War and Love

While models represented by coupled nonlinear equations are commonplace in the physical sciences (e.g. the BZ reaction), they are useful also in the biological sciences (e.g. epidemiology) and social sciences. An example of the last is a model of the defense expenditure of two competing nations. Richardson found that the expenditures $X$ and $Y$ of the two competing European groups of nations prior to World War I could be modelled well by the equations

$${ d X \over d t} = a_{1} Y ,$$ (5.7)

$${ d Y \over d t} = a_{2} X ,$$ (5.8)

which means that the rate of growth of the defense spending of each group is proportional to the actual spending by the competitor. This simplistic linear model represents an unbounded growth in the total budgets of the nations which in reality did not happen because of the eventual outbreak of war. A more complicated model is that due to Rapoport, in which two additional terms are added to each of Richardson's equations: One term (negative feedback) which tends to decrease defense spending (due perhaps to internal needs of nations to take care of other matters) and a nonlinear positive feedback term that tends to accelerate the arms spending. This model is investigated in the exercises. Others have suggested that the outbreak of war is signalled in such nonlinear models by the occurence of chaos in the solutions. See for example Refs.[12, 13]. An old saying goes: "All is fair in love and war". Indeed love sometimes seems like a battle. Thus it is not surprising that complicated human behaviour arising from feelings of love can also be modelled by time-dependent differential equations. Strogatz introduced a linear "love" model of two coupled differential equations, and more recently some authors have described a nonlinear model too. See the references if you are aroused!