doi: 10.3934/dcdsb.2020283

Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions

1. 

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

2. 

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

* Corresponding author: lam.184@math.ohio-state.edu

Received  March 2020 Revised  August 2020 Published  September 2020

Fund Project: The first author is supported by NSF grant DMS-1818769. The second and third authors are supported by NSF grant DMS-1853561

We consider the ecological and evolutionary dynamics of a reaction-diffusion-advection model for populations residing in a one-dimensional advective homogeneous environment, with emphasis on the effects of boundary conditions and domain size. We assume that there is a net loss of individuals at the downstream end with rate $ b \geq 0 $, while the no-flux condition is imposed on the upstream end. For the single species model, it is shown that the critical patch size is a decreasing function of the dispersal rate when $ b \leq 3/2 $; whereas it first decreases and then increases when $ b >3/2 $.

For the two-species competition model, we show that the infinite dispersal rate is evolutionarily stable for $ b < 3/2 $ and, when dispersal rates of both species are large, the population with larger dispersal rate always displaces the population with the smaller rate. For certain specific population loss rate $ b<3/2 $, it is also shown that there can be up to three evolutionarily stable strategies. For $ b>3/2 $, it is proved that the infinite random dispersal rate is not evolutionarily stable, and that, for some specific $ b>3/2 $, a finite dispersal rate is evolutionarily stable. Furthermore, for the intermediate domain size, this dispersal rate is optimal in the sense that the species adopting this rate is able to displace its competitor with a similar but different rate. Finally, nine qualitatively different pairwise invasibility plots are obtained by varying the parameter $ b $ and the domain size.

Citation: Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020283
References:
[1]

M. BallykL. DungD. A. Jones and H. L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1999), 573-596.  doi: 10.1137/S0036139997325345.  Google Scholar

[2]

H. BerestyckiO. DiekmannC. 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.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

R. S. CantrellC. Cosner and K.-Y. Lam, On resident-invader dynamics in infinite dimensional dynamical systems, J. Differential Equations, 263 (2017), 4565-4616.   Google Scholar

[5]

R. S. CantrellC. CosnerM. A. Lewis and Y. Lou, Evolution of dispersal in spatial population models with multiple timescales, J. Math. Biol., 80 (2020), 3-37.  doi: 10.1007/s00285-018-1302-2.  Google Scholar

[6]

R. S. CantrellC. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518.  doi: 10.1017/S0308210506000047.  Google Scholar

[7]

X. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst. A, 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.  Google Scholar

[8] F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes. The Adaptive Dynamics Approach and its Applications, Princeton University Press, Princeton, 2008.   Google Scholar
[9]

U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612.  doi: 10.1007/BF02409751.  Google Scholar

[10]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.  Google Scholar

[11]

S. A. H. GeritzE. KisdiG. Meszena and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evol. Ecol., 12 (1998), 35-57.   Google Scholar

[12]

M. GolubitskyW. HaoK.-Y. Lam and Y. Lou, Dimorphism by singularity theory in a model for river ecology, Bull. Math. Biol., 79 (2017), 1051-1069.  doi: 10.1007/s11538-017-0268-3.  Google Scholar

[13]

R. Hambrock and Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817.  doi: 10.1007/s11538-009-9425-7.  Google Scholar

[14]

W. Hao and C. Zheng, An adaptive homotopy method for computing bifurcations of nonlinear parametric systems, J. Sci. Comp., 82 (2020), 1-19.  doi: 10.1007/s10915-020-01160-w.  Google Scholar

[15]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theoretical Population Biology, 24 (1983), 244-251.   Google Scholar

[16]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.  Google Scholar

[17]

S. B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094. doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[18]

S.-B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974.  doi: 10.1137/100782358.  Google Scholar

[19]

J. HuismanM. ArrayàsU. Ebert and B. Sommeijer, How do sinking phytoplankton species manage to persist?, Amer. Nat., 159 (2002), 245-254.  doi: 10.1086/338511.  Google Scholar

[20]

T. KolokolnikovC. Ou and Y. Yuan, Profiles of self-shading, sinking phytoplankton with finite depth, J. Math. Biol., 59 (2009), 105-122.   Google Scholar

[21]

K.-Y. Lam and Y. Lou, Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal, Bull. Math. Biol., 76 (2014), 261-291.  doi: 10.1007/s11538-013-9901-y.  Google Scholar

