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Dynamic behaviors of the Ricker population model under a set of randomized perturbations. (English) Zbl 0952.92025
Summary: We studied the dynamics of the Ricker population model under perturbations by the discrete random variable $$\varepsilon$$ which follows distribution $$P\{\varepsilon= a_i\}=p_i$$, $$i=1,\dots,n$$, $$0<a_i\ll 1$$, $$n\geq 1$$. Under the perturbations, $$n+1$$ blurred orbits appeared in the bifurcation diagram. Each of the $$n+1$$ blurred orbits consisted of $$n$$ sub-orbits. The asymptotes of the $$n$$ sub-orbits in one of the $$n+1$$ blurred orbits were $$N_t= a_i$$ for $$i=1,\dots,n$$. For other $$n$$ blurred orbits, the asymptotes of the $$n$$ sub-orbits were $$N_t=a_i \exp[r(1-a_i)]+a_j$$, $$j=1,2,\dots,n$$, for $$i=1, \dots, n$$, respectively.
The effects of variances of the random variable $$\varepsilon$$ on the bifurcation diagrams were examined. As the variance value increased, the bifurcation diagram became more blurred. Perturbation effects of the approximate continuous uniform random variable and random error were compared. The effects of the two perturbations on dynamics of the Ricker model were similar, but with differences. Under different perturbations, the attracting equilibrium points and two-cycle periods in the Ricker model were relatively stable. However, some dynamic properties, such as the periodic windows and the $$n$$-cycle periods $$(n>4)$$, could not be observed even when the variance of a perturbation variable was very small. The process of reversal of the period-doubling, an important feature of the Ricker and other population models observed under constant perturbations, was relatively unstable under random perturbations.

##### MSC:
 92D25 Population dynamics (general) 37N25 Dynamical systems in biology 37H99 Random dynamical systems
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##### References:
 [1] Berryman, A.A.; Millstein, J.A., Are ecological systems chaotic – and if not, why not?, Trends in ecol. evol., 4, 26, (1989) [2] Turchin, P.; Taylor, A.D., Complex dynamics in ecological time series, Ecology, 73, 289, (1992) [3] Ellner, S.; Turchin, P., Chaos in a noisy world: new methods and evidence for time series analysis, Am. natural., 145, 343, (1995) [4] Higgins, K.; Hastings, A.; Sarvela, J.N.; Botsford, L.W., Stochastic dynamics and deterministic skeletons: population behavior of dungeness crab, Science, 276, 1431, (1997) · Zbl 1225.60113 [5] Pimm, S.L., The complexity and stability of ecosystems, Nature, 307, 321, (1984) [6] Scheuring, I.; Janosi, I.M., When two and two make four: a structured population without chaos, J. theoret. biol., 178, 89, (1996) [7] Stone, L., Coloured noise or low-dimensional chaos?, Proc. roy. soc. London B, 250, 77, (1992) [8] Sun, P.; Yang, X.B., Two properties of a gene-for-gene coevolution system under human perturbations, Phytopathology, 89, 811, (1999) [9] McCallum, H.I., Effects of immigration on population dynamics, J. theoret. biol., 154, 277, (1992) [10] Stone, L., Period-doubling reversals and chaos in simple ecological models, Nature, 36, 617, (1993) [11] Rohani, P.; Miramontes, O.; Hassell, M.P., Quasiperiodicity and chaos in population models, Proc. roy. soc. London B, 258, 17, (1994) [12] Rohani, P.; Miramontes, O., Immigration and the persistence of chaos in population models, J. theoret. biol., 175, 203, (1995) [13] Doebeli, M., Dispersal and dynamics, Theoret. populat. biol., 47, 82, (1995) · Zbl 0814.92015 [14] Sun, P.; Yang, X.B., Deterministic property changes in population models under random error perturbations, Ecological modelling, 119, 239, (1999) [15] Laha, R.G.; Rohatgi, V.K., Probability theory, (1979), Wiley New York [16] Serfling, R.J., Approximation theorems of mathematical statistics, (1980), Wiley New York · Zbl 0456.60027 [17] Drake, V.A.; Gatehouse, A.G., Insect migration: tracking resources through space and time, (1995), Cambridge University Cambridge [18] Wiktelius, S., Long-range migration of aphids into Sweden, Int. J. biometerol., 28, 200, (1984) [19] P. Sun, The studies of aphid population dynamics and its control in the fields of winter wheat in Hebei Province, PhD Thesis, Peking University, 1991 [20] Yang, X.B.; Zeng, S.M., Detecting patterns of wheat stripe rust pandemics in time and space, Phytopathology, 82, 571, (1992) [21] Zeng, S.M., Pancrin a prototype model of the pandemic cultivar-race interaction of yellow rust of wheat in China, Plant pathol., 40, 287, (1991) [22] Beaumont, G.P., Probability and random variables, (1986), Halstte New York · Zbl 0588.60002 [23] Cruthfield, J.P.; Farmer, J.D.; Huberman, B.A., Fluctuations and simple chaotic dynamics, Phys. rep., 92, 45, (1982) [24] M Schaffer, W.; Ellner, S.; Kot, M., Effects of noise on some dynamical models in ecology, J. math. biol., 24, 479, (1986) · Zbl 0626.92021 [25] Ripa, J.; Lundberg, P., Noise colour and the risk of population extinctions, Proc. roy. soc. London B, 263, 1751, (1996) [26] Rossler, J.; Kiwi, M.; Hess, B.; Markus, M., Modulated non-linear and a novel mechanism to induce chaos, Phys. rev. A, 39, 5954, (1989) [27] Sun, P.; Yang, X.B., The stability of gene-for-gene coevolution model under constant perturbations, Phytopathology, 88, 592, (1998) [28] Barnsley, M., Fractals everywhere, (1993), Academic Press Boston · Zbl 0691.58001 [29] May, R.M., Detecting density dependence in imaginary world, Nature, 338, 16, (1989) [30] Jin, Z., A brief introduction to chaos, Math. prac. theor., 12, 81, (1991) [31] Levin, S.A.; Grenfell, B.; Hastings, A.; Perelson, A.S., Mathematical and computational challenges in population biology and ecosystems science, Science, 275, 334, (1997) · Zbl 1225.92058
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