American Institute of Mathematical Sciences

Advanced
August  2018, 15(4): 1033-1054. doi: 10.3934/mbe.2018046

Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model

Stephen A. Gourley 1, , Xiulan Lai 2, , Junping Shi 3, , Wendi Wang 4, , Yanyu Xiao 5, and  Xingfu Zou 6,,

1. 

Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK
2. 

Institute for Mathematical Sciences, Renmin University of China, Beijing, China
3. 

Department of Mathematics, College of William and Mary, Williamsburg, VA, USA
4. 

Key Laboratory of Eco-environments in Three Gorges Reservoir Region, and School of Mathematics and Statistics, Southwest University, Chongqing, China
5. 

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA
6. 

Department of Applied Mathematics, University of Western Ontario, London, ON, Canada

* Corresponding author: Xingfu Zou

Received  November 15, 2017 Accepted  November 15, 2017 Published  March 2018

Lyme disease is transmitted via blacklegged ticks, the spatial spread of which is believed to be primarily via transport on white-tailed deer. In this paper, we develop a mathematical model to describe the spatial spread of blacklegged ticks due to deer dispersal. The model turns out to be a system of differential equations with a spatially non-local term accounting for the phenomenon that a questing female adult tick that attaches to a deer at one location may later drop to the ground, fully fed, at another location having been transported by the deer. We first justify the well-posedness of the model and analyze the stability of its steady states. We then explore the existence of traveling wave fronts connecting the extinction equilibrium with the positive equilibrium for the system. We derive an algebraic equation that determines a critical value $c^*$ which is at least a lower bound for the wave speed in the sense that, if $c < c^*$, there is no traveling wave front of speed $c$ connecting the extinction steady state to the positive steady state. Numerical simulations of the wave equations suggest that $c^*$ is the minimum wave speed. We also carry out some numerical simulations for the original spatial model system and the results seem to confirm that the actual spread rate of the tick population coincides with $c^*$. We also explore the dependence of $c^*$ on the dispersion rate of the white tailed deer, by which one may evaluate the role of the deer's dispersion in the geographical spread of the ticks.

Keywords: Lyme disease, black-legged tick, white-tailed deer, spatial model, delay, dispersal, traveling wavefront, spread rate.
Mathematics Subject Classification: Primary: 92D20; Secondary: 34K20.
Citation: Stephen A. Gourley, Xiulan Lai, Junping Shi, Wendi Wang, Yanyu Xiao, Xingfu Zou. Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 1033-1054. doi: 10.3934/mbe.2018046
References:
[1]

R. M. Bacon, K. J. Kugeler, K. S. Griffith and P. S. Mead, Lyme disease -United States, 2003-2005, Journal of the American Medical Association, 298 (2007), 278-279.   Google Scholar

[2]

A. G. Barbour and D. Fish, The biological and social phenomenon of Lyme disease, Science, 260 (1993), 1610-1616.  doi: 10.1126/science.8503006.  Google Scholar

[3]

R. J. Brinkerhoff, C. M. Folsom-O'Keefe, K. Tsao and M. A. Diuk-Wasser, Do birds affect Lyme disease risk? Range expansion of the vector-borne pathogen Borrelia burgdorferi, Front. Ecol. Environ, 9 (2011), 103-110.  doi: 10.1890/090062.  Google Scholar

[4]

S. G. Caraco, S. Glavanakov, G. Chen, J. E. Flaherty, T. K. Ohsumi and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: a model for Lyme disease, Am. Nat., 160 (2002), 348-359.   Google Scholar

[5]

M. R. Cortinas and U. Kitron, County-level surveillance of white-tailed deer infestation by Ixodes scapularis and Dermacentor albipictus (Acari: Ixodidae) along the Illinois River, J. Med. Entomol., 43 (2006), 810-819.   Google Scholar

[6]

D. T. Dennis, T. S. Nekomoto, J. C. Victor, W. S. Paul and J. Piesman, Reported distribution of Ixodes scapularis and Ixodes pacificus (Acari: Ixodidae) in the United States, J. Med. Entomol., 35 (1998), 629-638.  doi: 10.1093/jmedent/35.5.629.  Google Scholar

[7]

G. Fan, H. R. Thieme and H. Zhu, Delay differential systems for tick population dynamics, J. Math. Biol., 71 (2015), 1017-1048.  doi: 10.1007/s00285-014-0845-0.  Google Scholar

[8]

S. A. Gourley and S. Ruan, A delay equation model for oviposition habitat selection by mosquitoes, J. Math. Biol., 65 (2012), 1125-1148.  doi: 10.1007/s00285-011-0491-8.  Google Scholar

