2014, 19(10): 3359-3378. doi: 10.3934/dcdsb.2014.19.3359

Optimal migratory behavior in spatially-explicit seasonal environments

1. 

Department of Mathematics, Penn State University, University Park, PA 16802, United States

2. 

Division of Evolution, Ecology, and Genetics, The Australian National University, Canberra ACT 0200, Australia

Received  July 2013 Revised  February 2014 Published  October 2014

Mass migrations of vertebrate and arthropod species have long been perceived as some of the most mystical phenomena in nature. And for eons, we have been asking ourselves why animals migrate. Ecologically, migration provides benefits in currencies of survival, growth, and reproduction, allowing animals to exploit environmental heterogeneities in space and time. Yet for a given environment, different species respond with different behaviors -- some travelling large distances, while others shelter in place. Part of the explanation of this distinction is the physiological differences between species and their ability to move. But is physiological difference a necessary pre-condition? Or can environmental heterogeneity itself be sufficient for bifurcations in movement behavior?
    In this paper, we address this last question using a model for the evolution of migration in a density-independent, spatially-explicit setting when movement is costly based on the harvesting a single resource that varies in space and time. We use optimal control methods to calculate the optimal movement patterns in several different situations. In this framework, optimal movement strategies can be classified into six different regimes, based on the cost of movement, the strength and scale of seasonal resource variation, and the degree of trade-off between short-term and long-term benefits. We show that a migratory niche emerges in response to inseparable spatio-temporal environmental heterogeneity, and that this niche can bifurcate from changes to the resource distribution without need for physiological divergence.
Citation: Timothy C. Reluga, Allison K. Shaw. Optimal migratory behavior in spatially-explicit seasonal environments. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3359-3378. doi: 10.3934/dcdsb.2014.19.3359
References:
[1]

T. Alerstam, A. Hedenström and S. Åkesson, Long-distance migration: Evolution and determinants,, Oikos, 103 (2003), 247. doi: 10.1034/j.1600-0706.2003.12559.x.

[2]

R. M. Alexander, When is migration worthwhile for animals that walk, swim or fly?,, Journal of Avian Biology, 29 (1998), 387. doi: 10.2307/3677157.

[3]

M. Bulmer, Theoretical Evolutionary Ecology,, Sinauer, (1994).

[4]

R. S. Cantrell, C. Cosner, D. L. Deangelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy,, Journal of Biological Dynamics, 1 (2007), 249. doi: 10.1080/17513750701450227.

[5]

R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations,, Journal of Differential Equations, 245 (2008), 3687. doi: 10.1016/j.jde.2008.07.024.

[6]

S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes,, in Spatial Ecology (eds. S. Cantrell, (2009), 213. doi: 10.1201/9781420059861.ch11.

[7]

C. W. Clark, Mathematical Bioeconomics,, John Wiley & Sons, (2010).

[8]

D. Cohen, Optimization of seasonal migratory behavior,, American Naturalist, 101 (1967), 5. doi: 10.1086/282464.

[9]

K. A. Cresswell, W. H. Satterthwaite and G. A. Sword, Understanding the evolution of migration through empirical examples,, in Animal migration: A synthesis, (2011), 7. doi: 10.1093/acprof:oso/9780199568994.003.0002.

[10]

D. L. DeAngelis, G. S. Wolkowicz, Y. Lou, Y. Jiang, M. Novak, R. Svanbäck, M. S. Araujo, Y. S. Jo and E. A. Cleary, The effect of travel loss on evolutionarily stable distributions of populations in space,, The American Naturalist, 178 (2011), 15. doi: 10.1086/660280.

[11]

H. Dingle, Ecology and evolution of migration,, in Animal migration, (1980), 1. doi: 10.1016/B978-0-08-091833-4.50006-7.

[12]

H. Dingle and V. A. Drake, What is migration?,, Bioscience, 57 (2007), 113.

[13]

S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat distribution in birds,, Acta Biotheoretica, 19 (1969), 37. doi: 10.1007/BF01601954.

[14]

J. M. Fryxell, E. Milner-Gulland and A. R. Sinclair, Introduction,, in Animal migration: A synthesis, (2011), 1.

[15]

N. J. Gales, A. J. Cheal, G. J. Pobar and P. Williamson, Breeding biology and movements of Australian sea-lions, Neophoca cinerea, off the west coasst of Western Australia,, Wildlife Research, 19 (1992), 405. doi: 10.1071/WR9920405.

