A narrow maturation kernel approaches a delayed birth term

Direct integration gives $\int {\rho }_{\epsilon }(a)da=1$ and symmetry about $a_{\ast }$ gives $\int a{\rho }_{\epsilon }(a)da=a_{\ast }$. Its second centered moment is $\int (a-a_{\ast }{)}^2{\rho }_{\epsilon }(a)da={\epsilon }^2\mathrm{/}6$.

Let $b$ be Lipschitz on $[0,T]$ and take $t\ge a_{\ast }+\epsilon$. The support of ${\rho }_{\epsilon }$ is contained in $[0,t]$, so the convolution in [F2] is $H_{\epsilon }(t)=R\int {\rho }_{\epsilon }(a)e^{-ma}b(t-a)da$. For each supported $a$, $\mathrm{\mid }{e}^{-ma}b(t-a)-e^{-m{a}_{\ast }}b(t-a_{\ast })\mathrm{\mid }\le (\mathrm{Lip}(b)+m\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{\infty }})\mathrm{\mid }a-a_{\ast }\mathrm{\mid }$. Since $\mathrm{\mid }a-a_{\ast }\mathrm{\mid }\le \epsilon$ and the kernel has unit mass, $\mathrm{\mid }{H}_{\epsilon }(t)-R{e}^{-m{a}_{\ast }}b(t-a_{\ast })\mathrm{\mid }\le R\epsilon (\mathrm{Lip}(b)+m\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{\infty }})$.

A finite-width fertility is not an exact delay: set $m=0$ and $b(s)=s^2$ on the relevant nonnegative interval. By step 1.1, $H_{\epsilon }(t)=R((t-a_{\ast }{)}^2+{\epsilon }^2\mathrm{/}6)$, while the point-delay term is $Rb(t-a_{\ast })=R(t-a_{\ast }{)}^2$; their difference is $R{\epsilon }^2\mathrm{/}6>0$.