[22]

K.-Y. Lam and Y. Lou, Persistence, competition, and evolution, The Dynamics of Biological Systems, Math. Planet Earth, Springer, Cham, 4 (2019), 205–238.  Google Scholar

[23]

K.-Y. Lam, Y. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), Suppl. 1,188–212. doi: 10.1080/17513758.2014.969336.  Google Scholar

[24]

K.-Y. Lam and D. Munther, A remark on the global dynamics of competitive systems on ordered Banach spaces, Proc. Amer. Math. Soc., 144 (2016), 1153-1159.  doi: 10.1090/proc12768.  Google Scholar

[25]

Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.  Google Scholar

[26]

Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), 141-171.  doi: 10.1016/j.jde.2015.02.004.  Google Scholar

[27]

D. LudwigD. G. Aronson and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol., 8 (1979), 217-258.  doi: 10.1007/BF00276310.  Google Scholar

[28]

F. LutscherM. A. Lewis and E. McCauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160.  doi: 10.1007/s11538-006-9100-1.  Google Scholar

[29]

F. LutscherE. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772.  doi: 10.1137/050636152.  Google Scholar

[30]

J. Maynard-Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18.   Google Scholar

[31]

B. J. McGill and J. S. Brown, Evolutionary game theory and adaptive dynamics of continuous traits, Annu. Rev. Ecol. Evol. Syst., 38 (2007), 403-435.  doi: 10.1146/annurev.ecolsys.36.091704.175517.  Google Scholar

[32]

K. Müller, Investigations on the Organic Drift in North Swedish Streams, Tech. Report 34, Institute of Freshwater Research, Drottningholm, Sweden, 1954. Google Scholar

[33]

K. Müller, The colonization cycle of freshwater insects, Oecologica, 53 (1982), 202-207.   Google Scholar

[34]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[35]

A. B. Potapov and M. A. Lewis, Climate and competition: The effect of moving range boundaries on habitat invasibility, Bull. Math. Biol., 66 (2004), 975-1008.  doi: 10.1016/j.bulm.2003.10.010.  Google Scholar

[36] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, New York, Tokyo, 1997.   Google Scholar
[37]

D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237.   Google Scholar

[38]

O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Quart., 18 (2010), 439-469.   Google Scholar

[39]

A. Vutha and M. Golubitsky, Normal forms and unfoldings of singular strategy functions, Dyn. Games Appl., 5 (2015), 180-213.  doi: 10.1007/s13235-014-0116-0.  Google Scholar

[40]

D. Waxman and S. Gavrilets, 20 questions on adaptive dynamics, J Evol. Biol., 18 (2005), 1139-1154.  doi: 10.1111/j.1420-9101.2005.00948.x.  Google Scholar

show all references

References:
[1]

M. BallykL. DungD. A. Jones and H. L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1999), 573-596.  doi: 10.1137/S0036139997325345.  Google Scholar

[2]

H. BerestyckiO. DiekmannC. 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.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

R. S. CantrellC. Cosner and K.-Y. Lam, On resident-invader dynamics in infinite dimensional dynamical systems, J. Differential Equations, 263 (2017), 4565-4616.   Google Scholar

[5]

R. S. CantrellC. CosnerM. A. Lewis and Y. Lou, Evolution of dispersal in spatial population models with multiple timescales, J. Math. Biol., 80 (2020), 3-37.  doi: 10.1007/s00285-018-1302-2.  Google Scholar

[6]

R. S. CantrellC. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518.  doi: 10.1017/S0308210506000047.  Google Scholar

[7]

X. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst. A, 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.  Google Scholar

[8] F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes. The Adaptive Dynamics Approach and its Applications, Princeton University Press, Princeton, 2008.   Google Scholar
[9]

U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612.  doi: 10.1007/BF02409751.  Google Scholar

[10]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.  Google Scholar

[11]

S. A. H. GeritzE. KisdiG. Meszena and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evol. Ecol., 12 (1998), 35-57.   Google Scholar

[12]

M. GolubitskyW. HaoK.-Y. Lam and Y. Lou, Dimorphism by singularity theory in a model for river ecology, Bull. Math. Biol., 79 (2017), 1051-1069.  doi: 10.1007/s11538-017-0268-3.  Google Scholar

[13]

