Dynamics of the nonlocal KPP equation: Effects of a new free boundary condition

In this paper, we examine the effect of a new free boundary condition on the propagation dynamics of the nonlocal diffusion model considered in [9], which describes the spreading of a species with density $u(t,x)$ and population range $\left[g\right(t),h(t\left)\right]\subset R$. The existing free boundary condition can be written as$\{\begin{array}{c}{h}^{\prime }\left(t\right)=\mu \underset{g\left(t\right)}{\overset{h\left(t\right)}{\int }}u(t,x)W_J\left(h\right(t)-x)dx,\\ g^{\prime }\left(t\right)=-\mu \underset{g\left(t\right)}{\overset{h\left(t\right)}{\int }}u(t,x)W_J(x-g(t\left)\right)dx,\end{array}$ where $W_J\left(x\right)={\int }_x^{+\infty }J\left(y\right)dy$, and J is the kernel function of the nonlocal diffusion operator in the model. In the new free boundary condition, we replace $W_J$ by a general nonnegative locally Lipschitz continuous function W with $W\left(0\right)>0$, independent of J. This represents a very different assumption that the movement of the range boundary of the species is independent of its dispersal strategy, as in [20]. Our analysis shows that the dynamics of the model with the new free boundary condition resembles that of the old model except in the case that J is thin-tailed and ${\int }_0^{\infty }W\left(x\right)dx=\infty$, where new propagation phenomena appear.