Kliah Soto Jorge Munoz Francisco Hernandez dx x 0 0 dt dy x 0 cy dt and dx y 0 ax dt dy y 0 0 dt dx ax bxy 0 dt dy cy dxy 0 dt results in (0,0) and c a ( , ) d b Solve the system of equations: dy dy dx cy dxy y(c dx) / dx dt dt ax bxy x(a by) a by c dx dy dx y x a by dy y c dx dx x aln y by c ln x dx k Solution curve with all parameters = 1 Pink: prey x Blue: predator y dx ax bxy dt dy cy dxy eyz dt dz fz gyz dt Case 1: if z=0 then we have the 2 dimensional case Case 2: y=0 dx ax dt dy 0 dt dz fz dt In the absence of the middle predator y, we are left with: dx ax dt dz fz dt We combine it to one fraction and use separation of variables: dz dz dx fz / dx dt dt ax 1 1 fz dz ax dx z Kx f a species z approaches zero as t goes to infinity, and species x exponentially grows as t approaches infinity. The blue curve represents the prey, while the red curve represents the predator. 4 3 2 1 0 2 4 6 8 10 Case 3: x=0 dx 0 dt dy cy eyz dt dz fz gyz dt In the absence of the prey x, we are left with: dy cy eyz dt dz fz gyz dt We combine it to one fraction and use separation of variables: dz dz dy z ( f gy) / dy dt dt y (c ez) c ez f gy dz dy z y c ez f gy d z z y dy c ln z ez f ln y gy K species y and z will approach zero as t approaches infinity. The blue curve represents the top predator, while the red curve represents the middle predator. yz 1.0 0.8 0.6 0.4 0.2 t 1 2 3 4 5 Set all three equations equal to zero to determine the equilibria of the system: dx ax bxy 0 dt dy cy dxy eyz 0 dt dz fz gyz 0 dt dx ax bxy dt dy cy dxy eyz dt dz fz gyz dt When x=0: Either y=0 or z=-c/e z has to be positive so we conclude that y=0 making the last equation z=0. Equilibrium at (0,0,0) When y=0 System reduces to: dx ax dt dz fz dt x=0 and y=0 since a and f are positive. Again equilibrium (0,0,0). When we consider: dz fz gyz z ( f gy ) dt Either z= 0 or –f+gy =0. Taking the first case will result in the trivial solution again as well as the equilibrium from the two dimensional case. (c/d,a/b,0) Using parameterization we set x=s and the last equilibrium is: dx a as bsy s(a by) y dt b dy ds c cy dsy eyz y(c ds ez) z dt e dz f fz gyz z( f gy) y dt g Equilibrium point at (s,a/b=f/g,(ds-c)/e) dx ax bxy f ( x, y, z ) dt dy cy dxy eyz g ( x, y, z ) dt dz fz gyz h( x, y, z ) dt dx dt dy dt dy dt f f f x y z x y z g g g x y z x y z h h h x y z x y z Where the partial derivatives are evaluated at the equilibrium point xb 0 a by J ( x, y, z ) yd c dx ez ye 0 zg f gy Real part of the eigenvalues ◦ Positive: Unstable ◦ Negative: Stable ◦ Zero: Center Number of eigenvalues: ◦ Dimension of the manifold Manifold is tangent to the eigenspace spanned by the eigenvectors of their corresponding eigenvalues Eigenvalues: ◦ a, -c, -f Eigenvectors: {1,0,0}, {0,1,0}, {0,0,1} 0 a 0 J (0,0,0) 0 c 0 0 0 f One-dimensional unstable manifold: curve x-axis Two-dimensional stable manifold: surface yz- Plane 10 50 000 5 40 000 30 000 1 2 20 000 5 10 000 5 10 Unstable x-axis 15 20 10 Stable yz-Plane 3 4 5 ac 0 0 bc / d J (c / d , a / b,0) ad / b 0 ae / b 0 0 f ga / b Eigenvectors: {1, ( fb ag) {1, d 1 2 cd 2 2 2 2 , ( ab b df 2 abdfg a dg ) } b 2c ab2ce id ac ,0} bc Eigenvalues ( ga fb) / b i ac ( ga fb) / b i ac One-Dimensional invariant curve: ◦ Stable if ga<fb ◦ Unstable ga>fb Two-Dimensional center manifold Three-dimensional center manifold ◦ If ga=fb Blue represents the prey. Pink is the middle predator Yellow is the top predator (2,2,2) All parameters equal 1 a = 0.8 Blue represents the prey. Yellow is the middle predator Pink is the top predator (2,2,2) a=1.2 , b=c=d=e=f=g=1 Blue represents the prey. Pink is the middle predator Yellow is the top predator All parameters 1 initial condition (1,2,4) dx ax bxy dt The only parameters that have an effect on the top predator are a, g, f and b. ◦ Large values of a and g are beneficial while large values of f and b represent extinction. The parameters that affect the middle predator are c, d and e. They do not affect the survival of z. The survival of the middle predator is guaranteed as long as the prey is present. The top predator is the only one tha faces extinction when all species are present. dy cy dxy eyz dt dz fz gyz dt Eigenvalues for (c/d, a/b,0) ( ga fb) / b i ac