Abstract
Positive density-dependence occurs when individuals experience increased survivorship, growth, or reproduction with increased population densities. Mechanisms leading to these positive relationships include mate limitation, saturating predation risk, and cooperative breeding and foraging. Individuals within these populations may differ in age, size, or geographic location and thereby structure these populations. Here, I study structured population models accounting for positive density-dependence and environmental stochasticity i.e. random fluctuations in the demographic rates of the population. Under an accessibility assumption (roughly, stochastic fluctuations can lead to populations getting small and large), these models are shown to exhibit a dynamical trichotomy: (i) for all initial conditions, the population goes asymptotically extinct with probability one, (ii) for all positive initial conditions, the population persists and asymptotically exhibits unbounded growth, and (iii) for all positive initial conditions, there is a positive probability of asymptotic extinction and a complementary positive probability of unbounded growth. The main results are illustrated with applications to spatially structured populations with an Allee effect and age-structured populations experiencing mate limitation.
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References
Arnold, L., Gundlach, V.M., Demetrius, L.: Evolutionary formalism for products of positive random matrices. Ann. Appl. Probab. 4, 859–901 (1994)
Assas, L., Dennis, B., Elaydi, S., Kwessi, E., Livadiotis, G.: A stochastic modified beverton-holt model with the allee effect. J. Differ. Equ. Appl. 22(1), 37–54 (2016)
Benaïm, M., Schreiber, S.J.: Persistence of structured populations in random environments. Theor. Popul. Biol. 76, 19–34 (2009)
Caswell, H.: Matrix Population Models. Sinauer, Sunderland, Massachuesetts (2001)
Courchamp, F., Berec, L., Gascoigne, J.: Allee effects in ecology and conservation. Environ. Conserv. 36(1), 80–85 (2008)
Courchamp, F., Clutton-Brock, T., Grenfell, B.: Inverse density dependence and the Allee effect. Trends Ecol. Evol. 14, 405–410 (1999)
Dennis, B.: Allee effects: Population growth, critical density, and the chance of extinction. Natural Resour. Model. 3, 481–538 (1989)
Dennis, B.: Allee effects in stochastic populations. Oikos 96, 389–401 (2002)
Dennis, B., Assas, L., Elaydi, S., Kwessi, E., Livadiotis, G.: Allee effects and resilience in stochastic populations. Theor. Ecol. 1–13 (2015)
Evans, S.N., Ralph, P., Schreiber, S.J., Sen, A.: Stochastic growth rates in spatio-temporal heterogeneous environments. J. Math. Biol. 66, 423–476 (2013)
Gascoigne, J., Lipcius, R.N.: Periodic dynamics in a two-stage allee effect model are driven by tension between stage equilibria. Theor. Popul. Biol. 68(4), 237–241 (2005)
Gascoigne, J.C., Lipcius, R.N.: Allee effects driven by predation. J. Appl. Ecol. 41, 801–810 (2004)
Hardin, D.P., Takáč, P., Webb, G.F.: Asymptotic properties of a continuous-space discrete-time population model in a random environment. J. Math. Biol. 26, 361–374 (1988)
Hening, A., Nguyen, D.H., Yin, G.: Stochastic population growth in spatially heterogeneous environments: The density-dependent case. arXiv:1605.02027 (arXiv preprint) (2016)
Jansen, V.A.A., Yoshimura, J.: Populations can persist in an environment consisting of sink habitats only. Proc. Natl. Acad. Sci. USA 95, 3696–3698 (1998)
Kingman, J.F.C.: Subaddit. Ergod. Theory. Ann. Prob. 1, 883–909 (1973)
Liebhold, A., Bascompte, J.: The Allee effect, stochastic dynamics and the eradication of alien species. Ecol. Lett. 133–140 (2003)
McCarthy, M.A.: The Allee effect, finding mates and theoretical models. Ecol. Model. 103, 99–102 (1997)
Metz, J.A.J., de Jong, T.J., Klinkhamer, P.G.L.: What are the advantages of dispersing; a paper by Kuno extended. Oecologia 57, 166–169 (1983)
Roth, G., Schreiber, S.J.: Persistence in fluctuating environments for interacting structured populations. J. Math. Biol. 68, 1267–1317 (2014)
Roth, G., Schreiber, S.J.: Pushed to brink: Allee effects, environmental stochasticity, and extinction, special issue on allee effects for. J. Biol. Dyn. 8, 187–205 (2014)
Scheuring, I.: Allee effect increases dynamical stability in populations. J. Theor. Biol. 199, 407–414 (1999)
Schreiber, S.J.: Allee effects, chaotic transients, and unexpected extinctions. Theor. Popul. Biol. (2003)
Schreiber, S.J.: On Allee effects in structured populations. Proc. Amer. Math. Soc. 132(10):3047–3053 (electronic) (2004)
Schreiber, S.J.: Interactive effects of temporal correlations, spatial heterogeneity, and dispersal on population persistence. Proc. Royal Soc. Biol. Sci. 277, 1907–1914 (2010)
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Schreiber, S.J. (2017). A Dynamical Trichotomy for Structured Populations Experiencing Positive Density-Dependence in Stochastic Environments. In: Elaydi, S., Hamaya, Y., Matsunaga, H., Pötzsche, C. (eds) Advances in Difference Equations and Discrete Dynamical Systems. ICDEA 2016. Springer Proceedings in Mathematics & Statistics, vol 212. Springer, Singapore. https://doi.org/10.1007/978-981-10-6409-8_3
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