[9]

B. H. Hahn, C. S. Jarnevich, A. J. Monaghan and R. J. Eisen, Modeling the Geographic Distribution of Ixodes scapularis and Ixodes pacificus (Acari: Ixodidae) in the Contiguous United States, Journal of Medical Entomology, 53 (2016), 1176-1191.   Google Scholar

[10]

S. Hamer, G. Hickling, E. Walker and J. I. Tsao, Invasion of the Lyme disease vector Ixodes scapularis: Implications for Borrelia burgdorferi endemicity, EcoHealth, 7 (2010), 47-63.  doi: 10.1007/s10393-010-0287-0.  Google Scholar

[11]

X. Lai and X. Zou, Spreading speed and minimal traveling wave speed in a spatially nonlocal model for the population of blacklegged tick Ixodes scapularis, in preparation. Google Scholar

[12]

J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079.  doi: 10.1007/s11538-009-9457-z.  Google Scholar

[13]

D. Liang, J. W.-H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numeric computations, Diff. Eqns. Dynam. Syst., 11 (2003), 117-139.   Google Scholar

[14]

K. Liu, Y. Lou and J. Wu, Analysis of an age structured model for tick populations subject to seasonal effects, J. Diff. Eqns., 263 (2017), 2078-2112.  doi: 10.1016/j.jde.2017.03.038.  Google Scholar

[15]

N. K. Madhav, J. S. Brownstein, J. I. Tsao and D. Fish, A dispersal model for the range expansion of blacklegged tick (Acari: Ixodidae), J. Med. Entomol., 41 (2004), 842-852.  doi: 10.1603/0022-2585-41.5.842.  Google Scholar

[16]

M. G. Neubert and I. M. Parker, Projecting rates of spread for invasive species, Risk Analysis, 24 (2004), 817-831.  doi: 10.1111/j.0272-4332.2004.00481.x.  Google Scholar

[17]

N. H. Ogden, M. Bigras-Poulin, C. J. O'Callaghan, I. K. Barker, L. R. Lindsay, A. Maarouf, K. E. Smoyer-Tomic, D. Waltner-Toews and D. Charron, A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis, Int J. Parasitol., 35 (2005), 375-389.  doi: 10.1016/j.ijpara.2004.12.013.  Google Scholar

[18]

N. H. Ogden, L. R. Lindsay, K. Hanincova, I. K. Barker, M. Bigras-Poulin, D. F. Charron, A. Heagy, C. M. Francis, C. J. O'Callaghan, I. Schwartz and R. A. Thompson, Role of migratory birds in introduction and range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi and Anaplasma phagocytophilum in Canada, Applied and Environmental Microbiology, 74 (2008), 1780-1790.   Google Scholar

[19]

J. W.-H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure —I. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.  Google Scholar

[20]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.  Google Scholar

[21]

J. Van Buskirk and R. S. Ostfeld, Controlling Lyme disease by modifying the density and species composition of tick hosts, Ecological Applications, 5 (1995), 1133-1140.  doi: 10.2307/2269360.  Google Scholar

[22]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar

[23]

X. Wu, G. Röst and X. Zou, Impact of spring bird migration on the range expansion of Ixodes scapularis tick population, Bull. Math. Biol., 78 (2016), 138-168.  doi: 10.1007/s11538-015-0133-1.  Google Scholar

show all references

References:
[1]

R. M. Bacon, K. J. Kugeler, K. S. Griffith and P. S. Mead, Lyme disease -United States, 2003-2005, Journal of the American Medical Association, 298 (2007), 278-279.   Google Scholar

[2]

A. G. Barbour and D. Fish, The biological and social phenomenon of Lyme disease, Science, 260 (1993), 1610-1616.  doi: 10.1126/science.8503006.  Google Scholar

[3]

R. J. Brinkerhoff, C. M. Folsom-O'Keefe, K. Tsao and M. A. Diuk-Wasser, Do birds affect Lyme disease risk? Range expansion of the vector-borne pathogen Borrelia burgdorferi, Front. Ecol. Environ, 9 (2011), 103-110.  doi: 10.1890/090062.  Google Scholar

[4]

S. G. Caraco, S. Glavanakov, G. Chen, J. E. Flaherty, T. K. Ohsumi and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: a model for Lyme disease, Am. Nat., 160 (2002), 348-359.   Google Scholar

[5]

