the relative growth rates of the two components remain constant







If the two components are named Y and X




1/Y dy/dt = b/X dx/dt


or d log Y/dt  = b * d log X/dt


b is the ratio of the relative growth rates of the variables Y and X


Integrating both sides



log Y = log A + b * log X





Y = A * Xb



·              The size of one component is proportional to the size of the other component raised to some power b.

·              B is a constant with no units of measurements

·              A measures the size of Y when X is of unit size and is therefore affected by the units of measurement