On predator-prey models

Transcription

1 Predator-prey On models ddd Department of Mathematics College of William and Mary Math 41/CSUMS Talk February 3, 1

2 Collaborators Sze-Bi Hsu (Tsinghua University, Hsinchu, Taiwan) Junjie Wei (Harbin Institute of Technology, Harbin/Weihai, China) Rui Peng (University of New England, Armidale, Australia) Fengqi Yi (Harbin Engineering University, Harbin, China) Ph.D. 8 Jinfeng Wang (Harbin Institute of Technology/Harbin Normal University, Harbin, China) Ph.D. expected 1 Yuanyuan Liu (William & Mary, Williamsburg, USA) B.S. expected (math and econ) 1

3 Fur trade of Hudson Bay Company ( )

4 Hudson Bay Company lynx-hare data Charles Elton (194), Periodic fluctuations In the numbers of animals: their causes and effects, British Journal of Experimental Biology, was first (of MANY) publications to analyze this data set Rebecca Tyson, et.al. (9), Modelling the Canada lynx and snowshoe hare population cycle: the role of specialist predators, Theoretical Ecology, one of the latest.

5 Lotka-Volterra Predator-Prey Model Predator-prey Alfred Lotka ( ) Vito Volterra ( ) du dt = au buv, dv dt = cv + duv. see Math 3 Chapter

6 Predator-prey system with functional response du = u(a bu) cφ(u)v, dt dv = dv + f φ(u)v. dt φ(u): predator functional response φ(u) = u (Lotka-Volterra) u φ(u) = (Holling type II, m: prey handling ) 1 + mu [Holling, 1959](Michaelis-Menton biochemical kinetics) Biological work: [Rosenzweig-MacArthur, American Naturalist 1963] [Rosenzweig, Science, 1971] (Paradox of enrichment) [May, Science, 197] (Existence/uniqueness of limit cycle)

7 Rosenzweig-MacArthur model du (1 dt = u u ) muv K Nullcline: u =, v = 1 + u, dv muv = dv + dt 1 + u (K u)(1 + u) ; v =, d = mu m 1 + u. Solving d = mu d, one have u = λ 1 + u m d. Equilibria: (,), (1,), (λ,v λ ) where v λ = We take λ as a bifurcation parameter (K λ)(1 + λ) m Case 1: λ K: (K, ) is globally asymptotically stable Case : (K 1)/ < λ < K: (K,) is a saddle, and (λ,v λ ) is a locally stable equilibrium Case 3: < λ < (K 1)/: (K,) is a saddle, and (λ,v λ ) is a locally unstable equilibrium (λ = (K 1)/ is a Hopf bifurcation point)

8 Phase Portraits Left: (K 1)/ < λ < K: (K,) is a saddle, and (λ,v λ ) is a locally stable equilibrium Right: < λ < (K 1)/: (K,) is a saddle, and (λ,v λ ) is a locally unstable equilibrium; there exists a limit cycle A supercritical Hopf bifurcation occurs.

9 Global stability [Hsu, Hubble, Waltman, SIAM J. Appl. Math., 1978] [Hsu, Math. Biosci., 1978] (λ,v λ ) is globally asymptotically stable if K 1, or K > 1 and (K 1)/ < λ < K. [Cheng, SIAM J. Math. Anal., 1981] If < λ < (K 1)/, then (λ,v λ ) is unstable, and there is a unique periodic orbit which is globally asymptotically stable. More on uniqueness of limit cycle: [Zhang, 1986], [Kuang-Freedman, 1988] [Hsu-Hwang, 1995,1998], [Xiao-Zhang, 3, 8]

10 Summary of ODE du (1 dt = u u ) muv K Nullcline: u =, v = 1 + u, dv muv = dv + dt 1 + u (K u)(1 + u) ; v =, d = mu m 1 + u. Solving d = mu d, one have u = λ 1 + u m d. Equilibria: (,), (K,), (λ,v λ ) where v λ = We take λ as a bifurcation parameter (K λ)(1 + λ) m Case 1: λ K: (K, ) is globally asymptotically stable Case : (K 1)/ < λ < K: (λ,v λ ) is globally asymptotically stable Case 3: < λ < (K 1)/: unique limit cycle is globally asymptotically stable (λ = (K 1)/: Hopf bifurcation point)

11 Phase portrait (1) du dt = u (1 u K ) muv 1 + u, dv dt = dv + muv 1 + u Case 1: λ K: (K, ) is globally asymptotically stable v=f(u) d=g(u) v u (,) (k,)

12 Phase portrait () du dt = u (1 u K ) muv 1 + u, dv dt = dv + muv 1 + u Case : (K 1)/ < λ < K: (K,) is a saddle, and (λ,v λ ) is a locally stable equilibrium v=f(u) d=g(u) v (λ,f(λ)) u (,) (k,)

13 Phase portrait (3) du dt = u (1 u K ) muv 1 + u, dv dt = dv + muv 1 + u Case 3: < λ < (K 1)/: (K,) is a saddle, and (λ,v λ ) is a locally unstable equilibrium d=g(u) (λ,f(λ)) v v=f(u) u (,) (k,)

14 v A new result of this ODE [Hsu-Shi, Disc. Cont. Dyna. Syst.-B, 9] Relaxation oscillator profile of limit cycle in system. (Motivated by numerical observation) du dt = u (1 u) muv a + u, dv dt = dv + muv a + u u

15 Graph of limit cycle Parameters: a =.5, m = 1, d =.1, λ = 1/18.56, period T u and v t

16 v Small d u

17 Graph of limit cycle Parameters: a =.5, m = 1, d =.1, λ = 1/198.5, period T u and v t

18 Illustration of limit cycle v O 1 O 4 v 5 (u) v 4 (u) a/m O 5 v 6 (u) O O 3

19 Relaxation oscillation Theorem[Hsu-Shi, 9] If < a < 1 and m > are fixed, and as d (thus λ ), then C 1 λ 1 T(O 1 O ) C λ 1, T(O O 3 ) = O( lnλ ), T(O 4 O 1 ) = O( lnλ ), and T(O 3 O 4 ) = O(1). In particular, the period T as d. The shape of the graph of the limit cycle is a relaxation oscillator. Other known relaxation oscillators: Van der Pol oscillator in electrical circuits FitzHugh-Nagumo oscillator in action potentials of neurons Many other physiology models: heart beat, calcium signaling [Liu-Xiao-Yi, JDE, 3] oscillations in singularly perturbed 3-D system [Zhang-Wang-Wang-Shi, preprint] extend it to more general system

20 Relaxation oscillation (left) Van der Pol oscillator; (right) FitzHugh-Nagumo oscillator.

21 More general system du dt = g(u)(f (u) v), dv dt = v (g(u) d). (a1) f C 3 (R + ), f () >, there exists K >, such that for any u >, u K, f (u)(u K) < and f (K) = ; there exists λ (,K) such that f (u) > on [, λ), f (u) < on ( λ,k]; (a) g C (R + ), g() = ; g(u) > for u > and g (u) > for u ; there exists a unique λ (,K) such that g(λ) = d. g(u): functional response, f (u): prey isocline Rosenzweig-MacArthur model: g(u) = mu f (u) = (K u)(1 + u) m 1 + u and

22 Phase portraits du dt = g(u)(f (u) v), dv dt = v (g(u) d). v=f(u) d=g(u) d=g(u) v (λ,f(λ)) v (λ,f(λ)) v=f(u) (,) (k,) u (,) (k,) u Left: (λ,f (λ)) stable; Right: (λ,f (λ)) unstable, exist limit cycle

23 History du dt = g(u)(f (u) v), dv dt = v (g(u) d). [Hsu, 1979] If λ > λ and f (u) is concave in [,K], then (λ,f (λ)) is globally stable. ( λ is the top of the hump) [Kuang-Freedman, ( 1988] If < λ < λ, and d f ) (u)g(u) for all x [,K], then the limit cycle du g(u) d is unique and globally stable. [Hofbauer-So, 1989] counterexample: f (u) is concave, but Hopf bifurcation is subcritical, so there are two periodic orbits for λ ( λ, λ + ǫ), one of them is locally stable. More work: [Ruan-Xiao, 1] [Xiao-Zhang, 3] [Liu, 5]

