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]

Wendi Wang. Population dispersal and disease spread. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 797-804. doi: 10.3934/dcdsb.2004.4.797
[2]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082
[3]

Yang Yang, Yun-Rui Yang, Xin-Jun Jiao. Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence. Electronic Research Archive, 2020, 28 (1) : 1-13. doi: 10.3934/era.2020001
[4]

Jingdong Wei, Jiangbo Zhou, Wenxia Chen, Zaili Zhen, Lixin Tian. Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2853-2886. doi: 10.3934/cpaa.2020125
[5]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973
[6]

Rongsong Liu, Jiangping Shuai, Jianhong Wu, Huaiping Zhu. Modeling spatial spread of west nile virus and impact of directional dispersal of birds. Mathematical Biosciences & Engineering, 2006, 3 (1) : 145-160. doi: 10.3934/mbe.2006.3.145
[7]

Holly Gaff. Preliminary analysis of an agent-based model for a tick-borne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463-473. doi: 10.3934/mbe.2011.8.463
[8]

Shangbing Ai. Global stability of equilibria in a tick-borne disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 567-572. doi: 10.3934/mbe.2007.4.567
[9]

Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 633-659. doi: 10.3934/mbe.2007.4.633
[10]

W. E. Fitzgibbon, J. J. Morgan. Analysis of a reaction diffusion model for a reservoir supported spread of infectious disease. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6239-6259. doi: 10.3934/dcdsb.2019137
[11]

Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043
[12]

Min Zhu, Xiaofei Guo, Zhigui Lin. The risk index for an SIR epidemic model and spatial spreading of the infectious disease. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1565-1583. doi: 10.3934/mbe.2017081
[13]

Zhi-An Wang. Wavefront of an angiogenesis model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2849-2860. doi: 10.3934/dcdsb.2012.17.2849
[14]

Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139
[15]

Fei-Ying Yang, Yan Li, Wan-Tong Li, Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1969-1993. doi: 10.3934/dcdsb.2013.18.1969
[16]

Wan-Tong Li, Wen-Bing Xu, Li Zhang. Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2483-2512. doi: 10.3934/dcds.2017107
[17]

Jinling Zhou, Yu Yang, Cheng-Hsiung Hsu. Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1469-1495. doi: 10.3934/dcdsb.2019236
[18]

Yu-Xia Hao, Wan-Tong Li, Fei-Ying Yang. Traveling waves in a nonlocal dispersal predator-prey model. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020340
[19]

Zengji Du, Zhaosheng Feng. Existence and asymptotic behaviors of traveling waves of a modified vector-disease model. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1899-1920. doi: 10.3934/cpaa.2018090
[20]

Zhaosheng Feng, Goong Chen. Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 763-780. doi: 10.3934/dcds.2009.24.763

2018 Impact Factor: 1.313

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences