Probability of extinction: An exponential model of growth follows from the assumption that the yearly rate of change in a population is (b − d)N , where b is births per year, d is deaths per year, and N is current population. The increase is in fact to some degree probabilistic in nature. If we assume that population increase is normally distributed around rN, where r = b − d, then we can discuss the probability of extinction of a population.
a. If the population begins with a single individual, then the probability of extinction by time t is given by
If d = 0.24 and b = 0.72, what is the probability that this population will eventually become extinct? (Hint: The probability that the population will eventually become extinct is the limiting value for P.)
b. If the population starts with k individuals, then the probability of extinction by time t is
where P is the function in part a. Use function composition to obtain a formula for Q in terms of t, b, d, r, and k.
c. If b>d (births greater than deaths), so that r > 0, then the formula obtained in part b can be rewritten as
where a <>
d. If b is twice as large as d, what is the probability of eventual extinction if the population starts with k individuals?
e. What is the limiting value of the expression you found in part d as a function of k? Explain what this means in practical terms.