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Survival and extinction in a locally regulated population. (English) Zbl 1043.92030

Author’s abstract: B. M. Bolker and S. W. Pacala [Theor. Popul. Biol. 52, 179–197 (1997; Zbl 0890.92020)] recently introduced a model of an evolving population in which an individual’s fecundity is reduced in proportion to the “local population density.” We consider two versions of this model and prove complementary extinction/persistence results, one for each version. Roughly, if individuals in the population disperse sufficiently quickly relative to the range of the interaction induced by the density dependent regulation, then the population has positive chance of survival, whereas, if they do not, then the population will die out.

MSC:

92D25 Population dynamics (general)
60J85 Applications of branching processes

Citations:

Zbl 0890.92020
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References:

[1] Barton, N. H., Depaulis, F. and Etheridge, A. M. (2002). Neutral evolution in spatially continuous populations. Theoretical Population Biology 61 31–48. · Zbl 1038.92028
[2] Bolker, B. M. and Pacala, S. W. (1997). Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theoretical Population Biology 52 179–197. · Zbl 0890.92020
[3] Bolker, B. M. and Pacala, S. W. (1999). Spatial moment equations for plant competition: Understanding spatial strategies and the advantages of short dispersal. American Naturalist 153 575–602.
[4] Dawson, D. A. (1993). Measure-valued Markov processes. Ecole d’été de probabilités de Saint Flour XXI. Lecture Notes in Math. 1541 1–260. Springer, New York. · Zbl 0799.60080
[5] Dawson, D. A., Iscoe, I. and Perkins, E. A. (1989). Super-Brownian motion: Path properties and hitting probabilities. Probab. Theory Related Fields 83 135–205. · Zbl 0692.60063
[6] Durrett, R. (1995). Ten lectures on particle systems. Ecole d’été de probabilités de Saint Flour XXIII. Lecture Notes in Math. 1608 97–201. Springer, New York. · Zbl 0840.60088
[7] Durrett, R. and Perkins, E. A. (1999). Rescaled contact processes converge to super-Brownian motion in two or more dimensions. Probab. Theory Related Fields 114 309–399. · Zbl 0953.60093
[8] Etheridge, A. M. (2000). An Introduction to Superprocesses 20 . AMS, Providence, RI. · Zbl 0971.60053
[9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes : Characterization and Convergence . Wiley, New York. · Zbl 0592.60049
[10] Evans, S. N. and Perkins, E. A. (1994). Measure-valued branching diffusions with singular interactions. Canad. J. Math. 46 120–168. · Zbl 0806.60039
[11] Evans, S. N. and Perkins, E. A. (1998). Collision local times, historical stochastic calculus and competing species. Electron. J. Probab. 3 1–120. · Zbl 0899.60081
[12] Feller, W. (1951). Diffusion processes in genetics. Proc. Second Berkeley Symp. Math. Statist. Probab. 227–246. Univ. California Press, Berkeley. · Zbl 0045.09302
[13] Felsenstein, J. (1975). A pain in the torus: Some difficulties with the model of isolation by distance. American Naturalist 109 359–368.
[14] Iscoe, I. (1986). A weighted occupation time for a class of measure-valued critical branching Brownian motions. Probab. Theory Related Fields 71 85–116. · Zbl 0555.60034
[15] Kimura, M. (1953). Stepping stone model of population. Ann. Rep. Nat. Inst. Genetics Japan 3 62–63.
[16] Knight, F. B. (1981). Essentials of Brownian Motion and Diffusion . Amer. Math. Soc., Providence, RI. · Zbl 0458.60002
[17] Law, R., Murrell, D. J. and Dieckmann, U. (2003). On population growth in space and time: Spatial logistic equations. Ecology 84 252–262.
[18] Le Gall, J.-F. (1999). Spatial Branching Processes , Random Snakes and Partial Differential Equations . Birkhäuser, Basel. · Zbl 0938.60003
[19] Mueller, C. and Tribe, R. (1994). A phase-transition for a stochastic pde related to the contact process. Probab. Theory Related Fields 100 131–156. · Zbl 0809.60072
[20] Perkins, E. A. (2002). Dawson–Watanabe superprocesses and measure-valued diffusions. Ecole d’été de probabilités de Saint Flour. Lecture Notes in Math. 1781 125–329. Springer, New York. · Zbl 1020.60075
[21] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion . Springer, New York. · Zbl 0917.60006
[22] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. Ecole d’été de probabilités de Saint Flour. Lecture Notes in Math. 1180 265–439. Springer, New York. · Zbl 0608.60060
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