R. Hambrock and Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817.  doi: 10.1007/s11538-009-9425-7.  Google Scholar

[14]

W. Hao and C. Zheng, An adaptive homotopy method for computing bifurcations of nonlinear parametric systems, J. Sci. Comp., 82 (2020), 1-19.  doi: 10.1007/s10915-020-01160-w.  Google Scholar

[15]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theoretical Population Biology, 24 (1983), 244-251.   Google Scholar

[16]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.  Google Scholar

[17]

S. B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094. doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[18]

S.-B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974.  doi: 10.1137/100782358.  Google Scholar

[19]

J. HuismanM. ArrayàsU. Ebert and B. Sommeijer, How do sinking phytoplankton species manage to persist?, Amer. Nat., 159 (2002), 245-254.  doi: 10.1086/338511.  Google Scholar

[20]

T. KolokolnikovC. Ou and Y. Yuan, Profiles of self-shading, sinking phytoplankton with finite depth, J. Math. Biol., 59 (2009), 105-122.   Google Scholar

[21]

K.-Y. Lam and Y. Lou, Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal, Bull. Math. Biol., 76 (2014), 261-291.  doi: 10.1007/s11538-013-9901-y.  Google Scholar

[22]

K.-Y. Lam and Y. Lou, Persistence, competition, and evolution, The Dynamics of Biological Systems, Math. Planet Earth, Springer, Cham, 4 (2019), 205–238.  Google Scholar

[23]

K.-Y. Lam, Y. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), Suppl. 1,188–212. doi: 10.1080/17513758.2014.969336.  Google Scholar

[24]

K.-Y. Lam and D. Munther, A remark on the global dynamics of competitive systems on ordered Banach spaces, Proc. Amer. Math. Soc., 144 (2016), 1153-1159.  doi: 10.1090/proc12768.  Google Scholar

[25]

Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.  Google Scholar

[26]

Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), 141-171.  doi: 10.1016/j.jde.2015.02.004.  Google Scholar

[27]

D. LudwigD. G. Aronson and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol., 8 (1979), 217-258.  doi: 10.1007/BF00276310.  Google Scholar

[28]

F. LutscherM. A. Lewis and E. McCauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160.  doi: 10.1007/s11538-006-9100-1.  Google Scholar

[29]

F. LutscherE. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772.  doi: 10.1137/050636152.  Google Scholar

[30]

J. Maynard-Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18.   Google Scholar

[31]

B. J. McGill and J. S. Brown, Evolutionary game theory and adaptive dynamics of continuous traits, Annu. Rev. Ecol. Evol. Syst., 38 (2007), 403-435.  doi: 10.1146/annurev.ecolsys.36.091704.175517.  Google Scholar

[32]

K. Müller, Investigations on the Organic Drift in North Swedish Streams, Tech. Report 34, Institute of Freshwater Research, Drottningholm, Sweden, 1954. Google Scholar

[33]

K. Müller, The colonization cycle of freshwater insects, Oecologica, 53 (1982), 202-207.   Google Scholar

[34]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[35]

A. B. Potapov and M. A. Lewis, Climate and competition: The effect of moving range boundaries on habitat invasibility, Bull. Math. Biol., 66 (2004), 975-1008.  doi: 10.1016/j.bulm.2003.10.010.  Google Scholar

[36] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, New York, Tokyo, 1997.   Google Scholar
[37]

D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237.   Google Scholar

[38]

O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Quart., 18 (2010), 439-469.   Google Scholar

[39]

A. Vutha and M. Golubitsky, Normal forms and unfoldings of singular strategy functions, Dyn. Games Appl., 5 (2015), 180-213.  doi: 10.1007/s13235-014-0116-0.  Google Scholar

[40]

D. Waxman and S. Gavrilets, 20 questions on adaptive dynamics, J Evol. Biol., 18 (2005), 1139-1154.  doi: 10.1111/j.1420-9101.2005.00948.x.  Google Scholar

