The number of fish in a lake decreases by 25% each year since anglers take out m

Question

The number of fish in a lake decreases by 25% each year since anglers take out more than fish reproduce. To prevent a collapse of the population, the municipality decides to restock 500 fish at the end of each year. The DTDS for the number of fish in the lake in year t, denoted by x_t, is given by x_t+1 = 0.75 x_t 500. The updating function of the DTDS is f (x) The general solution of the DTDS is x_t The steady state of the DTDS is x* = Early one year, after a particularly cold and long winter, the municipality counts only x_0 = 800 fish in the lake. Calculate x_1, x_2, x_3. Draw the graph of the updating function and cobweb the solution of the DTDS starting from x_0 = 800. Is the steady state stable? Justify your answer through two different ways of reasoning. How long will it take for a population of x_0 = 800 fish to grow up to 80% of the steady state value? The municipality wants to increase the steady state level of fish in the lake to 3000, but they cannot afford a larger restocking program. Therefore, they decide to limit fishing. What is the percentage of annual decrease (in percent) that they can allow to get the steady state to be 3000?

Explanation / Answer

a) updating function f(x) = 0.75xt + 500

d) given x0 = 800 ;

x1 = 0.75 x 800 + 500 = 1100;

x2 = 0.75 x 1100 + 500 = 1325;

x3 = 0.75 x 1325 + 500 = 1493.75 ;

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