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Extinction in discrete, competitive, multi-species patch models
Spatial population dynamics in a producer-scrounger model
1. | Department of Mathematics, University of Miami, Coral Gables, FL 33146, United States |
2. | Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States |
References:
[1] |
H. Amann, Global existence for semilinear parabolic systems,, J. Reine Angew. Math., 360 (1985), 47.
doi: 10.1515/crll.1985.360.47. |
[2] |
H. Amann, Quasilinear evolution equations and parabolic systems,, Trans. Amer. Math. Soc, 293 (1986), 191.
doi: 10.1090/S0002-9947-1986-0814920-4. |
[3] |
H. Amann, Dynamic theory of quasilinear parabolic equations I: Abstract evolution equations,, Nonlinear Analysis TMA, 12 (1988), 895.
doi: 10.1016/0362-546X(88)90073-9. |
[4] |
H. Amann, Dynamic theory of quasilinear parabolic systems III: Global existence,, Math. Z., 202 (1989), 219.
doi: 10.1007/BF01215256. |
[5] |
H. Amann, Dynamic theory of quasilinear parabolic equations II: Reaction-diffusion systems,, Differential and Integral Equations, 3 (1990), 13.
|
[6] |
C. J. Barnard, Producers and Scroungers: Strategies of Exploitation and Parasitism,, Chapman and Hall, (1984). Google Scholar |
[7] |
C. J. Barnard and R. M. Sibly, Producers and scroungers: A general model and its application to captive flocks of house sparrows,, Anim. Behav., 29 (1981), 543.
doi: 10.1016/S0003-3472(81)80117-0. |
[8] |
Z. Barta, R. Flynn and L.-A. Giraldeau, Geometry for a selfish foraging group: A genetic algorithm approach,, Proc Roy Soc Lond B., 264 (1997), 1233.
doi: 10.1098/rspb.1997.0170. |
[9] |
G. Beauchamp, Learning rules for social foragers: Implications for the producer-scrounger game and ideal free distribution theory,, J. Theor. Biol., 207 (2000), 21.
doi: 10.1006/jtbi.2000.2153. |
[10] |
G. Beauchamp, A spatial model of producing and scrounging,, Anim. Behav., 76 (2008), 1935.
doi: 10.1016/j.anbehav.2008.08.017. |
[11] |
J. Bilotti and J. P. LaSalle, Periodic dissipative processes,, Bull. Amer. Math. Soc., 77 (1971), 1082.
doi: 10.1090/S0002-9904-1971-12879-3. |
[12] |
H. J. Brockmann and C. J. Barnard, Kleptoparasitism in birds,, Anim. Behav., 27 (1979), 487.
doi: 10.1016/0003-3472(79)90185-4. |
[13] |
M. Broom and G. D. Ruxton, Evolutionarily stable stealing: Game theory applied to kleptoparasitism,, Behav. Ecol., 9 (1998), 397. Google Scholar |
[14] |
M. Broom and G. D. Ruxton, Evolutionarily stable kleptoparasitism: Consequences of different prey type,, Behav. Ecol., 14 (2003), 23.
doi: 10.1093/beheco/14.1.23. |
[15] |
M. Broom, R. M. Luther and G. D. Ruxton, Resistance is useless? - Extensions to the game theory of kleptoparasitism,, Bull. Math. Biol., 66 (2004), 1645.
doi: 10.1016/j.bulm.2004.03.009. |
[16] |
M. Broom, R. M. Luther, G. D. Ruxton and J. Rychtar, A game-theoretic model of kleptoparasitic behavior in polymorphic populations,, J. Theor. Biol., 255 (2008), 81.
doi: 10.1016/j.jtbi.2008.08.001. |
[17] |
M. Broom, M. Crowe, M. Fitzgerald and J. Rychtar, The stochastic modelling of kleptoparasitism using a Markov process,, J. Theor. Biol., 264 (2010), 266.
doi: 10.1016/j.jtbi.2010.01.012. |
[18] |
K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function,, J. Math. Anal. Appl., 75 (1980), 112.
