Applicable nonlinear difference equations

Nonlinear phenomena, in some of its areas, have already been shown to be amenable to mathematization through differential equations. Their evolutionary characteristics have also been dealt with. The purpose of this appendix is to show, in brief, and once again, how a nonlinear phenomenon becomes not only mathematized through what are called nonlinear ‘difference equations’, but also revealing.

We choose again a situation from a biological system. It relates, in particular, to population biology. Let N represent some quantitative measure for a biological population. We take it that essential changes of N take place at certain discrete moments t1, t2 … which usually correspond to the birth of a new generation. Let N0 be the initial value of N at time t0; let the change of the variable N at a given time ti+1 (i = 0, 1, 2, …) be determined by its value at time ti and by externalities like food, habitation, etc. Let Ni be the value of N at time ti. We have then the difference equation of the type

where µ∈∧, the parameter space containing the information conditioned by externalities.

It is a matter of history that Malthus, way back in 1798, proposed a linear dependence of f on N, assuming the change of population to be proportional to the number of individuals:

where C(µ)>–1 is the difference between the birth rate B(µ) and the death rate D(µ).