3.2 Nonlinear Models
2/9/2011
MTH 225 Differential Equations
Notes 3.2 Nonlinear Applications
P. Seeburger
A more accurate model for many populations is given by the Logistic Population Model. This model assumes there is some maximum carrying capacity (population) of the environment, M. The logistic DE (or the logistic equation) arises from the statement:
The rate of change in the population varies jointly as the size of the population and the difference between the maximum carrying capacity M and the actual size of the population.
This gives us the DE:
that can be rewritten as: If P < M, If P is small compared to M, If P > M,
If P is close to M,
In the text, they use r = k and K = M, and they write the DE: Then they choose to make a = r and b = to rewrite the DE as: Solve: 1
3.2 Nonlinear Models
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3.2 Nonlinear Models
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Ex. 1: A pond is stocked with 100 goldfish. The carrying capacity of the pond is determined to be 1000 fish. In five weeks, the population is 200 fish. Determine the Logistic DE and Logistic Population Model for the goldfish in this pond.
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3.2 Nonlinear Models
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Ex. 2: Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. If it is assumed that the rate at which the virus spreads is proportional only to the number x of infected students and also the number of students not infected, determine the number of students who are infected after 6 days if there were 50 infected students after 4 days.
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3.2 Nonlinear Models
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2nd­Order Chemical Reaction: a grams of chemical A are mixed with b grams of chemical B. If the chemical product formed (chemical C) is made up of M parts of chemical A and N parts of chemical B, and X(t) is the number of grams of chemical C formed at time t,
Then the number of grams of each chemical (A and B) remaining at time t is:
Chemical A:
Chemical B:
The law of mass action states that when no temperature change is involved, the rate at which the new substance is formed is proportional to the product of the amount of A and B that are untransformed (remaining) at time t. That is:
Factoring out from the first term and from the second term yields:
This reaction is called a 2nd­order chemical reaction since it involves a 2nd­degree function of the dependent variable X.
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3.2 Nonlinear Models
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Ex. 3: Two chemicals A and B combine in the ratio 1:2 (by weight) to form a third substance C. The rate of reaction is proportional to the product of the instantaneous remaining amounts of the reactants. It is observed that 10 minutes after 10 grams of substance A are mixed with 20 grams of substance B, 5 grams of substance C has been formed.
a. Find an expression for the amount of the product (substance C) present at time t.
b. How long will it take for one­half the final amount of the product to form?
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3.2 Nonlinear Models
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Other nonlinear DE application examples:
Ex. 4: Snow starts to fall in the forenoon and falls at a constant rate r (ft/hr) all day. At noon a snowplow starts to clear a highway. The velocity of the plow is such that it removes a constant volume v of snow per unit of time (ft3/hr). The plow goes 2 miles during the first hour and 1 mile during the second hour. What time did it start to snow?
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3.2 Nonlinear Models
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Ex. 5: A person P, starting at the origin, moves in the direction of the positive x­axis, pulling a weight along the curve C, called a tractrix. The weight, initially located on the y­axis at (0, s), is pulled by a rope of constant length s, which is kept taut throughout the motion. Assume that the rope is always tangent to C.
a. Determine a differential equation for the path of motion.
b. Solve the DE. 8