June  2020, 40(6): 3571-3593. 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 and Continuous Dynamical Systems, 2020, 40 (6) : 3571-3593. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

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 and Continuous Dynamical Systems, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031

[2]

Yuanshi Wang. Asymmetric diffusion in a two-patch mutualism system characterizing exchange of resource for resource. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 963-985. doi: 10.3934/dcdsb.2020149

[3]

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

[4]

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

[5]

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

[6]

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

[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]

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

[9]

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

[10]

Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6173-6184. doi: 10.3934/dcdsb.2021012

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Qun Liu, Lihua Fu, Meng Zhang, Wanjuan Zhang. Two-dimensional seismic data reconstruction using patch tensor completion. Inverse Problems and Imaging, 2020, 14 (6) : 985-1000. doi: 10.3934/ipi.2020052

[17]

Li Ma, Youquan Luo. Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2555-2582. doi: 10.3934/dcdsb.2020022

[18]

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

[19]

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

[20]

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

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (313)
  • HTML views (559)
  • Cited by (0)

Other articles
by authors

[Back to Top]