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Asymptotic speed of spread for a nonlocal evolutionary-epidemic system
1. | Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France |
2. | CNRS, IMB, UMR 5251, F-33400 Talence, France |
3. | Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France |
We investigate spreading properties of solutions for a spatially distributed system of equations modelling the evolutionary epidemiology of plant-pathogen interactions. In this work the mutation process is described using a non-local convolution operator in the phenotype space. Initially equipped with a localized amount of infection, we prove that spreading occurs with a definite spreading speed that coincides with the minimal speed of the travelling wave solutions discussed in [
References:
[1] |
L. Abi Rizk, J. -B. Burie and A. Ducrot, Travelling wave solutions for a non-local evolutionary-epidemic system, J. Differential Equations, 267 (2019), 1467-1509.
doi: 10.1016/j. jde. 2019.02.012. |
[2] |
O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73.
doi: 10.1016/0022-0396(79)90080-9. |
[3] |
R. Djidjou-Demasse, A. Ducrot and F. Fabre, Steady state concentration for a phenotypic structured problem modeling the evolutionary epidemiology of spore producing pathogens, Math. Models Methods Appl. Sci., 27 (2017), 385-426.
doi: 10.1142/S0218202517500051. |
[4] |
A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.
doi: 10.1016/j. jde. 2016.02.023. |
[5] |
A Ducrot, T. Giletti, J.-S. Guo and M. Shimojo,
Asymptotic spreading speeds for a predator-prey system with two predators and one prey, Nonlinearity, 34 (2021), 669-705.
doi: 10.1088/1361-6544/abd289. |
[6] |
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), Paper No. 137, 34pp.
doi: 10.1007/s00526-019-1576-2. |
[7] |
A. Ducrot, J. -S. Guo, G. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), Paper No. 146, 25pp.
doi: 10.1007/s00033-019-1188-x. |
[8] |
L. Girardin, Non-cooperative Fisher-KPP systems: Asymptotic behavior of traveling waves, Math. Models Methods Appl. Sci., 28 (2018), 1067-1104.
doi: 10.1142/S0218202518500288. |
[9] |
—————, Non-cooperative Fisher-KPP systems: Traveling waves and long-time behavior, Nonlinearity, 31 (2018), 108-164.
doi: 10.1088/1361-6544/aa8ca7. |
[10] |
G. L. Iacono, F. Van den Bosch and N. Paveley, The evolution of plant pathogens in response to host resistance: factors affecting the gain from deployment of qualitative and quantitative resistance, J. Theo. Biol., 304 (2012), 152-163.
doi: 10.1016/j. jtbi. 2012.03.033. |
[11] |
Q. Griette and G. Raoul, Existence and qualitative properties of travelling waves for an epidemiological model with mutations, J. Differential Equations, 260 (2016), 7115-7151.
doi: 10.1016/j. jde. 2016.01.022. |
[12] |
J. -S. Guo, A. A. L. Poh and M. Shimojo, The spreading speed of an SIR epidemic model with nonlocal dispersal, Asymptotic Analysis, 120 (2020), 163-174.
doi: 10.3233/ASY-191584. |
[13] |
J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.
doi: 10.1137/0520025. |
[14] |
W. -T. Li, W. -B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst., 37 (2017), 2483-2512.
doi: 10.3934/dcds. 2017107. |
[15] |
F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM J. Appl. Math., 65 (2005), 1305-1327.
doi: 10.1137/S0036139904440400. |
[16] |
P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.
doi: 10.1137/S0036141003439173. |
[17] |
P. Meyer-Nieberg, Banach Lattices, Universitext, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-3-642-76724-1. |
[18] |
A. Morris, L. Borger and E. Crooks, Individual variability in dispersal and invasion speed, Mathematics, 7 (2019), p. 795. |
[19] |
S. Pan, Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407 (2013), 230-236.
doi: 10.1016/j. jmaa. 2013.05.031. |
[20] |
L. Rimbaud, J. Papaïx, J. -F. Rey, L. G. Barrett and P. H. Thrall, Assessing the durability and efficiency of landscape-based strategies to deploy plant resistance to pathogens, PLOS Computational Biology, 14 (2018), 1-33. |
[21] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for life science and medicine, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007, 97-122. |
[22] |
H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York-Heidelberg, 1974. |
[23] |
H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201.
doi: 10.1016/S0025-5564(00)00018-3. |
[24] |
—————, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801.
doi: 10.1016/j. jde. 2011.01.007. |
[25] |
Z. -C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.
doi: 10.1098/rspa. 2009.0377. |
[26] |
C. Wu, The spreading speed for a predator-prey model with one predator and two preys, Appl. Math. Lett., 91 (2019), 9-14.
doi: 10.1016/j. aml. 2018.11.022. |
[27] |
G. -B. Zhang and X. -Q. Zhao, Propagation phenomena for a two species Lotka-Volterra strong competition system with nonlocal dispersal, Calc. Var., 59 (2019), Paper No. 10, 34 pp.