[16]

A. M. Hein, C. Hou and J. F. Gillooly, Energetic and biomechanical constraints on animal migration distance,, Ecology letters, 15 (2012), 104. doi: 10.1111/j.1461-0248.2011.01714.x.

[17]

D. E. Hiebeler and B. R. Morin, The effect of static and dynamic spatially structured disturbances on a locally dispersing population,, Journal of Theoretical Biology, 246 (2007), 136. doi: 10.1016/j.jtbi.2006.12.024.

[18]

R. D. Holt and J. M. Fryxell, Theoretical reflections on the evolution of migration,, in Animal migration: A synthesis (eds. E. Milner-Gulland, (): 17. doi: 10.1093/acprof:oso/9780199568994.003.0003.

[19]

J. D. Hunter, Matplotlib: A 2d graphics environment,, Computing In Science & Engineering, 9 (2007), 90. doi: 10.1109/MCSE.2007.55.

[20]

E. Jones, T. Oliphant and P. Peterson, et al., SciPy: Open source scientific tools for Python (2001),, , ().

[21]

A. Kaitala, V. Kaitala and P. Lundberg, A theory of partial migration,, American Naturalist, 142 (1993), 59. doi: 10.1086/285529.

[22]

I. Kovacic and M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and Their Behaviour,, Wiley, (2011). doi: 10.1002/9780470977859.

[23]

S. Lenci and G. Rega, Forced harmonic vibration in a Duffing oscillator with negative linear stiffness and linear viscous damping,, in The Duffing Equation: Nonlinear Oscillators and their Behaviour (eds. I. Kovacic and M. J. Brennan), (2011), 219. doi: 10.1002/9780470977859.ch7.

[24]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC, (2007).

[25]

P. Lundberg, Partial bird migration and evolutionarily stable strategies,, Journal of Theoretical Biology, 125 (1987), 351. doi: 10.1016/S0022-5193(87)80067-X.

[26]

J. M. McNamara, A. I. Houston and E. J. Collins, Optimality models in behavioral biology,, SIAM Review, 43 (2001), 413. doi: 10.1137/S0036144500385263.

[27]

B. H. McRae, B. G. Dickson, T. H. Keitt, and V. B. Shah, Using Circuit Theory to Model Connectivity in Ecology, Evolution, and Conservation,, Ecology, 89 (2008), 2712.

[28]

A. Mertz and W. Slough, Graphics with pgf and tikz,, PracTeX Journal, 1 (): 28.

[29]

T. Mueller and W. F. Fagan, Search and navigation in dynamic environments-from individual behaviors to population distributions,, Oikos, 117 (2008), 654. doi: 10.1111/j.0030-1299.2008.16291.x.

[30]

R. Nathan, An emerging movement ecology paradigm,, Proceedings of the National Academy of Sciences, 105 (2008), 19050. doi: 10.1073/pnas.0808918105.

[31]

R. Nathan, W. M. Getz, E. Revilla, M. Holyoak, R. Kadmon, D. Saltz and P. E. Smouse, A movement ecology paradigm for unifying organismal movement research,, Proceedings of the National Academy of Sciences, 105 (2008), 19052. doi: 10.1073/pnas.0800375105.

[32]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations,, Wiley, (1995).

[33]

C. J. Pennycuick, The mechanics of bird migration,, Ibis, 111 (1969), 525. doi: 10.1111/j.1474-919X.1969.tb02566.x.

[34]

, Python Software Foundation,, Python Language Reference, (2010).

[35]

T. C. Reluga, J. Medlock and A. P. Galvani, The discounted reproductive number for epidemiology,, Mathematical Biosciences and Engineering, 6 (2009), 377. doi: 10.3934/mbe.2009.6.377.

[36]

D. E. Schindler, R. Hilborn, B. Chasco, C. P. Boatright, T. P. Quinn, L. A. Rogers and M. S. Webster, Population diversity and the portfolio effect in an exploited species,, Nature, 465 (2010), 609. doi: 10.1038/nature09060.

[37]

A. K. Shaw and S. A. Levin, To breed or not to breed: A model of partial migration,, Oikos, 120 (2011), 1871. doi: 10.1111/j.1600-0706.2011.19443.x.

[38]

A. K. Shaw and I. D. Couzin, Migration or residency? the evolution of movement behavior and information usage in seasonal environments,, The American Naturalist, 181 (2013), 114. doi: 10.1086/668600.