doi: 10.1016/0022-247X(80)90309-1. |
[19] |
T. Bugnyar and K. Kotrschal, Scrounging tactics in free-ranging ravens, Corvus corax,, Ethology, 108 (2002), 993.
doi: 10.1046/j.1439-0310.2002.00832.x. |
[20] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley Ser. Math. Comput. Biol., (2003).
doi: 10.1002/0470871296. |
[21] |
R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion,, Math. Biosci., 204 (2006), 199.
doi: 10.1016/j.mbs.2006.09.003. |
[22] |
R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinb. A, 137 (2007), 497.
doi: 10.1017/S0308210506000047. |
[23] |
R.S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes,, in Spatial Ecology (eds. R. S. Cantrell, (2009), 213.
doi: 10.1201/9781420059861.ch11. |
[24] |
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution,, Math. Biosci. Eng., 7 (2010), 17.
doi: 10.3934/mbe.2010.7.17. |
[25] |
R. S. Cantrell, C. Cosner and V. Hutson, Permanence in ecological systems with diffusion,, Proc. Royal Soc. Edinburgh, 123 (1993), 533.
doi: 10.1017/S0308210500025877. |
[26] |
X. Chen, K-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species,, Disc. Cont. Dyn. Syst. A, 32 (2012), 3841.
doi: 10.3934/dcds.2012.32.3841. |
[27] |
C. Cosner, Reaction-diffusion equations and ecological modeling,, in Tutorials in Mathematical Biosciences. IV, (2008), 77.
doi: 10.1007/978-3-540-74331-6_3. |
[28] |
C. Cosner and Y. Lou, When does movement toward better environment benefit a population?,, J. Math. Anal. Appl., 277 (2003), 489.
doi: 10.1016/S0022-247X(02)00575-9. |
[29] |
C. Cosner, A. L. Nevai and Z. Shuai, In, preparation., (). Google Scholar |
[30] |
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.
doi: 10.1007/s002850050120. |
[31] |
L. C. Evans, Partial Differential Equations,, Grad. Stud. Math., (1998).
|
[32] |
L.-A. Giraldeau and T. Caraco, Social Foraging Theory,, Princeton University Press, (2000). Google Scholar |
[33] |
B. S. Goh, Global stability in many-species systems,, Am. Nat., 111 (1977), 135.
doi: 10.1086/283144. |
[34] |
J. K. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388.
doi: 10.1137/0520025. |
[35] |
I. M. Hamilton, Kleptoparasitism and the distribution of unequal competitors,, Behav. Ecol., 13 (2002), 260.
doi: 10.1093/beheco/13.2.260. |
[36] |
A. Hastings, Can spatial variation alone lead to selection for dispersal?,, Theor. Pop. Biol., 24 (1983), 244.
doi: 10.1016/0040-5809(83)90027-8. |
[37] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).
|
[38] |
E. Iyengar, Kleptoparasitic interactions throughout the animal kingdom and a re-evaluation, based on participant mobility, of the conditions promoting the evolution of kleptoparasitism,, Biol. J. Linnean Soc., 93 (2008), 745.
doi: 10.1111/j.1095-8312.2008.00954.x. |
[39] |
R. M. Luther and M. Broom, Rapid convergence to an equilibrium state in kleptoparasitic populations,, J. Math. Biol., 48 (2004), 325.
doi: 10.1007/s00285-003-0237-3. |
[40] |
R. M. Luther, M. Broom and G. D. Ruxton, Is food worth fighting for? ESS's in mixed populations of kleptoparasites and foragers,, Bull. Math. Biol., 69 (2007), 1121.
doi: 10.1007/s11538-005-9052-x. |
[41] |
U. Luttge, Vascular Plants as Epiphytes,, Springer-Verlag, (1989). Google Scholar |
[42] |
X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces,, Trans. Amer. Math. Soc., 278 (1983), 21.
doi: 10.2307/1999300. |
[43] |
J. Morand-Ferron, G.-M. Wu and L.-A. Giraldeau, Persistent individual differences in tactic use in a producer-scrounger game are group dependent,, Anim. Behav., 82 (2011), 811. Google Scholar |
[44] |
J. Morgan, Boundedness and decay results for reaction-diffusion systems,, SIAM J. Math. Anal., 21 (1990), 1172.
doi: 10.1137/0521064. |
[45] |
K. Mottley and L.-A. Giraldeau, Experimental evidence that group foragers can converge on predicted producer-scrounger equilibria,, Anim. Behav., 60 (2000), 341.
