The Stochastic Beverton-Holt Equation and the M. Neubert Conjecture |
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Authors: | Email author" target="_blank">Cymra?HaskellEmail author Robert?J?Sacker |
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Institution: | (1) Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA |
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Abstract: | In the Beverton-Holt difference equation of population biology with intrinsic growth parameter above its critical value, any
initial non-zero population will approach an asymptotically stable fixed point, the carrying capacity of the environment.
When this carrying capacity is allowed to vary periodically it is known that there is a globally asymptotically stable periodic
solution and the average of the state variable along this solution is strictly less than the average of the carrying capacities,
i.e. the varying environment has a deleterious effect on the state average. In this work we consider the case of a randomly
varying environment and show that there is a unique invariant density to which all other density distributions on the state
variable converge. Further, for every initial non-zero state variable and almost all random sequences of carrying capacities,
the averages of the state variable along an orbit and the carrying capacities exist and the former is strictly less than the
latter.
2000 MSC: 37H10; 39A11; 92D25. |
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Keywords: | Population biology skew-product dynamical system stochastic difference equation |
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