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5. (5 pts) In 1798, Thomas Robert Malthus introduced a mathematical principle for the growth...

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5. (5 pts) In 1798, Thomas Robert Malthus introduced a mathematical principle for the growth of populations. The Mathusian Model assumes that the rate of growth in a population is proportional to the number of individuals in the population, that is: N'(1) = kN(t) where N is the population size, t time in the positive direction and k a growth/decline constant. A flaw of this model is that growth, k >0, is unbounded as time extends. The model was improved by Pierre Francois Verhulst in 1845 to include a carrying capacity for the population. In modern times, Verhulst's model is commonly called the Logistic Model for population growth: N'() = rN(t) (1- M) N'(t) = rN(t) (1- K where K is the carrying capacity of the population and r is a growth/decline constant. A study of livestock in Mongolia found that historical data indicated a carrying capacity of 105,000,000 sheep with a growth rate r= 0.64% per year. Suppose there are 80,000,000 sheep in Mongolia on Jan 1, 2019 which we will call t0. Use Euler's method on the logistic model of the sheep population with a step size of h = 0.2 to estimate the growth in the Mongolian sheep population by Jan 1, 2020. Demonstrate each step in the Euler's method explicitly for full credit

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