[39]

A. K. Shaw and S. A. Levin, The evolution of intermittent breeding,, Journal of Mathematical Biology, 66 (2013), 685. doi: 10.1007/s00285-012-0603-0.

[40]

S. C. Stearns, The Evolution of Life Histories,, Oxford University Presss, (1992).

[41]

, SymPy Development Team,, SymPy: Python library for symbolic mathematics, (2013).

[42]

W. Szempli'nska-Stupnicka and K. L. Janicki, Basin boundary bifurcations and boundary crisis in the twin-well Duffing oscillator: Scenarios related to the saddle of the large resonant orbit,, International Journal of Bifurcation and Chaos, 07 (1997), 129. doi: 10.1142/S0218127497000091.

[43]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM Journal on Applied Mathematics, 70 (2009), 188. doi: 10.1137/080732870.

[44]

J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos,, John Wiley and Sons, (2002).

[45]

J. M. 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. doi: 10.1098/rspb.1999.0696.

[46]

P. Wiener and S. Tuljapurkar, Migration in variable environments: Exploring life-history evolution using structured population models,, Journal of Theoretical Biology, 166 (1994), 75. doi: 10.1006/jtbi.1994.1006.

[47]

T. Williams, C. Kelley and {many others}, Gnuplot 4.4: an interactive plotting program,, , (2010).

[48]

S. Wright, Evolution and the Genetics of Populations,, Volume 2: Theory of Gene Frequencies, (1969).

show all references

References:
[1]

T. Alerstam, A. Hedenström and S. Åkesson, Long-distance migration: Evolution and determinants,, Oikos, 103 (2003), 247. doi: 10.1034/j.1600-0706.2003.12559.x.

[2]

R. M. Alexander, When is migration worthwhile for animals that walk, swim or fly?,, Journal of Avian Biology, 29 (1998), 387. doi: 10.2307/3677157.

[3]

M. Bulmer, Theoretical Evolutionary Ecology,, Sinauer, (1994).

[4]

R. S. Cantrell, C. Cosner, D. L. Deangelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy,, Journal of Biological Dynamics, 1 (2007), 249. doi: 10.1080/17513750701450227.

[5]

R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations,, Journal of Differential Equations, 245 (2008), 3687. doi: 10.1016/j.jde.2008.07.024.

[6]

S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes,, in Spatial Ecology (eds. S. Cantrell, (2009), 213. doi: 10.1201/9781420059861.ch11.

[7]

C. W. Clark, Mathematical Bioeconomics,, John Wiley & Sons, (2010).

[8]

D. Cohen, Optimization of seasonal migratory behavior,, American Naturalist, 101 (1967), 5. doi: 10.1086/282464.

[9]

K. A. Cresswell, W. H. Satterthwaite and G. A. Sword, Understanding the evolution of migration through empirical examples,, in Animal migration: A synthesis, (2011), 7. doi: 10.1093/acprof:oso/9780199568994.003.0002.

[10]

D. L. DeAngelis, G. S. Wolkowicz, Y. Lou, Y. Jiang, M. Novak, R. Svanbäck, M. S. Araujo, Y. S. Jo and E. A. Cleary, The effect of travel loss on evolutionarily stable distributions of populations in space,, The American Naturalist, 178 (2011), 15. doi: 10.1086/660280.

[11]

H. Dingle, Ecology and evolution of migration,, in Animal migration, (1980), 1. doi: 10.1016/B978-0-08-091833-4.50006-7.

[12]

H. Dingle and V. A. Drake, What is migration?,, Bioscience, 57 (2007), 113.

[13]

S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat distribution in birds,, Acta Biotheoretica, 19 (1969), 37. doi: 10.1007/BF01601954.

[14]

J. M. Fryxell, E. Milner-Gulland and A. R. Sinclair, Introduction,, in Animal migration: A synthesis, (2011), 1.

[15]

N. J. Gales, A. J. Cheal, G. J. Pobar and P. Williamson, Breeding biology and movements of Australian sea-lions, Neophoca cinerea, off the west coasst of Western Australia,, Wildlife Research, 19 (1992), 405. doi: 10.1071/WR9920405.

[16]

A. M. Hein, C. Hou and J. F. Gillooly, Energetic and biomechanical constraints on animal migration distance,, Ecology letters, 15 (2012), 104. doi: 10.1111/j.1461-0248.2011.01714.x.

