Review of Dynamic complexity in predator-prey models framed in difference equations

Transcription

1 Review of Dynamic complexity in predator-prey models framed in difference eqations J. Robert Bchanan November 10, 005 Millersville University of Pennsylvania Review of Dynamic complexity in predator-prey models framed in difference eqations p. 1

2 Backgrond and History Fll citation: J.R. Beddington, C.A. Free, and J.H. Lawton, Dynamic complexity in predator-prey models framed in difference eqations, Natre, 55, pp , May 1, Single species, first-order, nonlinear difference eqation models typically exhibit behavior which contains stable eqilibria cycles of period n, n N chaos Review of Dynamic complexity in predator-prey models framed in difference eqations p.

3 Main Reslt Two species, first-order, nonlinear predator/prey difference eqation models may exhibit the same behavior. Model is an extension of the Nicholson-Bailey host/parasite model: t+1 = t e r(1 t/k) av t v t+1 = t (1 e av t ) Prey species: t, predator species: v t Review of Dynamic complexity in predator-prey models framed in difference eqations p.

4 Absence of Predators When v t = 0 the model becomes one-dimensional. t+1 = t e r(1 t/k) May a has determined the behavior of the model as a fnction of r. 0 < r <, stable eqilibrim < r <.69, cycles.69 < r, chaos a R.M. May, Science, 186, pp (197) Review of Dynamic complexity in predator-prey models framed in difference eqations p.

5 Bifrcation Diagram r.5 Review of Dynamic complexity in predator-prey models framed in difference eqations p. 5

6 Presence of Predators Eqilibria: (,v ) = (0, 0) (,v ) = (K, 0) Possibly a third non-trivial eqilibrim when ) r (1 av = 0 K v = ( 1 e av ). Solve the first eqation for and sbstitte into the second eqation. v ( K = 1 a ( r v ) 1 e av ). Review of Dynamic complexity in predator-prey models framed in difference eqations p. 6

7 Intersecting Crves Define the two fnctions: f(x) = g(x) = x ( K 1 a ) (1 r x e ax ). Note that f(0) = g(0) = g(r/a) = 0. A sfficient condition for f and g to intersect for some x > 0 is that f (0) < g (0). This implies that when 1 K < a there exists v > 0 sch that f(v ) = g(v ). Review of Dynamic complexity in predator-prey models framed in difference eqations p. 7

8 Stability of Eqilibrim Sbstitte t = + δ t and v t = v + δ t v into the model. + δ t+1 = ( + δ)e t r(1 ( +δ)/k) a(v t +δv) t v + δv t+1 = ( + δ) (1 ) t e a(v +δv) t Retain only the terms linear in δ and δ v, then [ ] [ ] [ δ t+1 1 r + av a δv t+1 = v a( v ) δ t δv t ]. Review of Dynamic complexity in predator-prey models framed in difference eqations p. 8

9 Eigenvales Since then = v 1 e av, λ ± = 1 ± ( 1 r + (1 r + γ 1 e γ γ 1 + (γ 1 e γ ) r)e γ γ 1 e γ ) ) where γ = av. Review of Dynamic complexity in predator-prey models framed in difference eqations p. 9

10 Stability Region r 5 nstable stable 1 nstable Γ Review of Dynamic complexity in predator-prey models framed in difference eqations p. 10

11 Simlated Rns Fix K = 10, a = 0., ( 0,v 0 ) = (, ). v r = 0.5 Review of Dynamic complexity in predator-prey models framed in difference eqations p. 11

12 Simlated Rns v r = 0.75 Review of Dynamic complexity in predator-prey models framed in difference eqations p. 1

13 Simlated Rns v r = 1.5 Review of Dynamic complexity in predator-prey models framed in difference eqations p. 1

14 Simlated Rns 5 v r = 1.8 Review of Dynamic complexity in predator-prey models framed in difference eqations p. 1

15 Simlated Rns 5 v r =.1 Review of Dynamic complexity in predator-prey models framed in difference eqations p. 15

16 Simlated Rns 5 v r =.15 Review of Dynamic complexity in predator-prey models framed in difference eqations p. 16

17 Simlated Rns 5 v r =. Review of Dynamic complexity in predator-prey models framed in difference eqations p. 17

18 Simlated Rns 6 5 v r =.88 Review of Dynamic complexity in predator-prey models framed in difference eqations p. 18

19 Simlated Rns 6 5 v r =.55 Review of Dynamic complexity in predator-prey models framed in difference eqations p. 19

20 Simlated Rns 6 5 v r =.75 Review of Dynamic complexity in predator-prey models framed in difference eqations p. 0