Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 6B (Mathematical Biology)

Integrating the population equation over the line and using vanishing boundary flux gives $\dot{M}=-\mu M$, where $M(t)=\int n(x,t)dx$. Since $M(0)=Q$, one has $M(t)=Q{e}^{-\mu t}$. On the other hand the similarity substitution $x=\lambda (t)\eta$ givesConsequentlyThe decay condition is on the nonlinear diffusive flux $n{n}_x$; compact support also satisfies it.

For the similarity solution, differentiation givesChoose ${\lambda }^2\dot{\lambda }=e^{-\mu t}$. The remaining equation is $-(\eta f{)}^{\prime }=(f{f}^{\prime }{)}^{\prime }$. Its integration constant is zero for the required decaying or compactly supported profile, so $f(f^{\prime }+\eta )=0$. Where $f>0$, this gives $f^{\prime }=-\eta$. For a nonnegative population the complete profile is thereforeNormalization gives $Q=2b^3/3$, hence $b=(3Q/2{)}^{1/3}$. Integrating the scale equation with zero initial scale yieldsThusIts support satisfiesThe profile is a weak solution at its moving edges, where $n{n}_x$ vanishes. As $t\downarrow 0$ its support shrinks to the origin and its mass tends to $Q$, so its initial distribution is $Q{\delta }_0$. The bounded limiting range assumes $\mu >0$; at zero death rate the scale instead grows as $(3t{)}^{1/3}$.