Set the right side of the differential equation equal to zero N e r a N e b 2 0

Set the right side of the differential equation equal

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Set the right side of the differential equation equal to zero: N e ( r - a ( N e - b ) 2 ) = 0 One solution is the trivial or extinction equilibrium, N e = 0 When ( r - a ( N e - b ) 2 ) = 0, then ( N e - b ) 2 = r a or N e = b ± r r a Three distinct equilibria unless r = 0 or b = p r/a With the parameters r = 0 . 04, a = 10 - 8 , and b = 2200, the equilibria are N e = 0 N e = 200 4200 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (46/50)
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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Allee Effect 6 Phase Portrait : Graph of right hand side of differential equation showing equilibria and their stability 0 1000 2000 3000 4000 5000 -50 0 50 100 > > > < N dN/dt 0 100 200 300 -0.5 0 0.5 1 > < < > N dN/dt (zoom near origin) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (47/50)
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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Allee Effect 7 Solutions : For dN dt = N ( r - a ( N - b ) 2 ) Allee Effect 0 1000 2000 3000 4000 5000 N(t) 0 20 40 60 80 100 t Allee Effect (zoom near origin) –200 –100 0 100 200 300 N(t) 0 200 400 600 800 t Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (48/50)
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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Allee Effect 8 Interpretation : Model of Allee Effect From the phase portrait, the equilibria at 4200 and 0 are stable The threshold equilibrium at 200 is unstable If the population is above 200, it approaches the carrying capacity of this region with the stable population of 4200 If the population falls below 200, the model predicts extinction , N e = 0 This agrees with the description for these social birds, which require a critical number of birds to avoid predation Below this critical number, the predation increases above reproduction, and the population of parrots goes to extinction If the parrot population is larger than 4200, then their numbers will be reduced by starvation (and predation) to the carrying capacity, N e = 4200 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (49/50)
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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Maple Commands for Direction Fields with ( DEtools ): de := diff ( P ( t ) , t ) = 0 . 05 · P ( t ) · ( 1 - 1 2000 P ( t ) ) ; DEplot ( de , P ( t ), t = 0 .. 100, P = 0 .. 2500, [[ P (0) = 0] , [ P (0) = 100] , [ P (0) = 2000] , [ P (0) = 2500]], color = blue, linecolor = t ); Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (50/50)
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