M. R. Cortinas and U. Kitron, County-level surveillance of white-tailed deer infestation by Ixodes scapularis and Dermacentor albipictus (Acari: Ixodidae) along the Illinois River, J. Med. Entomol., 43 (2006), 810-819.   Google Scholar

[6]

D. T. Dennis, T. S. Nekomoto, J. C. Victor, W. S. Paul and J. Piesman, Reported distribution of Ixodes scapularis and Ixodes pacificus (Acari: Ixodidae) in the United States, J. Med. Entomol., 35 (1998), 629-638.  doi: 10.1093/jmedent/35.5.629.  Google Scholar

[7]

G. Fan, H. R. Thieme and H. Zhu, Delay differential systems for tick population dynamics, J. Math. Biol., 71 (2015), 1017-1048.  doi: 10.1007/s00285-014-0845-0.  Google Scholar

[8]

S. A. Gourley and S. Ruan, A delay equation model for oviposition habitat selection by mosquitoes, J. Math. Biol., 65 (2012), 1125-1148.  doi: 10.1007/s00285-011-0491-8.  Google Scholar

[9]

B. H. Hahn, C. S. Jarnevich, A. J. Monaghan and R. J. Eisen, Modeling the Geographic Distribution of Ixodes scapularis and Ixodes pacificus (Acari: Ixodidae) in the Contiguous United States, Journal of Medical Entomology, 53 (2016), 1176-1191.   Google Scholar

[10]

S. Hamer, G. Hickling, E. Walker and J. I. Tsao, Invasion of the Lyme disease vector Ixodes scapularis: Implications for Borrelia burgdorferi endemicity, EcoHealth, 7 (2010), 47-63.  doi: 10.1007/s10393-010-0287-0.  Google Scholar

[11]

X. Lai and X. Zou, Spreading speed and minimal traveling wave speed in a spatially nonlocal model for the population of blacklegged tick Ixodes scapularis, in preparation. Google Scholar

[12]

J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079.  doi: 10.1007/s11538-009-9457-z.  Google Scholar

[13]

D. Liang, J. W.-H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numeric computations, Diff. Eqns. Dynam. Syst., 11 (2003), 117-139.   Google Scholar

[14]

K. Liu, Y. Lou and J. Wu, Analysis of an age structured model for tick populations subject to seasonal effects, J. Diff. Eqns., 263 (2017), 2078-2112.  doi: 10.1016/j.jde.2017.03.038.  Google Scholar

[15]

N. K. Madhav, J. S. Brownstein, J. I. Tsao and D. Fish, A dispersal model for the range expansion of blacklegged tick (Acari: Ixodidae), J. Med. Entomol., 41 (2004), 842-852.  doi: 10.1603/0022-2585-41.5.842.  Google Scholar

[16]

M. G. Neubert and I. M. Parker, Projecting rates of spread for invasive species, Risk Analysis, 24 (2004), 817-831.  doi: 10.1111/j.0272-4332.2004.00481.x.  Google Scholar

[17]

N. H. Ogden, M. Bigras-Poulin, C. J. O'Callaghan, I. K. Barker, L. R. Lindsay, A. Maarouf, K. E. Smoyer-Tomic, D. Waltner-Toews and D. Charron, A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis, Int J. Parasitol., 35 (2005), 375-389.  doi: 10.1016/j.ijpara.2004.12.013.  Google Scholar

[18]

N. H. Ogden, L. R. Lindsay, K. Hanincova, I. K. Barker, M. Bigras-Poulin, D. F. Charron, A. Heagy, C. M. Francis, C. J. O'Callaghan, I. Schwartz and R. A. Thompson, Role of migratory birds in introduction and range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi and Anaplasma phagocytophilum in Canada, Applied and Environmental Microbiology, 74 (2008), 1780-1790.   Google Scholar

[19]

J. W.-H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure —I. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.  Google Scholar

[20]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.  Google Scholar

[21]

J. Van Buskirk and R. S. Ostfeld, Controlling Lyme disease by modifying the density and species composition of tick hosts, Ecological Applications, 5 (1995), 1133-1140.  doi: 10.2307/2269360.  Google Scholar

[22]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar

[23]

X. Wu, G. Röst and X. Zou, Impact of spring bird migration on the range expansion of Ixodes scapularis tick population, Bull. Math. Biol., 78 (2016), 138-168.  doi: 10.1007/s11538-015-0133-1.  Google Scholar

