Article Contents
Article Contents

# Asymmetric dispersal and evolutional selection in two-patch system

• * Corresponding author: Changwook Yoon
• 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.

Mathematics Subject Classification: Primary: 35F50, 92D25, 97M60, 74G30, 34D20; Secondary: 34D05.

 Citation:

• 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] 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. Arditi, C. 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. Cantrell, C. Cosner, D. 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. Choi, S. 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. Cressman, V. 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. DeAngelis, W.-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. DeAngelis, C. 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. Dieckman, B. 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. Dockery, V. Hutson, K. 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. Kim, O. 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. Kim, O. 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. Sullivan, B. Li, T. E. Miller, M. 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.
Open Access Under a Creative Commons license

Figures(6)