doi: 10.1006/anbe.2000.1474. |
[46] |
Y. Ohtsuka and Y. Toquenaga, The patch distributed producer-scrounger game,, J. Theor. Biol., 260 (2009), 261.
doi: 10.1016/j.jtbi.2009.06.002. |
[47] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[48] |
F. J. Richard, A. Dejean and J. P. Lachaud, Sugary food robbing in ants: A case of temporal cleptobiosis,, C. R. Biologies, 327 (2004), 509.
doi: 10.1016/j.crvi.2004.03.002. |
[49] |
G. D. Ruxton, Foraging in flocks: Non-spatial models may neglect important costs,, Ecol. Model., 82 (1995), 277.
doi: 10.1016/0304-3800(94)00098-3. |
[50] |
S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions,, Math. Ann., 258 (1982), 459.
doi: 10.1007/BF01453979. |
[51] |
N. Tania, B. Vanderlii, J. P. Heath and L. Edelstein-Keshet, Role of social interactions in dynamic patterns of resource patches and forager aggregation,, PNAS, 109 (2012), 11228.
doi: 10.1073/pnas.1201739109. |
[52] |
J. H. M. Thornley and I. R. Johnson, Plant and Crop Modelling - A Mathematical Approach to Plant and Crop Physiology,, The Blackburn Press, (2000). Google Scholar |
[53] |
W. L. Vickery, L.-A. Giraldeau, J. J. Templeton, D. L. Kramer and C. A. Chapman, Producers, scroungers, and group foraging,, Am. Nat., 137 (1991), 847.
doi: 10.1086/285197. |
[54] |
F. Vollrath, Behaviour of the kleptoparasitic spider Argyrodes elevatus (Araneae, Theridiidae),, Anim. Behav., 27 (1979), 519.
doi: 10.1016/0003-3472(79)90186-6. |
show all references
References:
[1] |
H. Amann, Global existence for semilinear parabolic systems,, J. Reine Angew. Math., 360 (1985), 47.
doi: 10.1515/crll.1985.360.47. |
[2] |
H. Amann, Quasilinear evolution equations and parabolic systems,, Trans. Amer. Math. Soc, 293 (1986), 191.
doi: 10.1090/S0002-9947-1986-0814920-4. |
[3] |
H. Amann, Dynamic theory of quasilinear parabolic equations I: Abstract evolution equations,, Nonlinear Analysis TMA, 12 (1988), 895.
doi: 10.1016/0362-546X(88)90073-9. |
[4] |
H. Amann, Dynamic theory of quasilinear parabolic systems III: Global existence,, Math. Z., 202 (1989), 219.
doi: 10.1007/BF01215256. |
[5] |
H. Amann, Dynamic theory of quasilinear parabolic equations II: Reaction-diffusion systems,, Differential and Integral Equations, 3 (1990), 13.
|
[6] |
C. J. Barnard, Producers and Scroungers: Strategies of Exploitation and Parasitism,, Chapman and Hall, (1984). Google Scholar |
[7] |
C. J. Barnard and R. M. Sibly, Producers and scroungers: A general model and its application to captive flocks of house sparrows,, Anim. Behav., 29 (1981), 543.
doi: 10.1016/S0003-3472(81)80117-0. |
[8] |
Z. Barta, R. Flynn and L.-A. Giraldeau, Geometry for a selfish foraging group: A genetic algorithm approach,, Proc Roy Soc Lond B., 264 (1997), 1233.
doi: 10.1098/rspb.1997.0170. |
[9] |
G. Beauchamp, Learning rules for social foragers: Implications for the producer-scrounger game and ideal free distribution theory,, J. Theor. Biol., 207 (2000), 21.