Figure 1.  The life-stage components of the model: questing larvae ($L$) find a host, feed and moult into questing nymphs ($N$), which then find a new host, feed and moult into questing adults ($A_q$). Adult females that find a deer host ($A_f$) feed, drop to the forest floor, lay $2000$ eggs and then die. Hatching eggs create the next generation of questing larvae. The $r$ parameters are the per-capita transition rates between each compartment
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  $H_1(\lambda, c)$ and $H_2(\lambda, c)$ for different $c$. (a) $c = 0.55$; (b) $c = c^* = 0.6176844021$, ($\lambda = 0.7081234538$); (c) $c = 0.7$. Here, the model parameters are taken as $ b = 3000$, $r_1 = 0.13$, $r_2 = 0.13$, $r_3 = 0.03$, $r_4 = 0.03$, $d_1 = 0.3$, $d_2 = 0.3$, $d_3 = 0.1$, $d_4 = 0.1$, $\tau_1 = 20$, $\tau_2 = 10$, $N_{cap} = 5000$, $h = 100$ and $D = 1$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Dependence of $c^*$ on $b$ and $D$ respectively: (a) with $D = 1$; (b) with $b = 3000$. Other parameters are taken as: $r_1 = 0.13$, $r_2 = 0.13$, $r_3 = 0.03$, $r_4 = 0.03$, $d_1 = 0.3$, $d_2 = 0.3$, $d_3 = 0.1$, $d_4 = 0.1$, $\tau_1 = 20$, $\tau_2 = 10$, $N_{cap} = 5000$ and $h = 100$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  There is no biologically relevant traveling wave front solution with speed $c = 0.1<c^* = 0.24$: $\phi_1$ may take negative values
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  There is a non-negative traveling wave front solution with speed $c = 0.4>c^* = 0.24$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  (a): time evolution of $L(x, t)$; (b): time evolution of $N(x, t)$; (c): contours of (a) with region where $L(x, t)>0.1$ shown in grey; (d): contours of (b) with region where $N(x, t)>0.1$ shown in grey
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  (a): time evolution of $A_q(x, t)$; (b): time evolution of $A_f(x, t)$; (c): contours of (a) with region where $A_q(x, t)>0.1$ shown in grey; (d): contours of (b) with region where $A_f(x, t)>0.1$ shown in grey
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Explanation of parameters
Parameters Meaning Value
$b$ Birth rate of tick $ 3000 $
$1/r_1$ average time that a questing larvae needs to feed and moult $1/0.13$
$1/r_2$ average time that a questing nymph needs to feed and moult $1/0.13$
$1/r_3$ average time that a questing adult needs to successfully attach to a deer $1/0.03$
$r_4$ Proportion of fed adults that can lay eggs 0.03
$d_1$ per-capita death rate of larvae 0.3
$d_2$ per-capita death rate of nymphs 0.3
$d_3$ per-capita death rate of questing adults 0.1
$d_4$ per-capita death rate of fed adults 0.1
$\tau_1$ average time between last blood feeding and hatch of laid eggs 20 days
$\tau_2$ average time tick is attached to a deer $10$ days
Table Options

Download as excel

Parameters Meaning Value
$b$ Birth rate of tick $ 3000 $
$1/r_1$ average time that a questing larvae needs to feed and moult $1/0.13$
$1/r_2$ average time that a questing nymph needs to feed and moult $1/0.13$
$1/r_3$ average time that a questing adult needs to successfully attach to a deer $1/0.03$
$r_4$ Proportion of fed adults that can lay eggs 0.03
$d_1$ per-capita death rate of larvae 0.3
$d_2$ per-capita death rate of nymphs 0.3
$d_3$ per-capita death rate of questing adults 0.1
$d_4$ per-capita death rate of fed adults 0.1
$\tau_1$ average time between last blood feeding and hatch of laid eggs 20 days
$\tau_2$ average time tick is attached to a deer $10$ days
[1]

Mahir Demir, Suzanne Lenhart. A spatial food chain model for the Black Sea Anchovy, and its optimal fishery. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 155-171. doi: 10.3934/dcdsb.2020373
[2]

Yu Jin, Xiang-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020362
[3]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[4]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032
[5]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159
[6]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017
[7]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
[8]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042
[9]

Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263
[10]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[11]

Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261
[12]

Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021005
[13]

Kengo Nakai, Yoshitaka Saiki. Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1079-1092. doi: 10.3934/dcdss.2020352
[14]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304
[15]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116
[16]

Yong-Jung Kim, Hyowon Seo, Changwook Yoon. Asymmetric dispersal and evolutional selection in two-patch system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3571-3593. doi: 10.3934/dcds.2020043
[17]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004
[18]

Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175
[19]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106
[20]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362

2018 Impact Factor: 1.313

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences