This is what is observed in nature suggesting that this model exhibits the

This is what is observed in nature suggesting that

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This is what is observed in nature suggesting that this model exhibits the behavior of the evolution of snails This does not mean that the model is a good model! It simply means that the model exhibits the basic behavior observed experimentally from the biological experiments Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Portraits - 1D — (41/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Allee Effect 1 Thick-Billed Parrot : Rhynchopsitta pachyrhycha Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Port — (42/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Allee Effect 2 Thick-Billed Parrot : Rhynchopsitta pachyrhycha A gregarious montane bird that feeds largely on conifer seeds, using its large beak to break open pine cones for the seeds These birds used to fly in huge flocks in the mountainous regions of Mexico and Southwestern U. S. Largely because of habitat loss, these birds have lost much of their original range and have dropped to only about 1500 breeding pairs in a few large colonies in the mountains of Mexico The pressures to log their habitat puts this population at extreme risk for extinction Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Portraits - 1D — (43/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Allee Effect 3 Thick-Billed Parrot : Rhynchopsitta pachyrhycha The populations of these birds appear to exhibit a property known in ecology as the Allee effect These parrots congregate in large social groups for almost all of their activities The large group allows the birds many more eyes to watch out for predators When the population drops below a certain number, then these birds become easy targets for predators, primarily hawks, which adversely affects their ability to sustain a breeding colony Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Port — (44/50)
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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 4 Allee Effect : Suppose that a population study on thick-billed parrots in a particular region finds that the population, N ( t ), of the parrots satisfies the differential equation: dN dt = N ( r - a ( N - b ) 2 ) , where r = 0 . 04, a = 10 - 8 , and b = 2200 Find the equilibria for this differential equation Determine the stability of the equilibria Draw a phase portrait for the behavior of this model Describe what happens to various starting populations of the parrots as predicted by this model Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Portraits - 1D
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