• Previous Article
    Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere
  • DCDS Home
  • This Issue
  • Next Article
    A Hopf lemma and regularity for fractional $ p $-Laplacians
doi: 10.3934/dcds.2020043

Asymmetric dispersal and evolutional selection in two-patch system

1. 

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Korea

2. 

Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, Yongin, 446-701, South Korea

3. 

College of Science & Technology, Korea University, Sejong 30019, Republic of Korea

* Corresponding author: Changwook Yoon

Received  March 2019 Revised  May 2019 Published  October 2019

Biological organisms leave their habitat when the environment becomes harsh. The essence of a biological dispersal is not in the rate, but in the capability to adjust to the environmental changes. In nature, conditional asymmetric dispersal strategies appear due to the spatial and temporal heterogeneity in the environment. Authors show that such a dispersal strategy is evolutionary selected in the context two-patch problem of Lotka-Volterra competition model. They conclude that, if a conditional asymmetric dispersal strategy is taken, the dispersal is not necessarily disadvantageous even for the case that there is no temporal fluctuation of environment at all.

Citation: Yong-Jung Kim, Hyowon Seo, Changwook Yoon. Asymmetric dispersal and evolutional selection in two-patch system. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020043
References:
[1]

R. Arditi, L.-F. Bersier and R. P. Rohr, The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka, Ecosphere, 7 (2016), e01599. doi: 10.1002/ecs2.1599.  Google Scholar

[2]

R. ArditiC. Lobry and T. Sari, Asymmetric dispersal in the multi-patch logistic equation, Theoret. Popul. Biol., 120 (2018), 11-15.  doi: 10.1016/j.tpb.2017.12.006.  Google Scholar

[3]

R. S. CantrellC. CosnerD. L. Deangelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dynam., 1 (2007), 249-271.  doi: 10.1080/17513750701450227.  Google Scholar

[4]

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

[5]

E. Cho and Y.-J. Kim, Starvation driven diffusion as a survival strategy of biological organisms, Bull. Math. Biol., 75 (2013), 845-870.  doi: 10.1007/s11538-013-9838-1.  Google Scholar

[6]

W. ChoiS. Baek and I. Ahn, Intraguild predation with evolutionary dispersal in a spatially heterogeneous environment, J. Math. Biol., 78 (2019), 2141-2169.  doi: 10.1007/s00285-019-01336-5.  Google Scholar

[7]

D. Cohen and S. A. Levin, Dispersal in patchy environments: The effects of temporal and spatial structure, Theoret. Popul. Biol., 39 (1991), 63-99.  doi: 10.1016/0040-5809(91)90041-D.  Google Scholar

[8]

R. CressmanV. Křivan and J. Garay, Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments, The American Naturalist, 164 (2004), 473-489.  doi: 10.1086/423827.  Google Scholar

[9]

D. L. DeAngelisW.-M. Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theoretical Ecology, 9 (2016), 443-453.  doi: 10.1007/s12080-016-0302-3.  Google Scholar

[10]

D. L. DeAngelisC. C. Travis and W. M. Post, Persistence and stability of seeddispersed species in a patchy environment, J. Theoret. Biol., 16 (1979), 107-125.  doi: 10.1016/0040-5809(79)90008-x.  Google Scholar

[11]

U. DieckmanB. O'Hara and W. Weisser, The evolutionary ecology of dispersal, Trends Ecol. Evol., 14 (1999), 88-90.  doi: 10.1016/S0169-5347(98)01571-7.  Google Scholar

[12]

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

[13]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theoret. Popul. Biol., 24 (1983), 244-251.  doi: 10.1016/0040-5809(83)90027-8.  Google Scholar

[14]

P. H. Joachim and H. Thomas, Evolution of density-and patch-size-dependent dispersal rates, Proc. R. Soc. Lond. B, 269 (2002). doi: 10.1098/rspb.2001.1936.  Google Scholar

[15]

M. L. Johnson and M. S. Gaines, Evolution of dispersal: Theoretical models and empirical tests using birds and mammels, Ann. Rev. Ecol. Syst., 21 (1990), 449-480.  doi: 10.1146/annurev.es.21.110190.002313.  Google Scholar

[16]

Y.-J. Kim and O. Kwon, Evolution of dispersal with starvation measure and coexistence, Bull. Math. Biol., 78 (2016), 254-279.  doi: 10.1007/s11538-016-0142-8.  Google Scholar

[17]

Y.-J. KimO. Kwon and F. Li, Evolution of dispersal toward fitness, Bull. Math. Biol., 75 (2013), 2474-2498.  doi: 10.1007/s11538-013-9904-8.  Google Scholar

[18]

Y.-J. KimO. Kwon and F. Li, Global asymptotic stability and the ideal free distribution in a starvation driven diffusion, J. Math. Biol., 68 (2014), 1341-1370.  doi: 10.1007/s00285-013-0674-6.  Google Scholar

[19]

Y.-J. Kim, S. Seo and C. Yoon, Asymmetric dispersal and ecological coexistence in two-patch system, preprint. Google Scholar

[20]

Y.-J. Kim, S. Seo and C. Yoon, Two-patch system revisited: New perspectives, Bull. Math. Biol., submitted. Google Scholar

[21]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[22]

M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments, The American Naturalist, 140 (1992), 1000-1009.  doi: 10.1086/285453.  Google Scholar

[23]

T. Nagylaki, Introduction to Theoretical Population Genetics, Biomathematics, 21. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-76214-7.  Google Scholar

[24]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[25]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, 2$^nd$ edition, Interdisciplinary Applied Mathematics, 14. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[26]

R. Ramos-Jiliberto and P. M. de Espans, The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka: Comment, Ecosphere, 8 (2017), e01895. doi: 10.1002/ecs2.1895.  Google Scholar

[27]

A. M. M. Rodrigues and R. A. Johnstone, Evolution of positive and negative density-dependent dispersal, Proc. R. Soc. B, 281 (2014). doi: 10.1098/rspb.2014.1226.  Google Scholar

[28]

L. L. SullivanB. LiT. E. MillerM. G. Neubert and A. K. Shaw, Density dependence in demography and dispersal generates fluctuating invasion speeds, Proceedings of the National Academy of Sciences, 114 (2017), 5053-5058.  doi: 10.1073/pnas.1618744114.  Google Scholar

[29]

J. M. J. Travis and C. Dytham, Habitat persistence, habitat availability and the evolution of dispersal, Proceedings of the Royal Society of London. Series B: Biological Sciences, 266 (1999), 723-728.  doi: 10.1098/rspb.1999.0696.  Google Scholar

show all references

References:
[1]

R. Arditi, L.-F. Bersier and R. P. Rohr, The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka, Ecosphere, 7 (2016), e01599. doi: 10.1002/ecs2.1599.  Google Scholar

[2]

R. ArditiC. Lobry and T. Sari, Asymmetric dispersal in the multi-patch logistic equation, Theoret. Popul. Biol., 120 (2018), 11-15.  doi: 10.1016/j.tpb.2017.12.006.  Google Scholar

[3]

R. S. CantrellC. CosnerD. L. Deangelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dynam., 1 (2007), 249-271.  doi: 10.1080/17513750701450227.  Google Scholar

[4]

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

[5]

E. Cho and Y.-J. Kim, Starvation driven diffusion as a survival strategy of biological organisms, Bull. Math. Biol., 75 (2013), 845-870.  doi: 10.1007/s11538-013-9838-1.  Google Scholar

[6]

W. ChoiS. Baek and I. Ahn, Intraguild predation with evolutionary dispersal in a spatially heterogeneous environment, J. Math. Biol., 78 (2019), 2141-2169.  doi: 10.1007/s00285-019-01336-5.  Google Scholar

[7]

D. Cohen and S. A. Levin, Dispersal in patchy environments: The effects of temporal and spatial structure, Theoret. Popul. Biol., 39 (1991), 63-99.  doi: 10.1016/0040-5809(91)90041-D.  Google Scholar

[8]

R. CressmanV. Křivan and J. Garay, Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments, The American Naturalist, 164 (2004), 473-489.  doi: 10.1086/423827.  Google Scholar

[9]

D. L. DeAngelisW.-M. Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theoretical Ecology, 9 (2016), 443-453.  doi: 10.1007/s12080-016-0302-3.  Google Scholar

[10]

D. L. DeAngelisC. C. Travis and W. M. Post, Persistence and stability of seeddispersed species in a patchy environment, J. Theoret. Biol., 16 (1979), 107-125.  doi: 10.1016/0040-5809(79)90008-x.  Google Scholar

[11]

U. DieckmanB. O'Hara and W. Weisser, The evolutionary ecology of dispersal, Trends Ecol. Evol., 14 (1999), 88-90.  doi: 10.1016/S0169-5347(98)01571-7.  Google Scholar

[12]

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

[13]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theoret. Popul. Biol., 24 (1983), 244-251.  doi: 10.1016/0040-5809(83)90027-8.  Google Scholar

[14]

P. H. Joachim and H. Thomas, Evolution of density-and patch-size-dependent dispersal rates, Proc. R. Soc. Lond. B, 269 (2002). doi: 10.1098/rspb.2001.1936.  Google Scholar

[15]

M. L. Johnson and M. S. Gaines, Evolution of dispersal: Theoretical models and empirical tests using birds and mammels, Ann. Rev. Ecol. Syst., 21 (1990), 449-480.  doi: 10.1146/annurev.es.21.110190.002313.  Google Scholar

[16]

Y.-J. Kim and O. Kwon, Evolution of dispersal with starvation measure and coexistence, Bull. Math. Biol., 78 (2016), 254-279.  doi: 10.1007/s11538-016-0142-8.  Google Scholar

[17]

Y.-J. KimO. Kwon and F. Li, Evolution of dispersal toward fitness, Bull. Math. Biol., 75 (2013), 2474-2498.  doi: 10.1007/s11538-013-9904-8.  Google Scholar

[18]

Y.-J. KimO. Kwon and F. Li, Global asymptotic stability and the ideal free distribution in a starvation driven diffusion, J. Math. Biol., 68 (2014), 1341-1370.  doi: 10.1007/s00285-013-0674-6.  Google Scholar

[19]

Y.-J. Kim, S. Seo and C. Yoon, Asymmetric dispersal and ecological coexistence in two-patch system, preprint. Google Scholar

[20]

Y.-J. Kim, S. Seo and C. Yoon, Two-patch system revisited: New perspectives, Bull. Math. Biol., submitted. Google Scholar

[21]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[22]

M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments, The American Naturalist, 140 (1992), 1000-1009.  doi: 10.1086/285453.  Google Scholar

[23]

T. Nagylaki, Introduction to Theoretical Population Genetics, Biomathematics, 21. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-76214-7.  Google Scholar

[24]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[25]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, 2$^nd$ edition, Interdisciplinary Applied Mathematics, 14. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[26]

R. Ramos-Jiliberto and P. M. de Espans, The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka: Comment, Ecosphere, 8 (2017), e01895. doi: 10.1002/ecs2.1895.  Google Scholar

[27]

A. M. M. Rodrigues and R. A. Johnstone, Evolution of positive and negative density-dependent dispersal, Proc. R. Soc. B, 281 (2014). doi: 10.1098/rspb.2014.1226.  Google Scholar

[28]

L. L. SullivanB. LiT. E. MillerM. G. Neubert and A. K. Shaw, Density dependence in demography and dispersal generates fluctuating invasion speeds, Proceedings of the National Academy of Sciences, 114 (2017), 5053-5058.  doi: 10.1073/pnas.1618744114.  Google Scholar

[29]

J. M. J. Travis and C. Dytham, Habitat persistence, habitat availability and the evolution of dispersal, Proceedings of the Royal Society of London. Series B: Biological Sciences, 266 (1999), 723-728.  doi: 10.1098/rspb.1999.0696.  Google Scholar

Figure 5.  Asymptotic behavior of numerical computation of (31)-(32) with $ \epsilon = 0.02 $. The legends are ordered by the size of asymptotic limits
Figure 1.  A mega patch is a collection of many smaller patches. Dispersal across mega patches are counted in a two-patch system
Figure 2.  Steady state solutions of (14). In the left figure, $ (K_1,K_2) = (2,5) $ and $ \theta_i $'s are monotone. In the right one, $ (K_1,K_2) = (0.2,5) $ and $ \theta_1 $ has maximum at $ d = 0.6613 $
Figure 3.  Diagrams for $ y = -x(1-\frac{x}{K_1}) $ and $ y = x(1-\frac{x}{K_2}) $. Steady states are intersection points with $ y = R $. See (19)
Figure 4.  The graph of motility function $ \gamma(s) $ without uniqueness. We have chosen a piecewise linear motility $ \gamma $ which takes the values in (20). Since $ s_i $ and $ \tilde s_i $ are close to each other, $ \gamma $ increases steeply for $ s\in(s_i,\tilde s_i) $
Figure 6.  Asymptotic behavior ($ h = 0.1, \ell = 0.01, d = 0.02 $). The legends are ordered by the size of asymptotic limits
[1]

Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020031

[2]

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

[3]

Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences & Engineering, 2017, 14 (4) : 843-880. doi: 10.3934/mbe.2017046

[4]

Theodore E. Galanthay. Mathematical study of the effects of travel costs on optimal dispersal in a two-patch model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1625-1638. doi: 10.3934/dcdsb.2015.20.1625

[5]

Xiaoying Wang, Xingfu Zou. On a two-patch predator-prey model with adaptive habitancy of predators. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 677-697. doi: 10.3934/dcdsb.2016.21.677

[6]

Donald L. DeAngelis, Bo Zhang. Effects of dispersal in a non-uniform environment on population dynamics and competition: A patch model approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3087-3104. doi: 10.3934/dcdsb.2014.19.3087

[7]

Wei Feng, Jody Hinson. Stability and pattern in two-patch predator-prey population dynamics. Conference Publications, 2005, 2005 (Special) : 268-279. doi: 10.3934/proc.2005.2005.268

[8]

Qi Wang. On the steady state of a shadow system to the SKT competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2941-2961. doi: 10.3934/dcdsb.2014.19.2941

[9]

Komi Messan, Yun Kang. A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 947-976. doi: 10.3934/dcdsb.2017048

[10]

Robert Stephen Cantrell, Brian Coomes, Yifan Sha. A tridiagonal patch model of bacteria inhabiting a Nanofabricated landscape. Mathematical Biosciences & Engineering, 2017, 14 (4) : 953-973. doi: 10.3934/mbe.2017050

[11]

Henri Berestycki, Jean-Michel Roquejoffre, Luca Rossi. The periodic patch model for population dynamics with fractional diffusion. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 1-13. doi: 10.3934/dcdss.2011.4.1

[12]

Roberto Garra. Confinement of a hot temperature patch in the modified SQG model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2407-2416. doi: 10.3934/dcdsb.2018258

[13]

Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999

[14]

Toshikazu Kuniya, Yoshiaki Muroya, Yoichi Enatsu. Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1375-1393. doi: 10.3934/mbe.2014.11.1375

[15]

Jose-Luis Lisani, Antoni Buades, Jean-Michel Morel. How to explore the patch space. Inverse Problems & Imaging, 2013, 7 (3) : 813-838. doi: 10.3934/ipi.2013.7.813

[16]

Antonio Fasano, Marco Gabrielli, Alberto Gandolfi. Investigating the steady state of multicellular spheroids by revisiting the two-fluid model. Mathematical Biosciences & Engineering, 2011, 8 (2) : 239-252. doi: 10.3934/mbe.2011.8.239

[17]

Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531

[18]

Sze-Bi Hsu, Feng-Bin Wang. On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1479-1501. doi: 10.3934/cpaa.2011.10.1479

[19]

Qi Wang. On steady state of some Lotka-Volterra competition-diffusion-advection model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019193

[20]

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

2018 Impact Factor: 1.143

Article outline

Figures and Tables

[Back to Top]