[17]

D. E. Hiebeler and B. R. Morin, The effect of static and dynamic spatially structured disturbances on a locally dispersing population,, Journal of Theoretical Biology, 246 (2007), 136. doi: 10.1016/j.jtbi.2006.12.024.

[18]

R. D. Holt and J. M. Fryxell, Theoretical reflections on the evolution of migration,, in Animal migration: A synthesis (eds. E. Milner-Gulland, (): 17. doi: 10.1093/acprof:oso/9780199568994.003.0003.

[19]

J. D. Hunter, Matplotlib: A 2d graphics environment,, Computing In Science & Engineering, 9 (2007), 90. doi: 10.1109/MCSE.2007.55.

[20]

E. Jones, T. Oliphant and P. Peterson, et al., SciPy: Open source scientific tools for Python (2001),, , ().

[21]

A. Kaitala, V. Kaitala and P. Lundberg, A theory of partial migration,, American Naturalist, 142 (1993), 59. doi: 10.1086/285529.

[22]

I. Kovacic and M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and Their Behaviour,, Wiley, (2011). doi: 10.1002/9780470977859.

[23]

S. Lenci and G. Rega, Forced harmonic vibration in a Duffing oscillator with negative linear stiffness and linear viscous damping,, in The Duffing Equation: Nonlinear Oscillators and their Behaviour (eds. I. Kovacic and M. J. Brennan), (2011), 219. doi: 10.1002/9780470977859.ch7.

[24]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC, (2007).

[25]

P. Lundberg, Partial bird migration and evolutionarily stable strategies,, Journal of Theoretical Biology, 125 (1987), 351. doi: 10.1016/S0022-5193(87)80067-X.

[26]

J. M. McNamara, A. I. Houston and E. J. Collins, Optimality models in behavioral biology,, SIAM Review, 43 (2001), 413. doi: 10.1137/S0036144500385263.

[27]

B. H. McRae, B. G. Dickson, T. H. Keitt, and V. B. Shah, Using Circuit Theory to Model Connectivity in Ecology, Evolution, and Conservation,, Ecology, 89 (2008), 2712.

[28]

A. Mertz and W. Slough, Graphics with pgf and tikz,, PracTeX Journal, 1 (): 28.

[29]

T. Mueller and W. F. Fagan, Search and navigation in dynamic environments-from individual behaviors to population distributions,, Oikos, 117 (2008), 654. doi: 10.1111/j.0030-1299.2008.16291.x.

[30]

R. Nathan, An emerging movement ecology paradigm,, Proceedings of the National Academy of Sciences, 105 (2008), 19050. doi: 10.1073/pnas.0808918105.

[31]

R. Nathan, W. M. Getz, E. Revilla, M. Holyoak, R. Kadmon, D. Saltz and P. E. Smouse, A movement ecology paradigm for unifying organismal movement research,, Proceedings of the National Academy of Sciences, 105 (2008), 19052. doi: 10.1073/pnas.0800375105.

[32]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations,, Wiley, (1995).

[33]

C. J. Pennycuick, The mechanics of bird migration,, Ibis, 111 (1969), 525. doi: 10.1111/j.1474-919X.1969.tb02566.x.

[34]

, Python Software Foundation,, Python Language Reference, (2010).

[35]

T. C. Reluga, J. Medlock and A. P. Galvani, The discounted reproductive number for epidemiology,, Mathematical Biosciences and Engineering, 6 (2009), 377. doi: 10.3934/mbe.2009.6.377.

[36]

D. E. Schindler, R. Hilborn, B. Chasco, C. P. Boatright, T. P. Quinn, L. A. Rogers and M. S. Webster, Population diversity and the portfolio effect in an exploited species,, Nature, 465 (2010), 609. doi: 10.1038/nature09060.

[37]

A. K. Shaw and S. A. Levin, To breed or not to breed: A model of partial migration,, Oikos, 120 (2011), 1871. doi: 10.1111/j.1600-0706.2011.19443.x.

[38]

A. K. Shaw and I. D. Couzin, Migration or residency? the evolution of movement behavior and information usage in seasonal environments,, The American Naturalist, 181 (2013), 114. doi: 10.1086/668600.

[39]

A. K. Shaw and S. A. Levin, The evolution of intermittent breeding,, Journal of Mathematical Biology, 66 (2013), 685. doi: 10.1007/s00285-012-0603-0.

[40]

S. C. Stearns, The Evolution of Life Histories,, Oxford University Presss, (1992).

[41]

, SymPy Development Team,, SymPy: Python library for symbolic mathematics, (2013).

