Persistence in fluctuating environments
Tóm tắt
Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations’ invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka–Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.
One day is fine, the next is black.—The Clash
Tài liệu tham khảo
Abrams PA, Holt RD, Roth JD (1998) Apparent competition or apparent mutualism? Shared predation when populations cycle. Ecology 79(1): 201–212
Benaïm M, Hofbauer J, Sandholm W (2008) Robust permanence and impermanence for the stochastic replicator dynamics. J Biol Dyn 2: 180–195
Bjornstad ON, Grenfell BT (2001) Noisy clockwork: time series analysis of population fluctuations in animals. Science 293(5530): 638
Butler GJ, Waltman P (1986) Persistence in dynamical systems. J Differ Equ 63: 255–263
Chesson PL (1978) Predator–prey theory and variability. Annu Rev Ecol Syst 9: 323–347
Chesson PL (1982) The stabilizing effect of a random environment. J Math Biol 15(1): 1–36
Chesson PL (1994) Multispecies competition in variable environments. Theor Popul Biol 45(3): 227–276
Chesson PL, Ellner S (1989) Invasibility and stochastic boundedness in monotonic competition models. J Math Biol 27: 117–138
Chesson P, Kuang JJ (2008) The interaction between predation and competition. Nature 456(7219): 235–238
Chesson PL, Warner RR (1981) Environmental variability promotes coexistence in lottery competitive systems. Am Nat 117(6): 923
Durrett R (1996) Stochastic calculus. Probability and stochastics series. CRC Press, Boca Raton
Ellner SP (1984) Asymptotic behavior of some stochastic difference equation population models. J Math Biol 19: 169–200
Ellner S (1989) Convergence to stationary distributions in two-species stochastic competition models. J Math Biol 27(4): 451–462
Ellner S, Sasaki A (1996) Patterns of genetic polymorphism maintained by fluctuating selection with overlapping generations. Theor Popul Biol 50: 31–65
Fudenberg D, Harris C (1992) Evolutionary dynamics with aggregate shocks. J Econ Theory 57: 420–441
Garay BM, Hofbauer J (2003) Robust permanence for ecological differential equations, minimax, and discretizations. SIAM J Math Anal 34: 1007–1039
Gard TC (1984) Persistence in stochastic food web models. Bull Math Biol 46(3): 357–370
Gause GF (1934) The struggle for existence. Williams and Wilkins, Baltimore
Gillespie JH (1973) Polymorphism in random environments. Theor Popul Biol 4: 193–195
Gillespie JH, Guess HA (1978) The effects of environmental autocorrelations on the progress of selection in a random environment. Am Nat 112: 897–909
Gyllenberg M, Hognas G, Koski T (1994a) Null recurrence in a stochastic Ricker model. In: Analysis, algebra, and computers in mathematical research (Lulea, 1992). Lecture Notes in Pure and Applied Mathematics. Decker, New York, pp 147–164
Gyllenberg M, Hognas G, Koski T (1994b) Population models with environmental stochasticity. J Math Biol 32: 93–108
Hofbauer J (1981) A general cooperation theorem for hypercycles. Monatsh Math 91: 233–240
Hofbauer J, Schreiber SJ (2004) To persist or not to persist?. Nonlinearity 17: 1393–1406
Hofbauer J, Schreiber SJ (2010) Robust permanence for interacting structured populations. J Differ Equ 248: 1955–1971
Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge
Hofbauer J, So JWH (1989) Uniform persistence and repellors for maps. Proc Am Math Soc 107: 1137–1142
Hofbauer J, Hutson V, Jansen W (1987) Coexistence for systems governed by difference equations of Lotka-Volterra type. J Math Biol 25(5): 553–570
Holt RD (1977) Predation, apparent competition and the structure of prey communities. Theor Popul Biol 12: 197–229
Holt RD, Grover J, Tilman D (1994) Simple rules for interspecific dominance in systems with exploitative and apparent competition. Am Nat 144: 741–771
Hutson V (1984) A theorem on average Liapunov functions. Monatsh Math 98: 267–275
Hutson V, Schmitt K (1992) Permanence and the dynamics of biological systems. Math Biosci 111: 1–71
Jansen VAA, Sigmund K (1998) Shaken not stirred: on permanence in ecological communities. Theor Popul Biol 54: 195–201
Kimura M (1964) Diffusion models in population genetics. J Appl Probab 1: 177–232
Kuang JJ, Chesson P (2008) Predation-competition interactions for seasonally recruiting species. Am Nat 171: 119–133
Kuang JJ, Chesson P (2009) Coexistence of annual plants: generalist seed predation weakens the storage effect. Ecology 90: 170–182
Mañé R (1983) Ergodic theory and differentiable dynamics. Springer-Verlag, New York
May RM, Leonard W (1975) Nonlinear aspects of competition between three species. SIAM J Appl Math 29: 243–252
Meyn SP, Tweedie RL (1993) Markov Chains and stochastic stability. Springer, New York
Paine RT (1966) Food web complexity and species diversity. Am Nat 100: 65–75
Schreiber SJ (2000) Criteria for C r robust permanence. J Differ Equ 162: 400–426
Schreiber SJ (2006) Persistence despite perturbations for interacting populations. J Theor Biol 242: 844–852
Schreiber SJ (2007) On persistence and extinction of randomly perturbed dynamical systems. Discrete Contin Dyn Syst B 7: 457–463
Schuster P, Sigmund K, Wolff R (1979) Dynamical systems under constant organization 3:Cooperative and competitive behavior of hypercycles. J Differ Equ 32: 357–368
Simmons S (1998) Minimax and monotonicity. Springer-Verlag, Berlin
Turelli M (1978) Random environments and stochastic calculus. Theor Popul Biol 12: 140–178
Turelli M (1981) Niche overlap and invasion of competitors in random environments I. Models without demographic stochasticity. Theor Popul Biol 20: 1–56