Next generations, equilibria, stable, oscillating, extinction, carrying capacity

Question

Next generations, equilibria, stable, oscillating, extinction, carrying capacity, etc.

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If we start with an initial population of 9,000, then does this population of birds go to Extinction or Carrying Capacity?

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Think this means ecologically from a conservation perspective for a species that has this allee effect.

18. (1 point) The San Diego Zoo discovered that because their flamingo population was too small, it would not reproduce until they bor- rowed some from Sea World. Scientists have discovered that certain gregarious animals require a minimum number of an imals in a colony before they reproduce successfully. This is called the Allee effect. Consider the following model for the population of a gregarious bird species, where the population, Nn, is given in thousands of birds: a. Assume that the initial population is No 5, then deter- mine the population for the next two generations (N1 and Na)

Explanation / Answer

Given

a) Nn+1 = A(Nn) = Nn + 0.16N (1-1/9(Nn -y)2)

now let n = 0

N1 = A(N0) = N0 + 0.16N0 (1 -1/9 (N0 -7)^2)

=> N1 = 5 + 0.8(1 -1/9(4) = 5+ 0.8(5/9) = 5+4/9 = 5.44

now N2 = A(N1) = N1 + 0.16N1 ( 1 - 1/9(N1 -7)2) = 5.44 + 0.87 ( 1- 1/9(2.43)) = 6.08

b) Now we know A(Nn) = Nn + 0.16Nn ( 1 - 1/9(Nn -7)2) = Nn + 0.16Nn (( 9 - Nn2 -49 +14Nn)/9)

= N +(-6.4N - 0.16N3 +2.24N2) / 9

now A'(N) = 1 -6.4/9 - 0.16/3 N2 + 4.48/9 N = 0

9 - 6.4 - 0.48N2+ 4.48N /9 = 0

=> 0.48N2 - 4.48N - 2.6 = 0

=> N2 - 9.33N - 5.42 = 0

=> N = -0.55 ,9.88

N1e = 0

N2e = -0.55

N3e = 9.88

c) A(N) =  N + 0.16N ( 1 - 1/9(N -7)2)

= N+ 0.16N(( 9 - N2 -49 +14N)/9)

= N +(-6.4N - 0.16N3 +2.24N2) / 9

now A'(N) = 1 -6.4/9 - 0.16/3 N2 + 4.48/9 N

d) A'(N1e) = A'(0) = 1 - 6.4/9 - 0.16/3 (0)2 + 4.48/9(0) = -1<0.29<1 stable

A'(N2e) = A'(-0.55) = 1 - 6.4/9 - 0.16/3 (-0.55)2 + 4.48/9 (-0.55) = -1<0.001<1 Stable

A'(N3e) = A'(9.88) = 1 -6.4/9 - 0.16/3(9.88)2 + 4.48/9 (9.88) = -1<0.008<1 Stable

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