[42]

W. Szempli'nska-Stupnicka and K. L. Janicki, Basin boundary bifurcations and boundary crisis in the twin-well Duffing oscillator: Scenarios related to the saddle of the large resonant orbit,, International Journal of Bifurcation and Chaos, 07 (1997), 129. doi: 10.1142/S0218127497000091.

[43]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM Journal on Applied Mathematics, 70 (2009), 188. doi: 10.1137/080732870.

[44]

J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos,, John Wiley and Sons, (2002).

[45]

J. M. 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. doi: 10.1098/rspb.1999.0696.

[46]

P. Wiener and S. Tuljapurkar, Migration in variable environments: Exploring life-history evolution using structured population models,, Journal of Theoretical Biology, 166 (1994), 75. doi: 10.1006/jtbi.1994.1006.

[47]

T. Williams, C. Kelley and {many others}, Gnuplot 4.4: an interactive plotting program,, , (2010).

[48]

S. Wright, Evolution and the Genetics of Populations,, Volume 2: Theory of Gene Frequencies, (1969).

[1]

Eunha Shim. Optimal strategies of social distancing and vaccination against seasonal influenza. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1615-1634. doi: 10.3934/mbe.2013.10.1615

[2]

Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925

[3]

Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159

[4]

Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial & Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311

[5]

A. Chauviere, T. Hillen, L. Preziosi. Modeling cell movement in anisotropic and heterogeneous network tissues. Networks & Heterogeneous Media, 2007, 2 (2) : 333-357. doi: 10.3934/nhm.2007.2.333

[6]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076

[7]

Gengsheng Wang, Guojie Zheng. The optimal control to restore the periodic property of a linear evolution system with small perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1621-1639. doi: 10.3934/dcdsb.2010.14.1621

[8]

Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320

[9]

Oren Barnea, Rami Yaari, Guy Katriel, Lewi Stone. Modelling seasonal influenza in Israel. Mathematical Biosciences & Engineering, 2011, 8 (2) : 561-573. doi: 10.3934/mbe.2011.8.561

[10]

Julien Dambrine, Nicolas Meunier, Bertrand Maury, Aude Roudneff-Chupin. A congestion model for cell migration. Communications on Pure & Applied Analysis, 2012, 11 (1) : 243-260. doi: 10.3934/cpaa.2012.11.243

[11]

Russell Betteridge, Markus R. Owen, H.M. Byrne, Tomás Alarcón, Philip K. Maini. The impact of cell crowding and active cell movement on vascular tumour growth. Networks & Heterogeneous Media, 2006, 1 (4) : 515-535. doi: 10.3934/nhm.2006.1.515

[12]

Thomas Hillen, Peter Hinow, Zhi-An Wang. Mathematical analysis of a kinetic model for cell movement in network tissues. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1055-1080. doi: 10.3934/dcdsb.2010.14.1055

[13]

Sepideh Mirrahimi. Adaptation and migration of a population between patches. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 753-768. doi: 10.3934/dcdsb.2013.18.753

[14]

Zhenguo Bai, Yicang Zhou. Threshold dynamics of a bacillary dysentery model with seasonal fluctuation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 1-14. doi: 10.3934/dcdsb.2011.15.1

[15]

Yangjin Kim, Soyeon Roh. A hybrid model for cell proliferation and migration in glioblastoma. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 969-1015. doi: 10.3934/dcdsb.2013.18.969

[16]

Reinhard Bürger. A survey of migration-selection models in population genetics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 883-959. doi: 10.3934/dcdsb.2014.19.883

[17]

Avner Friedman, Yangjin Kim. Tumor cells proliferation and migration under the influence of their microenvironment. Mathematical Biosciences & Engineering, 2011, 8 (2) : 371-383. doi: 10.3934/mbe.2011.8.371

[18]

Yangjun Ma, Maoxing Liu, Qiang Hou, Jinqing Zhao. Modelling seasonal HFMD with the recessive infection in Shandong, China. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1159-1171. doi: 10.3934/mbe.2013.10.1159

[19]

Natalia L. Komarova. Spatial stochastic models of cancer: Fitness, migration, invasion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 761-775. doi: 10.3934/mbe.2013.10.761

[20]

Yilei Tang, Dongmei Xiao, Weinian Zhang, Di Zhu. Dynamics of epidemic models with asymptomatic infection and seasonal succession. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1407-1424. doi: 10.3934/mbe.2017073

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]