Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

Figure 1.  Illustration of $ \chi $ in cases (I) (left) and (II) (right)

Figure 2.  Numerically computed spreading speed $ c_u^* $ (pink circles) as a function of $ c_{het} $ for Case (I) (left) and Case (II) (right). The purple plain line is the theoretical spreading speed provided by Theorem 2.1 and Theorem 2.3. In both cases parameters are fixed with $ \alpha = 1 $, $ d_+ = 1 $ and $ d_- = 1/4 $, such that the corresponding linear speeds are $ c_+ = 2 $ and $ c_- = 1 $. The function $ \chi $ was set to $ \chi(x) = \frac{d_+ e^{-\lambda x} + d_-}{1+e^{-\lambda x}} $ in Case (I) and to $ \chi(x) = \frac{d_- e^{-\lambda x} + d_+}{1+e^{-\lambda x}} $ in Case (II) with $ \lambda = 2 $. Numerical simulations were performed by discretizing equation (1) via finite differences in space and a semi-implicit scheme in time. Typical discretization step sizes were set to $ \delta t = 0.02 $ in time and $ \delta x = 0.02 $ in  

Figure 3.  Numerically computed spreading speed $ c_u^* $ (pink circles) as a function of $ c_{het} $ for Case (II) in the degenerate case where $ d_- = 0 $. The purple plain line is the curve $ c_{het}\mapsto \frac{4 d_+ \alpha}{c_{het}} $ obtained by formally taking the limit $ d_- = 0 $ in Theorem 2.3. Other parameters are fixed with $ \alpha = 1 $ and $ d_+ = 1 $, such that the corresponding linear speed is $ c_+ = 2 $. The function $ \chi $ was set to $ \chi(x) = \frac{d_+}{1+e^{-\lambda x}} $ with $ \lambda = 2 $

Figure 4.  Illustration of the building block of the general sub-solution (10) (before its scaling by $ \epsilon $) which is composed of two parts $ \rho \Psi_+ $ (pink curve) and $ \Psi_- $ (blue curve) in the moving frame $ z = x-ct $. It is of class $ \mathscr{C}^2 $ and compactly supported on $ \left[-\frac{\pi}{2\omega}-z_+^*,\frac{\pi}{2\beta}-z_-^*\right] $

Figure 5.  Sketch of the super-solution $ u_\tau(t,x) $ given in Lemma 5.1 with $ C = 1 $ which is composed of three parts: it is constant and equal to $ 1 $ for $ x\leq ct-\tau $ (gray curve), and then it is the concatenation of two exponentials (blue and pink curves) for $ x\geq ct-\tau $ which are glued at $ x = c_{het}t-\tau $. Note that the factor $ \rho(t) $ is to ensure continuity between the two exponentials

Figure 6.  Sketch of the sub-solution given in Proposition 2 which is the concatenation of the sub-solution $ \underline{u}_{1,\tau}(x-ct) $ given in Lemma 5.2 (composed of the difference of two exponentials) and the function $ {\varphi}_{\lambda_\star-\epsilon} $ which solves $ \mathcal{L} \varphi = (\lambda_\star-\epsilon)\varphi $ with prescribed asymptotic behavior at $ -\infty $. Note that the factor $ \rho(t) $ is to ensure continuity at the matching point $ x = c_{het}t-\tau/2 $