http://sites.math.rutgers.edu/~zeilberg/Bio25/projects.html

Last Update: Nov. 1, 2025

All projects are due (.pdf and .txt and optionally .mw) Wed. Dec. 10, 8:00pm. The project leader should email

ShaloshBEkhad@gmail.com

Subject: Projects X

with attachments: projectX.pdf, projectX.txt, (optionally projectX.mw)

Projects and Teams

1. A full Maple implementation of

   Dynamic complexity in predator-prey models framed in difference equations by

   by J.R. Beddington et. al. , Nature v. 255 (1975), pp 58-60. Hopefully also study related papers.

   Team Leader: tbd

   Other members: tbd

   ---

2. Investigate numerically (and who knows, even solve) some of the conjectures in this intriguing paper "SI and SIR Epidemic models by Linda J.S. Allen.

   Team Leader: tbd

   Other members: tbd

   ---

3. Investigate numerically (and who knows, even solve), some of the conjectures, and similar ones, in this intriguing paper "Convergence to Periodic Solution" by A.M. Amleh and G. Ladas

   Team Leader: tbd

   Other member: tbd

   ---

4. Study numerically the ultimate perodic orbits for various parameters of the generalized discrete Logistic Equation

   x(n)=k x(n-1)(1-x(n-1)).

   These generalizations should have more parameters. For exampla:

   x(n)=k x(n-1)^a (1-x(n-1))^b ,

   that has three parameters (k, a, and b). Also study second- (and higher-) order difference equations, e.g.

   x(n)=k x(n-1)(1-x(n-1))(1-x(n-2)) ,

   and more generally

   x(n)=k x(n-1)^a(1-x(n-1))^b(1-x(n-2))^c ,

   etc.

   Team Leader: tbd

   Other team members: tbd

   ---

5. Investigate at depth, both numerically and analytically, generalizations of the Hardy-Weinberg rule with with more realistic assumptions.

   Team Leader: tbd

   Other member: tbd

   ---

6. Definining in Maple

   F:=proc(A,k,x) local i: x-mul(x-A[i],i=1..nops(A))/k:end:

   where A is a given increasing list of positive numbers (for the sake of simplicity make them integers), find the smallest k where it starts having non-empty steady-states, let's call it CutOff(A). for k above the cutoffs, find the "basin of attraction" of each of the stable steady-states. For k smaller than the Cutoffs find whether f(f(x)) (use Compk in DMB.txt) has stable steady-states.

   Team Leader: tbd

   Other member: tbd

   ---

7. For many choices of T:=RT([x,y],1000): Find what fraction have a stable equilibrium point and write a numerical procedure to find the basin of attraction for those that do.

   Team Leader: tbd

   Other member: tbd

   ---

8. Using procedures

   GeneNet(a0,a,b,n,m1,m2,m3,p1,p2,p3) and TimeSeries(F,x,pt,h,A,i)

   in the Maple package DMB.txt

   Investigate, at depth, the gene network model described in Chapter 4 of the Ellner-Guckenheimer book

   Draw may diagrams like those in figures 4.2,4.3, 4.4 in that book, and try to group the `parameter space' into the `periodic case', giving diagrams like figure 4.2, and the `stable equilibrium' case, giving diagrams like figure 4.3.

   Team Leader: tbd

   Other members: tbd