Question

1. (Model with a threshold level) Suppose that a given population can keep increasing just if the initial size is large enough and it dies out otherwise. The level that allows the increase of population is called a threshold level. In this case the rate of increase is proportional to the population size P and the difference of the present population size and the threshold level T (see class notes). Write the corresponding mathematical model for this case a Let Po be the initial population size. (a) Find the equilibrium solutions and classify them. (b) Solve the model explicitly for P(t). Find linn P(t) for two cases (i) P.> T, and (ii) P nd answer the following questions. (e) Find P(15) if F, = 2000, T = 2500 and k = 0.05.

Answer 1

Solution:

"Given that 1. Proportional to the population size 'p: The difference of the present population size and threshold level 'T' i.e, dp/dt = kp(p - T) k = 0 constant a) from equilibrium solution dp/dt = kp(p/T - 1) p = 0 & p = T { because p = T is unstable \r\n b) population size will decrease to zero if initial size rightarrow p_0 < T population size p without a bound p_0 > T dp/p(p - T) = kdt implies dp/T [1/p - T - 1/p] = dt.k implies dp/p - Tp - dp/p = kT dt implies