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Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response
Long-time dynamics of a diffusive epidemic model with free boundaries
School of Science and Technology, University of New England, Armidale, NSW 2351, Australia |
In this paper, we determine the long-time dynamical behaviour of a reaction-diffusion system with free boundaries, which models the spreading of an epidemic whose moving front is represented by the free boundaries. The system reduces to the epidemic model of Capasso and Maddalena [
References:
[1] |
I. Ahn, S. Baek and Z. Lin,
The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Model., 40 (2016), 7082-7101.
doi: 10.1016/j.apm.2016.02.038. |
[2] |
G. Bunting, Y. Du and K. Krakowski,
Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.
doi: 10.3934/nhm.2012.7.583. |
[3] |
J. Cao, W. T. Li and F. Yang,
Dynamics of a nonlocal SIS epidemic model with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 247-266.
doi: 10.3934/dcdsb.2017013. |
[4] |
V. Capasso and S. L. Paveri-fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Revue d'epidemiologie et de sante publique, 27 (1979), 121-132. Google Scholar |
[5] |
V. Capasso and L. Maddalena,
Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981/82), 173-184.
doi: 10.1007/BF00275212. |
[6] |
X. Chen and A. Friedman,
A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800.
doi: 10.1137/S0036141099351693. |
[7] |
X. Chen and A. Friedman,
A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986.
doi: 10.1137/S0036141002418388. |
[8] |
W. Ding, Y. Du and X. Liang,
Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1539-1573.
doi: 10.1016/j.anihpc.2019.01.005. |
[9] |
Y. Du, Z. M. Guo and R. Peng,
A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.
doi: 10.1016/j.jfa.2013.07.016. |
[10] |
Y. Du and Z. Lin,
Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
doi: 10.1137/090771089. |
[11] |
Y. Du and Z. Lin,
The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discret. Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.
doi: 10.3934/dcdsb.2014.19.3105. |
[12] |
Y. Du and B. Lou,
Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.
doi: 10.4171/JEMS/568. |
[13] |
Y. Du, M. Wang and M. Zhou,
Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.
doi: 10.1016/j.matpur.2016.06.005. |
[14] |
Y. Du and C.-H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries,, Calc. Var. Partial Differential Equations, 57 (2018), 36 pp.
doi: 10.1007/s00526-018-1339-5. |
[15] |
J. Fang and X-Q. Zhao,
Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dynam. Differential Equations, 21 (2009), 663-680.
doi: 10.1007/s10884-009-9152-7. |
[16] |
J. S. Guo and C. H. Wu,
On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.
doi: 10.1007/s10884-012-9267-0. |
[17] |
K. Hadeler,
Stefan problem, traveling fronts, and epidemic spread, Discrete Contin. Dyn. Syst. B., 21 (2016), 417-436.
doi: 10.3934/dcdsb.2016.21.417. |
[18] |
H. Huang and M. Wang,
The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. B., 20 (2015), 2039-2050.
doi: 10.3934/dcdsb.2015.20.2039. |
[19] |
Y. Kaneko, H. Matsuzawa and Y. Yamada,
Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity, SIAM J. Math. Anal., 52 (2020), 65-103.
doi: 10.1137/18M1209970. |
[20] |
K. Kim, Z. Lin and Q. Zhang,
An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992-2001.
doi: 10.1016/j.nonrwa.2013.02.003. |
[21] |
C. X. Lei, Z. Lin and Q. Y. Zhang,
The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166.
doi: 10.1016/j.jde.2014.03.015. |
[22] |
B. Li, H. F. Weinberger and M. A. Lewis,
Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosc., 196 (2005), 82-98.
doi: 10.1016/j.mbs.2005.03.008. |
[23] |
Z. Lin and H. Zhu,
Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409.
doi: 10.1007/s00285-017-1124-7. |
[24] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. |
[25] |
W. Merz and P. Rybka,
A free boundary problem describing reaction-diffusion problems in chemical vapor infiltration of pyrolytic carbon, J. Math. Anal. Appl., 292 (2004), 571-588.
doi: 10.1016/j.jmaa.2003.12.025. |
[26] |
R. Peng and X. Q. Zhao,
The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.
doi: 10.3934/dcds.2013.33.2007. |
[27] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[28] |
L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971. |
[29] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. |
[30] |
J.-B. Wang and X.-Q. Zhao,
Uniqueness and global stability of forced waves in a shifting environment, Proc. Amer. Math. Soc., 147 (2019), 1467-1481.
doi: 10.1090/proc/14235. |
[31] |
M. Wang,
Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 415-421.
doi: 10.3934/dcdsb.2018179. |
[32] |
Z. Wang, H. Nie and Y. Du,
Spreading speed for a West Nile virus model with free boundary, J. Math. Biol., 79 (2019), 433-466.
doi: 10.1007/s00285-019-01363-2. |
[33] |
J. Wu and X. Zou,
Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Diffrential Equations, 13 (2001), 651-687.
doi: 10.1023/A:1016690424892. |
[34] |
Y. Tao and M. Chen,
An elliptic-hyperbolic free boundary problem modelling cancer therapy, Nonlinearity, 19 (2006), 419-440.
doi: 10.1088/0951-7715/19/2/010. |
[35] |
M. Zhao, W. Li and W. J. Ni,
Spreading speed of a degenerate and cooperative epidemic model with free boundaries, Discrete Contin. Dyn. Syst. B, 25 (2020), 981-999.
doi: 10.3934/dcdsb.2019199. |
[36] |
X.-Q. Zhao and W. Wang, Fisher waves in an epidemic model,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004) 1117–1128.
doi: 10.3934/dcdsb.2004.4.1117. |
show all references
References:
[1] |
I. Ahn, S. Baek and Z. Lin,
The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Model., 40 (2016), 7082-7101.
doi: 10.1016/j.apm.2016.02.038. |
[2] |
G. Bunting, Y. Du and K. Krakowski,
Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.
doi: 10.3934/nhm.2012.7.583. |
[3] |
J. Cao, W. T. Li and F. Yang,
Dynamics of a nonlocal SIS epidemic model with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 247-266.
doi: 10.3934/dcdsb.2017013. |
[4] |
V. Capasso and S. L. Paveri-fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Revue d'epidemiologie et de sante publique, 27 (1979), 121-132. Google Scholar |
[5] |
V. Capasso and L. Maddalena,
Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981/82), 173-184.
doi: 10.1007/BF00275212. |
[6] |
X. Chen and A. Friedman,
A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800.
doi: 10.1137/S0036141099351693. |
[7] |
X. Chen and A. Friedman,
A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986.
doi: 10.1137/S0036141002418388. |
[8] |
W. Ding, Y. Du and X. Liang,
Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1539-1573.
doi: 10.1016/j.anihpc.2019.01.005. |
[9] |
Y. Du, Z. M. Guo and R. Peng,
A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.
doi: 10.1016/j.jfa.2013.07.016. |
[10] |
Y. Du and Z. Lin,
Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
doi: 10.1137/090771089. |
[11] |
Y. Du and Z. Lin,
The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discret. Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.
doi: 10.3934/dcdsb.2014.19.3105. |
[12] |
Y. Du and B. Lou,
Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.
doi: 10.4171/JEMS/568. |
[13] |
Y. Du, M. Wang and M. Zhou,
Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.
doi: 10.1016/j.matpur.2016.06.005. |
[14] |
Y. Du and C.-H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries,, Calc. Var. Partial Differential Equations, 57 (2018), 36 pp.
doi: 10.1007/s00526-018-1339-5. |
[15] |
J. Fang and X-Q. Zhao,
Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dynam. Differential Equations, 21 (2009), 663-680.
doi: 10.1007/s10884-009-9152-7. |
[16] |
J. S. Guo and C. H. Wu,
On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.
doi: 10.1007/s10884-012-9267-0. |
[17] |
K. Hadeler,
Stefan problem, traveling fronts, and epidemic spread, Discrete Contin. Dyn. Syst. B., 21 (2016), 417-436.
doi: 10.3934/dcdsb.2016.21.417. |
[18] |
H. Huang and M. Wang,
The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. B., 20 (2015), 2039-2050.
doi: 10.3934/dcdsb.2015.20.2039. |
[19] |
Y. Kaneko, H. Matsuzawa and Y. Yamada,
Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity, SIAM J. Math. Anal., 52 (2020), 65-103.
doi: 10.1137/18M1209970. |
[20] |
K. Kim, Z. Lin and Q. Zhang,
An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992-2001.
doi: 10.1016/j.nonrwa.2013.02.003. |
[21] |
C. X. Lei, Z. Lin and Q. Y. Zhang,
The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166.
doi: 10.1016/j.jde.2014.03.015. |
[22] |
B. Li, H. F. Weinberger and M. A. Lewis,
Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosc., 196 (2005), 82-98.
doi: 10.1016/j.mbs.2005.03.008. |
[23] |
Z. Lin and H. Zhu,
Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409.
doi: 10.1007/s00285-017-1124-7. |
[24] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. |
[25] |
W. Merz and P. Rybka,
A free boundary problem describing reaction-diffusion problems in chemical vapor infiltration of pyrolytic carbon, J. Math. Anal. Appl., 292 (2004), 571-588.
doi: 10.1016/j.jmaa.2003.12.025. |
[26] |
R. Peng and X. Q. Zhao,
The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.
doi: 10.3934/dcds.2013.33.2007. |
[27] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[28] |
L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971. |
[29] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. |
[30] |
J.-B. Wang and X.-Q. Zhao,
Uniqueness and global stability of forced waves in a shifting environment, Proc. Amer. Math. Soc., 147 (2019), 1467-1481.
doi: 10.1090/proc/14235. |
[31] |
M. Wang,
Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 415-421.
doi: 10.3934/dcdsb.2018179. |
[32] |
Z. Wang, H. Nie and Y. Du,
Spreading speed for a West Nile virus model with free boundary, J. Math. Biol., 79 (2019), 433-466.
doi: 10.1007/s00285-019-01363-2. |
[33] |
J. Wu and X. Zou,
Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Diffrential Equations, 13 (2001), 651-687.
doi: 10.1023/A:1016690424892. |
[34] |
Y. Tao and M. Chen,
An elliptic-hyperbolic free boundary problem modelling cancer therapy, Nonlinearity, 19 (2006), 419-440.
doi: 10.1088/0951-7715/19/2/010. |
[35] |
M. Zhao, W. Li and W. J. Ni,
Spreading speed of a degenerate and cooperative epidemic model with free boundaries, Discrete Contin. Dyn. Syst. B, 25 (2020), 981-999.
doi: 10.3934/dcdsb.2019199. |
[36] |
X.-Q. Zhao and W. Wang, Fisher waves in an epidemic model,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004) 1117–1128.
doi: 10.3934/dcdsb.2004.4.1117. |
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