January  2021, 26(1): 367-400. 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  January 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. 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.

[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.

[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.

[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. 

[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.

[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.

[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.

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

[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.

[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. 

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

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

[33]

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

[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.

[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.

[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. 
[37]

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

[38]

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

[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.

[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.

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.

[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.

[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.

[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. 

[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.

[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.

[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.

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

[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.

[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. 

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

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

[33]

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

[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.

[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.

[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. 
[37]

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

[38]

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

[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.

[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.

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 $
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