doi: 10.1006/jtbi.2000.2153. |
[10] |
G. Beauchamp, A spatial model of producing and scrounging,, Anim. Behav., 76 (2008), 1935.
doi: 10.1016/j.anbehav.2008.08.017. |
[11] |
J. Bilotti and J. P. LaSalle, Periodic dissipative processes,, Bull. Amer. Math. Soc., 77 (1971), 1082.
doi: 10.1090/S0002-9904-1971-12879-3. |
[12] |
H. J. Brockmann and C. J. Barnard, Kleptoparasitism in birds,, Anim. Behav., 27 (1979), 487.
doi: 10.1016/0003-3472(79)90185-4. |
[13] |
M. Broom and G. D. Ruxton, Evolutionarily stable stealing: Game theory applied to kleptoparasitism,, Behav. Ecol., 9 (1998), 397. Google Scholar |
[14] |
M. Broom and G. D. Ruxton, Evolutionarily stable kleptoparasitism: Consequences of different prey type,, Behav. Ecol., 14 (2003), 23.
doi: 10.1093/beheco/14.1.23. |
[15] |
M. Broom, R. M. Luther and G. D. Ruxton, Resistance is useless? - Extensions to the game theory of kleptoparasitism,, Bull. Math. Biol., 66 (2004), 1645.
doi: 10.1016/j.bulm.2004.03.009. |
[16] |
M. Broom, R. M. Luther, G. D. Ruxton and J. Rychtar, A game-theoretic model of kleptoparasitic behavior in polymorphic populations,, J. Theor. Biol., 255 (2008), 81.
doi: 10.1016/j.jtbi.2008.08.001. |
[17] |
M. Broom, M. Crowe, M. Fitzgerald and J. Rychtar, The stochastic modelling of kleptoparasitism using a Markov process,, J. Theor. Biol., 264 (2010), 266.
doi: 10.1016/j.jtbi.2010.01.012. |
[18] |
K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function,, J. Math. Anal. Appl., 75 (1980), 112.
doi: 10.1016/0022-247X(80)90309-1. |
[19] |
T. Bugnyar and K. Kotrschal, Scrounging tactics in free-ranging ravens, Corvus corax,, Ethology, 108 (2002), 993.
doi: 10.1046/j.1439-0310.2002.00832.x. |
[20] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley Ser. Math. Comput. Biol., (2003).
doi: 10.1002/0470871296. |
[21] |
R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion,, Math. Biosci., 204 (2006), 199.
doi: 10.1016/j.mbs.2006.09.003. |
[22] |
R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinb. A, 137 (2007), 497.
doi: 10.1017/S0308210506000047. |
[23] |
R.S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes,, in Spatial Ecology (eds. R. S. Cantrell, (2009), 213.
doi: 10.1201/9781420059861.ch11. |
[24] |
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution,, Math. Biosci. Eng., 7 (2010), 17.
doi: 10.3934/mbe.2010.7.17. |
[25] |
R. S. Cantrell, C. Cosner and V. Hutson, Permanence in ecological systems with diffusion,, Proc. Royal Soc. Edinburgh, 123 (1993), 533.
doi: 10.1017/S0308210500025877. |
[26] |
X. Chen, K-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species,, Disc. Cont. Dyn. Syst. A, 32 (2012), 3841.
doi: 10.3934/dcds.2012.32.3841. |
[27] |
C. Cosner, Reaction-diffusion equations and ecological modeling,, in Tutorials in Mathematical Biosciences. IV, (2008), 77.
doi: 10.1007/978-3-540-74331-6_3. |
[28] |
C. Cosner and Y. Lou, When does movement toward better environment benefit a population?,, J. Math. Anal. Appl., 277 (2003), 489.
doi: 10.1016/S0022-247X(02)00575-9. |
[29] |
C. Cosner, A. L. Nevai and Z. Shuai, In, preparation., (). Google Scholar |
[30] |
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.
