# American Institute of Mathematical Sciences

September  2022, 15(9): 2467-2496. doi: 10.3934/dcdss.2021146

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

 1 Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, UPS IMT, F-31062 Toulouse Cedex 9, France 2 IECL; UMR 7502, University of Lorraine, B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex, France 3 Department of Mathematical Sciences, Center for Mathematics and Artificial Intelligence (CMAI), George Mason University, Fairfax, VA, USA

*Corresponding author: Grégory Faye

Received  March 2021 Revised  September 2021 Published  September 2022 Early access  November 2021

Fund Project: GF acknowledges support from an ANITI (Artificial and Natural Intelligence Toulouse Institute) Research Chair and from Labex CIMI under grant agreement ANR-11-LABX-0040. GF and TG acknowledge support from the ANR project Indyana under grant agreement ANR-21-CE40-0008-01. The research of MH was partially supported by the National Science Foundation (DMS-2007759)

We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotonicity of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems at $x = \pm \infty$. Most notably, when the profile of the diffusion coefficient is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the diffusion coefficient is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.

Citation: Grégory Faye, Thomas Giletti, Matt Holzer. Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity. Discrete and Continuous Dynamical Systems - S, 2022, 15 (9) : 2467-2496. doi: 10.3934/dcdss.2021146
##### References:
 [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5. [2] H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.  doi: 10.1007/s11538-008-9367-5. [3] H. Berestycki and J. Fang, Forced waves of the Fisher–KPP equation in a shifting environment, J. Differential Equations, 264 (2018), 2157-2183.  doi: 10.1016/j.jde.2017.10.016. [4] H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl., 186 (2007), 469-507.  doi: 10.1007/s10231-006-0015-0. [5] H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous equations, Memoirs of the American Mathematical Society. [6] H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations, J. Math. Phys., 53 (2012), 115619, 23pp. doi: 10.1063/1.4764932. [7] H. Berestycki and L. Rossi, On the principal eigenvalue of elliptic operators in $\mathbb{R}^n$ and applications, J. Eur. Math. Soc. (JEMS), 8 (2006), 195-215.  doi: 10.4171/JEMS/47. [8] H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed I-the case of the whole space, Discrete Contin. Dyn. Syst., 21 (2008), 41-67.  doi: 10.3934/dcds.2008.21.41. [9] H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math., 68 (2015), 1014-1065.  doi: 10.1002/cpa.21536. [10] J. Bouhours and T. Giletti, Spreading and vanishing for a monostable reaction–diffusion equation with forced speed, J. Dynam. Differential Equations, 31 (2019), 247-286.  doi: 10.1007/s10884-018-9643-5. [11] W. A. Coppel, Dichotomies in Stability Theory, Vol. 629. Springer-Verlag, Berlin-New York, 1978. [12] C. Cosner, Challenges in modeling biological invasions and population distributions in a changing climate, Ecological Complexity, 20 (2014), 258-263.  doi: 10.1016/j.ecocom.2014.05.007. [13] A. Ducrot, T. Giletti and H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differential Equations, 58 (2019), 137, 34pp. doi: 10.1007/s00526-019-1576-2. [14] G. Faye and M. Holzer, Bifurcation to locked fronts in two component reaction-diffusion systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 545-584.  doi: 10.1016/j.anihpc.2018.08.001. [15] R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x. [16] T. Gallay and C. Mascia, Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivity, Nonlinear Anal. Real World Appl., 63 (2022), 103387, 28pp. doi: 10.1016/j.nonrwa.2021.103387. [17] R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753. [18] L. Girardin and K.-Y. Lam, Invasion of open space by two competitors: Spreading properties of monostable two-species competition-diffusion systems, Proc. Lond. Math. Soc., 119 (2019), 1279-1335.  doi: 10.1112/plms.12270. [19] M. Holzer and A. Scheel, Accelerated fronts in a two-stage invasion process, SIAM J. Math. Anal., 46 (2014), 397-427.  doi: 10.1137/120887746. [20] T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Applied Mathematical Sciences, 185. Springer, New York, 2013. [21] A. N. Kolmogorov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application àun problème biologique, Bull. Univ. Moskow, Ser. Internat., Sec. A, 1 (1937), 1-25. [22] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a banach space, Uspehi Matem. Nauk (N. S.), 3 (1948), 3-95. [23] K.-Y. Lam and X. Yu, Asymptotic spreading of kpp reactive fronts in heterogeneous shifting environments, preprint, arXiv: 2101.06698, 2021. [24] M. Lewis and J. Murray, Modelling territoriality and wolf–deer interactions, Nature, 366 (1993), 738-740.  doi: 10.1038/366738a0. [25] B. Li, S. Bewick, J. Shang and W. F. Fagan, Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math., 74 (2014), 1397-1417.  doi: 10.1137/130938463. [26] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Phys. D, 145 (2000), 233-277.  doi: 10.1016/S0167-2789(00)00114-7. [27] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

