A population subject to the Allee effect will decrease when population numbers are either very low...

A population subject to the Allee effect will decrease when population numbers are either very low (due to inability to find a mate) or very high (due to overcrowding), but will increase when population numbers are at intermediate levels (due to successful reproduction and not too much overcrowding). A model for a population subject to an Allee effect is given by

x_t+1= (4x²_t)/(3+x²_t), x_t is greater than and equal to 0

where ?_t is the population number in year t (measured in hundreds of individuals.

(a) Determine the equilibria.

(b) Use the Stability Criterion to classify the equilibria as asymptotically stable or unstable.

(c) Use cobwebbing to illustrate the dynamics of the difference equation for ?₀=0.5, ?₀=0.5, ?₀=1.5 and ?₀=4(it is OK to plot all three on the same diagram).

(d) Explain the significance of all equilibria and their stability in terms of the population size in the long run.

Expert Solution

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Colby Messinger answered 2 years ago

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