Figure 1.  Normal form diagrams of $ \ell^* $ against $ \mu $ for different cases of $ b $, as the illustrations of Proposition 1.3. The value of $ \mu_{\min} $ is given in (1.4)
Figure 2.  The above normal form diagrams summarize the analytical results from Theorems 1.7, 1.9 and 1.11. They illustrate the transition of 9 qualitatively different pairwise invasibility plots, i.e. nullclines of $ \Lambda(\xi,\tau) $, as parameters $ b $ and $ \ell $ vary. For a pair of strategies $ (\xi,\tau) $, if it lies on a region marked with a plus (resp. minus) sign, then it indicates that the species with strategy $ \tau $ can (resp. cannot) invade the species with strategy $ \xi $ when rare. A red circle stands for an ESS and CvSS; a red square stands for an ESS and non-CvSS; a green square stands for a non-ESS and non-CvSS
Figure 3.  Numerical simulation of the pairwise invasibility plots (i.e. the nullclines of $ \Lambda(\xi,\tau) $) for parameter values $ b = 1.49, 1.5, 1.51 $ and $ \ell = 10, 20, 50 $. The horizontal axis is $ \xi $ and the vertical axis is $ \tau $
Table 1.  Signs of the second derivatives of $ \Lambda $ at $ (\xi, \tau) = (0,0) $ when $ b = \frac{3}{2} $
$ \Lambda_{\tau\tau}(0,0) $ $ (\Lambda_{\tau\tau} + \Lambda_{\tau\xi})(0,0) $ $ \Lambda_{\xi\xi}(0,0) $
$ \ell> {51}/{2} $ $<0 $ (ESS) $>0 $ (not CvSS) $<0 $
$ {27}/{2}<\ell< {51}/{2} $ $<0 $ (ESS) $<0 $ (CvSS) $<0 $
$ {3}/{2}<\ell< {27}/{2} $ $<0 $ (ESS) $<0 $ (CvSS) $>0 $
$ \Lambda_{\tau\tau}(0,0) $ $ (\Lambda_{\tau\tau} + \Lambda_{\tau\xi})(0,0) $ $ \Lambda_{\xi\xi}(0,0) $
$ \ell> {51}/{2} $ $<0 $ (ESS) $>0 $ (not CvSS) $<0 $
$ {27}/{2}<\ell< {51}/{2} $ $<0 $ (ESS) $<0 $ (CvSS) $<0 $
$ {3}/{2}<\ell< {27}/{2} $ $<0 $ (ESS) $<0 $ (CvSS) $>0 $
[1]

Bo Duan, Zhengce Zhang. A two-species weak competition system of reaction-diffusion-advection with double free boundaries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 801-829. doi: 10.3934/dcdsb.2018208

[2]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[3]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[4]

Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701

[5]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989

[6]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841

[7]

Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020217

[8]

Mostafa Bendahmane, Kenneth H. Karlsen. Renormalized solutions of an anisotropic reaction-diffusion-advection system with $L^1$ data. Communications on Pure & Applied Analysis, 2006, 5 (4) : 733-762. doi: 10.3934/cpaa.2006.5.733

[9]

Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176

[10]

Linfeng Mei, Xiaoyan Zhang. On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 221-243. doi: 10.3934/dcdsb.2012.17.221

[11]

Peter Poláčik, Eiji Yanagida. Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 209-218. doi: 10.3934/dcds.2002.8.209

[12]

Alex Potapov, Ulrike E. Schlägel, Mark A. Lewis. Evolutionarily stable diffusive dispersal. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3319-3340. doi: 10.3934/dcdsb.2014.19.3319

[13]

Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191

[14]

Blaise Faugeras, Olivier Maury. An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical parameter estimation procedure: Application to the Indian Ocean skipjack tuna fishery. Mathematical Biosciences & Engineering, 2005, 2 (4) : 719-741. doi: 10.3934/mbe.2005.2.719

[15]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020197

[16]

E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39

[17]

Igor Pažanin, Marcone C. Pereira. On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption. Communications on Pure & Applied Analysis, 2018, 17 (2) : 579-592. doi: 10.3934/cpaa.2018031

[18]

Jing-Jing Xiang, Yihao Fang. Evolutionarily stable dispersal strategies in a two-patch advective environment. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1875-1887. doi: 10.3934/dcdsb.2018245

[19]

Qi Wang. On steady state of some Lotka-Volterra competition-diffusion-advection model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 859-875. doi: 10.3934/dcdsb.2019193

[20]

Qi Wang. Some global dynamics of a Lotka-Volterra competition-diffusion-advection system. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3245-3255. doi: 10.3934/cpaa.2020142

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (21)
  • HTML views (53)
  • Cited by (0)

Other articles
by authors

[Back to Top]