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##### References:
 [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5. [2] H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.  doi: 10.1007/s11538-008-9367-5. [3] H. Berestycki and J. Fang, Forced waves of the Fisher–KPP equation in a shifting environment, J. Differential Equations, 264 (2018), 2157-2183.  doi: 10.1016/j.jde.2017.10.016. [4] H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl., 186 (2007), 469-507.  doi: 10.1007/s10231-006-0015-0. [5] H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous equations, Memoirs of the American Mathematical Society. [6] H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations, J. Math. Phys., 53 (2012), 115619, 23pp. doi: 10.1063/1.4764932. [7] H. Berestycki and L. Rossi, On the principal eigenvalue of elliptic operators in $\mathbb{R}^n$ and applications, J. Eur. Math. Soc. (JEMS), 8 (2006), 195-215.  doi: 10.4171/JEMS/47. [8] H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed I-the case of the whole space, Discrete Contin. Dyn. Syst., 21 (2008), 41-67.  doi: 10.3934/dcds.2008.21.41. [9] H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math., 68 (2015), 1014-1065.  doi: 10.1002/cpa.21536. [10] J. Bouhours and T. Giletti, Spreading and vanishing for a monostable reaction–diffusion equation with forced speed, J. Dynam. Differential Equations, 31 (2019), 247-286.  doi: 10.1007/s10884-018-9643-5. [11] W. A. Coppel, Dichotomies in Stability Theory, Vol. 629. Springer-Verlag, Berlin-New York, 1978. [12] C. Cosner, Challenges in modeling biological invasions and population distributions in a changing climate, Ecological Complexity, 20 (2014), 258-263.  doi: 10.1016/j.ecocom.2014.05.007. [13] A. Ducrot, T. Giletti and H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differential Equations, 58 (2019), 137, 34pp. doi: 10.1007/s00526-019-1576-2. [14] G. Faye and M. Holzer, Bifurcation to locked fronts in two component reaction-diffusion systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 545-584.  doi: 10.1016/j.anihpc.2018.08.001. [15] R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x. [16] T. Gallay and C. Mascia, Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivity, Nonlinear Anal. Real World Appl., 63 (2022), 103387, 28pp. doi: 10.1016/j.nonrwa.2021.103387. [17] R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753. [18] L. Girardin and K.-Y. Lam, Invasion of open space by two competitors: Spreading properties of monostable two-species competition-diffusion systems, Proc. Lond. Math. Soc., 119 (2019), 1279-1335.  doi: 10.1112/plms.12270. [19] M. Holzer and A. Scheel, Accelerated fronts in a two-stage invasion process, SIAM J. Math. Anal., 46 (2014), 397-427.  doi: 10.1137/120887746. [20] T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Applied Mathematical Sciences, 185. Springer, New York, 2013. [21] A. N. Kolmogorov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application àun problème biologique, Bull. Univ. Moskow, Ser. Internat., Sec. A, 1 (1937), 1-25. [22] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a banach space, Uspehi Matem. Nauk (N. S.), 3 (1948), 3-95. [23] K.-Y. Lam and X. Yu, Asymptotic spreading of kpp reactive fronts in heterogeneous shifting environments, preprint, arXiv: 2101.06698, 2021. [24] M. Lewis and J. Murray, Modelling territoriality and wolf–deer interactions, Nature, 366 (1993), 738-740.  doi: 10.1038/366738a0. [25] B. Li, S. Bewick, J. Shang and W. F. Fagan, Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math., 74 (2014), 1397-1417.  doi: 10.1137/130938463. [26] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Phys. D, 145 (2000), 233-277.  doi: 10.1016/S0167-2789(00)00114-7. [27] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.
Illustration of $\chi$ in cases (I) (left) and (II) (right)
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
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 $\lambda = 2$
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 $\left[-\frac{\pi}{2\omega}-z_+^*,\frac{\pi}{2\beta}-z_-^*\right]$
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
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 $x = c_{het}t-\tau/2$
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