Speed selection, front locking, and threshold-attainment laws in a nonlocal cooperative Fisher–KPP system

We investigated leading-edge spectral speed candidates, conditional front locking, and threshold-attainment laws in a two-component nonlocal cooperative Fisher–KPP system. The model couples two Fisher–KPP fronts through normalized nonlocal exchange kernels and captures how cooperative exchange can entrain components with different native propagation scales. By linearizing the traveling-wave system at the leading edge, we derived an explicit Perron–Frobenius spectral branch and defined a variational spectral candidate speed $ c^* $. We proved that this spectral candidate is well defined, satisfies quantitative row-sum bounds, and recovers the local cooperative spectral candidate as the interaction kernels concentrate at the origin. We emphasize that $ c^* $ is a leading-edge spectral candidate rather than a fully proved nonlinear spreading or minimal wave speed in the absence of an additional linear-determinacy theorem. Under a stated front-convergence hypothesis, we then derived conditional level-set locking and phase-locking consequences: If a common-speed front with speed $ c^* $ is dynamically realized, then both components have the same asymptotic level-set speed, and their spatial offsets remain asymptotically constant. This conditional locking structure further yields an affine asymptotic law for threshold-attainment times. Finite-difference simulations with efficient nonlocal convolution compare fitted finite-time front speeds with the spectral candidate and illustrate how positive coupling collapses the late-time speed gap. The computations also show that weak coupling produces a sharply anisotropic transition layer whose local structure is modulated by kernel width and kernel shape.