# Deterministic Models of Growth

I Growth of individual organisms

## Von Bertalanffy’s Growth Model

Assumptions:

·        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

Thus,

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

Then,

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)

Or

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

Thus,

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

Flexibility is achieved by varying the parameter m

Models for Special cases

 Growth Model m Growth Equation Monomolecular 0 W(t) = Wmax (1- A* exp(-kt)) Von Bertalanffy 2/3 W(t) = Wmax (1 – A * exp(-kt))3 Gompertz 1 W(t) = Wmax exp(-A * exp(-kt)) Autocatalytic 2 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

Then,

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

Or

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