doi: 10.1007/s002850050120. |
[31] |
L. C. Evans, Partial Differential Equations,, Grad. Stud. Math., (1998).
|
[32] |
L.-A. Giraldeau and T. Caraco, Social Foraging Theory,, Princeton University Press, (2000). Google Scholar |
[33] |
B. S. Goh, Global stability in many-species systems,, Am. Nat., 111 (1977), 135.
doi: 10.1086/283144. |
[34] |
J. K. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388.
doi: 10.1137/0520025. |
[35] |
I. M. Hamilton, Kleptoparasitism and the distribution of unequal competitors,, Behav. Ecol., 13 (2002), 260.
doi: 10.1093/beheco/13.2.260. |
[36] |
A. Hastings, Can spatial variation alone lead to selection for dispersal?,, Theor. Pop. Biol., 24 (1983), 244.
doi: 10.1016/0040-5809(83)90027-8. |
[37] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).
|
[38] |
E. Iyengar, Kleptoparasitic interactions throughout the animal kingdom and a re-evaluation, based on participant mobility, of the conditions promoting the evolution of kleptoparasitism,, Biol. J. Linnean Soc., 93 (2008), 745.
doi: 10.1111/j.1095-8312.2008.00954.x. |
[39] |
R. M. Luther and M. Broom, Rapid convergence to an equilibrium state in kleptoparasitic populations,, J. Math. Biol., 48 (2004), 325.
doi: 10.1007/s00285-003-0237-3. |
[40] |
R. M. Luther, M. Broom and G. D. Ruxton, Is food worth fighting for? ESS's in mixed populations of kleptoparasites and foragers,, Bull. Math. Biol., 69 (2007), 1121.
doi: 10.1007/s11538-005-9052-x. |
[41] |
U. Luttge, Vascular Plants as Epiphytes,, Springer-Verlag, (1989). Google Scholar |
[42] |
X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces,, Trans. Amer. Math. Soc., 278 (1983), 21.
doi: 10.2307/1999300. |
[43] |
J. Morand-Ferron, G.-M. Wu and L.-A. Giraldeau, Persistent individual differences in tactic use in a producer-scrounger game are group dependent,, Anim. Behav., 82 (2011), 811. Google Scholar |
[44] |
J. Morgan, Boundedness and decay results for reaction-diffusion systems,, SIAM J. Math. Anal., 21 (1990), 1172.
doi: 10.1137/0521064. |
[45] |
K. Mottley and L.-A. Giraldeau, Experimental evidence that group foragers can converge on predicted producer-scrounger equilibria,, Anim. Behav., 60 (2000), 341.
doi: 10.1006/anbe.2000.1474. |
[46] |
Y. Ohtsuka and Y. Toquenaga, The patch distributed producer-scrounger game,, J. Theor. Biol., 260 (2009), 261.
doi: 10.1016/j.jtbi.2009.06.002. |
[47] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[48] |
F. J. Richard, A. Dejean and J. P. Lachaud, Sugary food robbing in ants: A case of temporal cleptobiosis,, C. R. Biologies, 327 (2004), 509.
doi: 10.1016/j.crvi.2004.03.002. |
[49] |
G. D. Ruxton, Foraging in flocks: Non-spatial models may neglect important costs,, Ecol. Model., 82 (1995), 277.
doi: 10.1016/0304-3800(94)00098-3. |
[50] |
S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions,, Math. Ann., 258 (1982), 459.
doi: 10.1007/BF01453979. |
[51] |
N. Tania, B. Vanderlii, J. P. Heath and L. Edelstein-Keshet, Role of social interactions in dynamic patterns of resource patches and forager aggregation,, PNAS, 109 (2012), 11228.
doi: 10.1073/pnas.1201739109. |
[52] |
J. H. M. Thornley and I. R. Johnson, Plant and Crop Modelling - A Mathematical Approach to Plant and Crop Physiology,, The Blackburn Press, (2000). Google Scholar |
[53] |
W. L. Vickery, L.-A. Giraldeau, J. J. Templeton, D. L. Kramer and C. A. Chapman, Producers, scroungers, and group foraging,, Am. Nat., 137 (1991), 847.
doi: 10.1086/285197. |
[54] |
F. Vollrath, Behaviour of the kleptoparasitic spider Argyrodes elevatus (Araneae, Theridiidae),, Anim. Behav., 27 (1979), 519.
doi: 10.1016/0003-3472(79)90186-6. |
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