3. Ricker model Consider the following density-dependent difference equation for

Question

3. Ricker model Consider the following density-dependent difference equation for population density Nn at time n: where r > 0 is the growth rate and K > 0 is the carrying capacity. This model is preferred by some over the discrete logistic model because N is always positive if No is positive. a. Find the equilibria N and discuss their linear stability as a function ofr. Show that there is a change in stability at r = 2 (r-2 is therefore called a bifurcation point). b. Find the 2-cycle equilibria for r 2 from

Explanation / Answer

if m<= 1 the trivial equlibrium u0(m) =1 is global attractor.

if 1<m<m2 =3 the nontrivial equlibrium u1(m) is a global attractor.

there exists an increasing sequence m2=3 of bifurication values (period -doubling bifurcation points)such that for m2<m<m2+1 there exists a globally attractive 2-periodic solution.

the sequence m2 converges to m is approximately 3.57 ; for m>m* the behaviour is complex, generally of chaotic type.

the previous results are summarized in the bifurcation ograph and we show outcome of cobwebbing method in case of chaotic solutions m>m*

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