Question

#2 Consider the following model for the dynamics of a population of size N (measured as number of individuals x 10) over time (in months) that is subject to harvesting: The population grows according to a logistic equation in the absence of harvesting and h is a constant per a) Find all equilibria and determine the values of h for which each is stable or unstable. 4aestng andcnstant capita harvest rate. b) Construct the bifurcation plot: plot the equilibria from part a) as function of h for 0 s h S 2, using solid curve to represent locally stable equilibria and dashed curves to represent unstable equilibria. Describe in words what happens to the predicted population size as h increases continuously from h 0 to h 2 (assume that the initial population is positive).

Answer 1

Consider the given modl for the dnamies f the population stye N (measured as not individuals x 104) Ovor tme Cn monts) C. uohe, h s a (parameler) censtant peor capita harvat vale 40 is autonomous mon-uncar dtferontal equalton volth one pariameter )Equiltbia słate tn ohtch sutem dotsn't cha Thectae, the opeN 2 equibrta at No ond N 21-h).

he have, fid,h) N(IN)-hAt df dN ON then eqvilibum is stale when , the equu J N ie ond the eaui libniumsunstabe when (o,h)>o d N ie. he l therefore, the equilibrium at 0s 0 slable ushen h>) (unstable whn hel uu a nonhyperbolic whenh-1

Equi libum at Ne 2ti-h). 1-2+h-h Then the qurium to stabe uoh (0 ie when h10 cnd the equilbium ts awstable shn te when h- d N d N(911-h.h)=o when and mon.htperholic t.e when h Therefore the equalibtum at N- a(i-h) fs stalble when h (x x) unstable when hl () non-hyperbolic uwhun h- The potnt (N,h) (o,1), at vohich the equilibrtums cellidef ohese the tansvithal bluratn ocurs Fxed pont of the sypale)

3 1 xed ppint 3 curve stable equi dashed cune- unstable equribia Curve

(C) when always regardless the an-o N=o, -o atua olt u the constaut h n f), s e bi furaton pott cant e a feld bikurcation polnt The bilucaton point in tuts model is tarsoutral biturcatten point) (지, h)-(0,) ahere the equuhia voltde aud ther dtakites get exbanped. Nau ne tefit analyye heo the predtided powalaton ore aliteys as na othrurady tom ation -Assume he inital populatbon ts pasttive. 2. At hl, do dt 2 2. dt- As, we can obseve , dN chanqes sign om postive. to negat We about h, so the populaton yeaches its maximum value and then decreases

Thenepore , as h twh to h-2 , the predrded pepdation stye decuascr Incuases continuous