Population Growth Models

By Michal Šára

Here I shall show calibration of two stochastic growth models by means of stochastic differential equations (SDE) and apply them on two countries: Belgium,Italy.

Death-Birth Process

Death-Birth process \( \{N(t);t\geq 0\} \) is driven by following SDE: \[ \mathrm{d}N(t)=(b-d)N(t)\mathrm{d}t+\sqrt{(b+d)N(t)}\mathrm{d}W(t), \] where \( b \),\( d \) are unknown parameters.It is also possible for \( b \),\( d \) to be functions of time \( t \) or even stochastic processes.

Continuous Calibration

Calibration of the model is made continuously for every 20 observations (always moving forward one observation at a time) and 21st value is predicted (based on this calibrated model).

Belgium

Compared to the real population size (in the thousands) in 2009, which was \( 10 414 \) predicted population size for year 2009 is:

## [1] 10427

Italy

Compared to the real population size (in the thousands) in 2009, which was \( 58 126 \) predicted population size for year 2009 is:

## [1] 58218

Calibration on real data

Belgium

Mean from 1000 trajectories at terminal time 2009 gives us prediction:

## [1] 9891

Italy

Mean from 1000 trajectories at terminal time 2009 gives us prediction:

## [1] 65766

Stochastic Verhulst Model

Stochastic Verhulst Model has following SDE: \[ \mathrm{d}N(t)=rN(t)\left(1-\frac{N(t)}{K}\right)\mathrm{d}t+\sigma N(t)\mathrm{d}W(t), \] where \( \sigma,\,K,\,r \) are unknown parameters.

Continuous Calibration

Belgium

Compared to the real population size (in the thousands) in 2009, which was \( 10414 \) predicted population size for year 2009 is:

## [1] 10430

Italy

Compared to the real population size (in the thousands) in 2009, which was \( 58 126 \) predicted population size for year 2009 is:

## [1] 58215

Calibration on real data (Growth rate)

Belgium

Mean from 1000 trajectories at terminal time 2009 gives us prediction:

## [1] 10432

Italy

Mean from 1000 trajectories at terminal time 2009 gives us prediction:

## [1] 58088