# Suppose we’re modeling the populationP of a species of animal aftert

Question 9: (1 point)A differential equation is an equation that includes both a function and one or more of its derivatives.Differential equations can be very useful tools for modelling real-world phenomena. Suppose we’re modeling the population P of a species of animal after t years spent in a wildlife reserve that can sustainably support M members of this species. Suppose the population can be described as somefunction of time, P (t)Let P’ (t) = P(t) 1 –be an equation modelling the rate of change of a population P as a function of t (in years). Note that the value of this derivative depends on the value of the population at that time, P (t).(a) If P (to) is much larger than M, what does P'(to) indicate will happen qualitatively to the size of the population as time passes shortly after to ?(a) decrease(b) increase(c) stay the same(b) If P(t1 ) is much smaller than M, what does P’ (t1 ) indicate will happen qualitatively to the size of the population as time passes shortly after ti?(a) decrease(b) increase(c) stay the samec) A population is stable if its value does not change over time. For which population sizes will the number of animals in this habitat remain unchanged as time passes? Be sure to separate your answers with a semi-colon. (Hint: There are two such values, and one would make conservationistspretty sad.)P(t) =(d) Each year, 5 animals are removed from a reserve that can support up to M = 500 animals (these animals are then sent to zoos around the globe). We can build this recurring loss of animals into our model as follows: P’ (t) = P(t) 1P(t)500-5Given this new model, for which values of P (t) rounded off to the nearest integer will the population remain stable? Again, there will be two answers. (Try your best to interpret each answer within the context of the model.)

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