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Logistics

Logistics. Logistic Equations. Logistics equations are meant to model the growth or decay of a population or species with a certain maximum or minimum population, that acts as an asymptote

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Logistics

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  1. Logistics

  2. Logistic Equations • Logistics equations are meant to model the growth or decay of a population or species with a certain maximum or minimum population, that acts as an asymptote • Populations often have a maximum capacity that could be limited by things such as food supplies, habitat, or other factors of nature.

  3. Equation Model • M=carrying capacity • is a graph with an asymptote at y=M and an inflection point at y=

  4. Things to consider • M is the carrying capacity and the asymptote • is the maximum growth or decay rate • must be in the form

  5. Examples •   Change the given differential calculus equation into calculus logistic equation. The given equation is • Solution- factor out 3P • Equation is now in logistic form. You win

  6. More Winning • A state game commission releases 40 elk into a game refuge. After five years, the elk population is 104. The commission believes that the environment can support no more than 4000 elk. The growth rate of the elk population “P” is • And P is between 40 and 4000 elk • Questions: What is the carrying capacity? • When will the population be growing the fastest?

  7. Solution • M=4000 therefore the carrying capacity is 4000 elk • The population will be growing fastest when

  8. Examples…with piranha! • In a pond that can hold a maximum of 1000 fish, I started one population off at 300 piranha, and the other with 301 --  I meant them to be the same, but later I discovered that I had miscounted.  In any event, on day 14, the pond that started with 301 fish was close to extinction, while the one that started with fewer fish was holding its own with about 300 fish. If the populations start off similarly-sized, doesn't it just stand to reason that they will more-or-less parallel each other in their population growth and decline patterns?

  9. NO • Populations grow and decline in a chaotic nature and cannot be predicted simply by an initial condition. But, it can be modeled by logistic calculus. • Carrying Capacity=M=1000 piranha • Given k value=4 • Even with a tiny change in initial population (y), the rate of change of the population is different and continues to effect the system.

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