Question

LOGISTI We know that if the number of individuals, N, in a population at time t follows an exponential law of growth, then N-N, exr where k >0 and No is the population when t -o. es that at time, t, the rate of growth, N, of the population is proportional to dt dN the number of individuals in the population. That is, kN Under exponential growth, a population would get infinitely large as time goes on. In reality, when a population gets large enough, there are environmental factors which slow down the rate of growth. These factors eventually cause N to decrease. dN dt If we assume that population size is limited to a maximum of M, then as N approaches M dN approaches zero and the rate of growth is proportional to the product of the population size dt and the difference between the maximum size and the population size. ha is,kN(M-N) at is, dN From this differential equation, it can be proved that the expression for the logistic growth function is: where b, k and M are constants for a particular situation N (t) , 1+be -Mkt MAPS- ROUTINE COMPLEX) Question 1 the logistie model to describe in two or three sentences what happens tote Btowth rate,-, as the population size nears its maximum (ie., as N-、M ), Sn s dt dN Is a solution to the logistic M-T- N# (b)Show that the population model I+ b.e Mkis solution dN equation,--kN ( M-N) dt

Answer 1

dt =0 Lr dt (t+4e、mkt)-("kmaje'.MAト 2 d t