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Extinction conditions for isolated populations with Allee effect. (English) Zbl 1217.92080
Summary: One of the main ecological phenomenons is the Allee effect [see {\it W. C. Allee}, The social life of animals. London: W. Heinemann (1938)], in which a positive benefit from the presence of conspecifics arises. We describe the dynamical behavior of a population with Allee effect in a finite domain that is surrounded by a completely hostile environment. Using spectral methods to rewrite the local density of habitants we are able to determine the critical patch size and the bifurcation diagram, hence characterizing the stability of possible solutions, for different ways to introduce the Allee effect in the reaction-diffusion equations.
92D50Animal behavior
37N25Dynamical systems in biology
35Q92PDEs in connection with biology and other natural sciences
Full Text: DOI
[1] Allee, W. C.: Animal aggregations, A study in general sociology, (1931)
[2] Allee, W. C.: The social life of animals, (1938)
[3] Allee, W. C.; Emerson, A. E.; Park, O.; Park, T.; Schmidt, K. P.: Principles of animal ecology, (1949)
[4] Stephens, P. A.; Sutherland, W. J.; Freckleton, R. P.: What is the allee effect?, Oikos 87, 185 (1999)
[5] Berec, L.; Angulo, E.; Courchamp, F.: Multiple allee effects and population management, Tree 22, 185 (2007)
[6] Boukal, D. S.; Berec, L.: Single-species models of the allee effect extinction boundaries, sex ratios and mate encounters, J. theor. Biol. 218, 375 (2002)
[7] Fowler, M. S.; Ruxton, G. D.: Population dynamic consequences on allee effects, J. theor. Biol. 215, 39 (2002)
[8] Taylor, C. M.; Hastings, A.: Allee effects in biological invasions, Ecol. lett. 8, 895 (2005)
[9] Courchamp, F.; Clutton-Brock, T.; Grenfell, B.: Inverse density dependence and the allee effect, Tree 14, 405 (1999)
[10] Stephens, P. A.; Sutherland, W. J.: Consequences of the allee effect for behaviours, ecology and conservation, Tree 14, 401 (1999)
[11] Courchamp, F.; Berec, L.; Gascoigne, J.: Allee effects in ecology and conservation, (2008)
[12] Wang, M.; Kot, M.: Speeds of invasion in a model with strong or weak allee effects, Math. biol. 171, 83 (2001) · Zbl 0978.92033 · doi:10.1016/S0025-5564(01)00048-7
[13] Lewis, M. A.; Kareiva, P.: Allee dynamics and the spread of invading organisms, Theor. popul. Biol. 43, 141 (1993) · Zbl 0769.92025 · doi:10.1006/tpbi.1993.1007
[14] Dennis, B.: Allee effects: population growth, critical density, and the chance of extinction, Nat. res. Mod. 3, 481 (1989) · Zbl 0850.92062
[15] Skellam, J. G.: Random dispersal in theoretical populations, Biometrika 38, 196 (1951) · Zbl 0043.14401
[16] Bradford, E.; Philip, J. P.: Stability of steady distributions of asocial populations dispersing in one dimension, J. theor. Biol. 29, 13 (1970)
[17] Méndez, V.; Campos, D.: Population extinction and survival in a hostile environment, Phys. rev. E 77, 022901 (2008)
[18] Kierstead, H.; Slobodkin, L. B.: The size of water masses containing plankton blooms, J. mar. Res. 12, 141 (1953)
[19] Martin, A. P.: On filament width in oceanic plankton distributions, J. plankton res. 22, 597 (2000)
[20] Malvadkar, U.; Hastings, A.: Persistence of mobile species in marine protected areas, Fish. res. 91, 69 (2008)
[21] Latore, J.; Gould, P.; Mortimer, A. M.: Spatial dynamics and critical patch size of annual plant populations, J. theor. Biol. 190, 277 (1998)
[22] Shnerb, N. M.: Extinction of a bacterial colony under forced convection in pie geometry, Phys. rev. E 63, 011906 (2000)
[23] Smoller, J.; Wasserman, A.: Global bifurcation of steady-state solutions, J. differ. Eqn. 39, 269 (1981) · Zbl 0425.34028 · doi:10.1016/0022-0396(81)90077-2
[24] Britton, N. F.: Reaction -- diffusion equations and their applications to biology, (1986) · Zbl 0602.92001
[25] Cantrell, R. S.; Cosner, C.: Spatial ecology via reaction -- diffusion equations, (2003) · Zbl 1059.92051
[26] Shi, J.; Shivaji, R.: Persistence in reaction -- diffusion models with weak allee effect, J. math. Biol. 52, 807 (2006) · Zbl 1110.92055 · doi:10.1007/s00285-006-0373-7
[27] Lee, Y. H.; Sherbakov, L.; Taber, J.; Shi, J.: Bifurcation diagrams of population models with nonlinear diffusion, J. comput. Appl. math. 194, 357 (2006) · Zbl 1122.35309 · doi:10.1016/j.cam.2005.08.004
[28] Chesson, P. L.: Models for spatially distributed populations: the effect of within-patch variability, Theor. popul. Biol. 19, 288 (1981) · Zbl 0472.92015 · doi:10.1016/0040-5809(81)90023-X
[29] Chesson, P. L.: Scale transition theory with special reference to species coexistence in a variable environment, J. biol. Dyn. 3, 149 (2009)
[30] Hesthaven, J. S.; Gottlieb, S.; Gottlieb, D.: Spectral methods for time-dependent problems, (2007) · Zbl 1111.65093
[31] Edelstein-Keshet, L.: Mathematical models in biology, (2005) · Zbl 1100.92001
[32] Conrad, J. M.; Clark, C. W.: Natural resource economics: notes and problems, (1987)
[33] Clark, C. W.: Mathematical bioeconomic: the optimal management of renewable resources, (1990) · Zbl 0712.90018
[34] Méndez, V.; Llebot, J. E.: Hyperbolic reaction -- diffusion equations for a forest fire model, Phys. rev. E 56, 6557 (1997)
[35] Brunet, E.; Derrida, B.: Shift in the velocity of a front due to a cutoff, Phys. rev. E 56, 2597 (1997)
[36] Lin, J.; Andreasen, V.; Casagrandi, R.; Levin, S. A.: Traveling waves in a model of influenza A drift, J. theor. Biol. 222, 437 (2003)
[37] Lutscher, F.; Mccauley, E.; Lewis, M. A.: Spatial patterns and coexistence mechanisms in systems with unidirectional flow, Theor. popul. Biol. 71, 267 (2007) · Zbl 1124.92050 · doi:10.1016/j.tpb.2006.11.006