doi: 10.1007/s00526-019-1662-5. |
[28] |
M. Zhao, W. Li and Y. Du, The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries, Commun. Pure Appl. Anal., 19 (2020), 4599-4620.
doi: 10.3934/cpaa. 2020208. |
show all references
References:
[1] |
L. Abi Rizk, J. -B. Burie and A. Ducrot, Travelling wave solutions for a non-local evolutionary-epidemic system, J. Differential Equations, 267 (2019), 1467-1509.
doi: 10.1016/j. jde. 2019.02.012. |
[2] |
O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73.
doi: 10.1016/0022-0396(79)90080-9. |
[3] |
R. Djidjou-Demasse, A. Ducrot and F. Fabre, Steady state concentration for a phenotypic structured problem modeling the evolutionary epidemiology of spore producing pathogens, Math. Models Methods Appl. Sci., 27 (2017), 385-426.
doi: 10.1142/S0218202517500051. |
[4] |
A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.
doi: 10.1016/j. jde. 2016.02.023. |
[5] |
A Ducrot, T. Giletti, J.-S. Guo and M. Shimojo,
Asymptotic spreading speeds for a predator-prey system with two predators and one prey, Nonlinearity, 34 (2021), 669-705.
doi: 10.1088/1361-6544/abd289. |
[6] |
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), Paper No. 137, 34pp.
doi: 10.1007/s00526-019-1576-2. |
[7] |
A. Ducrot, J. -S. Guo, G. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), Paper No. 146, 25pp.
doi: 10.1007/s00033-019-1188-x. |
[8] |
L. Girardin, Non-cooperative Fisher-KPP systems: Asymptotic behavior of traveling waves, Math. Models Methods Appl. Sci., 28 (2018), 1067-1104.
doi: 10.1142/S0218202518500288. |
[9] |
—————, Non-cooperative Fisher-KPP systems: Traveling waves and long-time behavior, Nonlinearity, 31 (2018), 108-164.
doi: 10.1088/1361-6544/aa8ca7. |
[10] |
G. L. Iacono, F. Van den Bosch and N. Paveley, The evolution of plant pathogens in response to host resistance: factors affecting the gain from deployment of qualitative and quantitative resistance, J. Theo. Biol., 304 (2012), 152-163.
doi: 10.1016/j. jtbi. 2012.03.033. |
[11] |
Q. Griette and G. Raoul, Existence and qualitative properties of travelling waves for an epidemiological model with mutations, J. Differential Equations, 260 (2016), 7115-7151.
doi: 10.1016/j. jde. 2016.01.022. |
[12] |
J. -S. Guo, A. A. L. Poh and M. Shimojo, The spreading speed of an SIR epidemic model with nonlocal dispersal, Asymptotic Analysis, 120 (2020), 163-174.
doi: 10.3233/ASY-191584. |
[13] |
J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.
doi: 10.1137/0520025. |
[14] |
W. -T. Li, W. -B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst., 37 (2017), 2483-2512.
doi: 10.3934/dcds. 2017107. |
[15] |
F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM J. Appl. Math., 65 (2005), 1305-1327.
doi: 10.1137/S0036139904440400. |
[16] |
P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.
doi: 10.1137/S0036141003439173. |
[17] |
P. Meyer-Nieberg, Banach Lattices, Universitext, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-3-642-76724-1. |
[18] |
A. Morris, L. Borger and E. Crooks, Individual variability in dispersal and invasion speed, Mathematics, 7 (2019), p. 795. |
[19] |
S. Pan, Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407 (2013), 230-236.
doi: 10.1016/j. jmaa. 2013.05.031. |
[20] |
L. Rimbaud, J. Papaïx, J. -F. Rey, L. G. Barrett and P. H. Thrall, Assessing the durability and efficiency of landscape-based strategies to deploy plant resistance to pathogens, PLOS Computational Biology, 14 (2018), 1-33. |
[21] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for life science and medicine, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007, 97-122. |
[22] |
H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York-Heidelberg, 1974. |
[23] |
H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201.
doi: 10.1016/S0025-5564(00)00018-3. |
[24] |
—————, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801.
doi: 10.1016/j. jde. 2011.01.007. |
[25] |
Z. -C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.
doi: 10.1098/rspa. 2009.0377. |
[26] |
C. Wu, The spreading speed for a predator-prey model with one predator and two preys, Appl. Math. Lett., 91 (2019), 9-14.
doi: 10.1016/j. aml. 2018.11.022. |
[27] |
G. -B. Zhang and X. -Q. Zhao, Propagation phenomena for a two species Lotka-Volterra strong competition system with nonlocal dispersal, Calc. Var., 59 (2019), Paper No. 10, 34 pp.
doi: 10.1007/s00526-019-1662-5. |
[28] |
M. Zhao, W. Li and Y. Du, The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries, Commun. Pure Appl. Anal., 19 (2020), 4599-4620.
doi: 10.3934/cpaa. 2020208. |
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