Question

[1] A certain forest has three creatures: rabbits, squirrels, and foxes. The rabbits eat green stuff, the squirrels mostly nuts and seeds and the foxes eat the rabbits and squirrels but they have a slightly easier chance of catching rabbits. Also, when food is short the rabbits eat the squirrel’s preferred food and vice versa. For the system

dR/ dt = 4R − R(S + 3F)

dS/ dt = 2S − S(R + 2F)

dF/ dt = −F + F(2R + S)

You should determine all the critical points of the system but concentrate on the one(s) where none of the species has no members. Next, you should compute the Jacobian at the relevant point(s) and from there the eigenvalues and eigenvectors. Of course, the interesting part is to interpret all of this form a rabbit, squirrel and fox perspective, Graphs would be helpful. Warning: the numbers in the equation here haven’t been carefully crafted to give nice round numbers so be prepared for this!

Answer 1

ds dt dt R-o, we have

nbal pent 3 pimt- istem gino (th) When F=o, かw (2,4,0)门 m,

speie ase-I(0,0, 0) o O 0

4-l+ 丁 10 )ら?) 2-(012) -2 、 1 2 2 7 ース 2

toti stille 4-(o+4) 1+2 o2 /s 4/3 9-16 0 3 Page-es

7. Ab And tu eigen velt") a

Find the eigenvalues for 0 2, -2 51 104 6_193 Eigenvectors for λ-: 6 104 3i Eigenvectors for λ-2: Eigenvectors for λ--2:

4-(t0)2 3x2 12-(2to) (23%*) 2,4,0 0 0 2 ersum ReR-7

0 -2 Find the eigenvalues for-4 0 -8 3, 22,2v2 Eigenvectors for λ-3 1v2 Eigenvectors for λ 2,12 : V2 Eigenvectors for-2v2: