Propagation from large localized populations without zero-state instability

Determine whether solutions of the KPP-bistable periodic patch model propagate when the initial data are sufficiently large and concentrated in a sufficiently large bistable patch, \(\int_0^{K_2} f_2(s)\,\mathrm{d}s>0\), and the trivial steady state is stable rather than unstable.

Background

Theorem 2.2 establishes local persistence for sufficiently large initial data occupying a sufficiently large bistable patch when the bistable reaction has positive integral, regardless of whether the zero steady state is stable or unstable. In the unstable case, the paper also proves propagation with positive spreading speed.

The unresolved issue concerns the stable-zero-state regime. The heterogeneous periodic interfaces and lack of translation invariance make it unclear whether a large population bump can cross patch boundaries and initiate propagation through the periodic environment.

References

For our patchy problem~eq-patch, under the hypotheses of Theorem~\ref{thmNOEXT}, but without the instability of the trivial solution $0$, whether the solutions exhibit propagation remains an open problem, one obstacle in the proof being due to the heterogeneous nature of~eq-patch and its non-invariance by translation: roughly speaking, it is not clear whether initial large bumps in a given patch can cross the interfaces and then propagate to the other patches.

eq:

{ut−d(x)uxx=f(x,u),t>0, x∈R∖S,u(t,x−)=u(t,x+),ux(t,x−)=σux(t,x+),t>0, x∈S1,u(t,x−)=u(t,x+),σux(t,x−)=ux(t,x+),t>0, x∈S2,\left\{ \begin{aligned} u_t - d(x) u_{xx} &= f(x,u), && t>0,\ x\in\mathbb{R}\setminus S, \\ u(t,x^-) &= u(t,x^+), \quad u_x(t,x^-) = \sigma u_x(t,x^+), && t>0,\ x\in S_1, \\ u(t,x^-) &= u(t,x^+), \quad \sigma u_x(t,x^-) = u_x(t,x^+), && t>0,\ x\in S_2, \end{aligned} \right.

patch:

d(x)={d1,x∈(nl−l1,nl),d2,x∈(nl,nl+l2),f(x,s)={f1(s),x∈(nl−l1,nl),f2(s),x∈(nl,nl+l2).d(x)= \begin{cases} d_1, & x \in (nl - l_1, nl), \\ d_2, & x \in (nl, nl + l_2), \end{cases} \qquad f(x,s)= \begin{cases} f_1(s), & x \in (nl - l_1, nl), \\ f_2(s), & x \in (nl, nl + l_2). \end{cases}

— Propagation phenomena in KPP-bistable periodic patchy environments  (2608.17474 - Griette et al., 18 Aug 2026) in Section 2, subsection “Discussion”