24 Our result du dt = g(u)(f (u) v), dv dt = v (g(u) d). Theorem[Wang-Shi-Wei, submitted] If λ > λ, and (a8) (uf (u)), (u/g(u)) for u [,K], and (uf (u)) for u ( λ,k); or (a9) f (u), (1/g(u)) for u [,K], and f (u) for u ( λ,k), then (λ,f (λ)) is globally stable. Sharpness: If (uf (u)) u= λ > and g(u) = u/(a + u) or g(u) = u, then Hopf bifurcation is subcritical and global stability does not hold. Realistic example: du (1 dt = ru u ) ( 1 A + C ) dv Buv, = dv + Buv. K u + C dt Prey growth: weak Allee effect when A < and C > A. C large: global stability; C small: subcritical Hopf bifurcation.

25 Global stability vs. Multiple periodic orbits du (1 dt = ru u ) ( 1 A + C ) Buv, K u + C dv dt = dv + Buv. v 1. v u u Both: r = B = 1, A =.8, K = 1; (left) d =.1199, C =.5; (subcritical Hopf) (right) d =.6, C = (supercritical Hopf).

26 Habitat for predator and prey Habitat fragmentation

27 Habitat Restoration: connecting patches Crossing bridges built in Banff, Canada; (for elks) under-cross built in Florida, USA (for deers)

28 Habitat network Habitat with a network structure

29 Habitats as graph theory Network connectivity

30 Effect of network structure on dynamics Matthew Holland, Alan Hastings, (8), Strong effect of dispersal network structure on ecological dynamics, Nature. David Vasseur, Jeremy Fox, (9), Phase-locking and environmental fluctuations generate synchrony in a predator prey community, Nature. dn i dt = N i ( 1 N i K dp i dt = mn ip i 1 + N i ep i + d p ) mn ip i 1 + N i + d n n d ij (N j N i ), j=1 n d ij (P j P i ). j=1 Homogeneous patches, dispersal matrix (d ij ) (d ij = d ji ) single patch: Rosenzweig MacArthur model [Holland-Hastings]: n = 1, random network; cluster solutions, long transient dynamics.

31 Stability in network model Local dynamics: du (1 dt = u u ) muv K 1 + u, dv dt = ev + muv 1 + u Case 1: λ K: (K,) is globally stable Case : (K 1)/ < λ < K: (λ,v λ ) is globally stable Case 3: < λ < (K 1)/: unique limit cycle is globally stable (λ = (K 1)/: Hopf bifurcation point) Theorem[Wang-Shi-Liu, in preparation] If λ K, then (K,) n is globally stable; and if K 1 < λ < K, then (λ,v λ ) n is globally stable. [Li-Shuai (1), JDE] Lyapunov funct. ODE on network Note: same result holds for reaction-diffusion model.

32 Two-patch model where F(u,v) = u u = F(u,v) + a(w u), v = G(u,v) + c(x v), w = F(w,x) a(w u), x = G(w,x) c(x v), ( 1 u ) muv muv, G(u,v) = K 1 + u 1 + u ev. where a,c > are the diffusion rates. Can we completely classify the dynamics?

33 Some Partial Results for Two-patch model [Liu-Shi, in preparation] Let U(t) = u(t) + w(t), V (t) = v(t) + x(t) (sum) and W (t) = u(t) w(t), X(t) = v(t) x(t) (difference) 1. When a, c are large, then W(t), X(t) as t so the system is synchronized.. For < a < a, there exists c = c (a) such that when c < c, there are two additional Hopf bifurcation points λ H, λ+ H, and there exists a non-symmetrical periodic orbit for λ (λ H, λ+ H ). (numerical result shows the non-symmetrical periodic orbit is unique and unstable) 3. For a < a 1, there exists c 1 = c 1 (a) such that when c > c 1, there are two additional bifurcation points λ S, λ+ S such that there exists two non-symmetrical equilibrium points for λ (λ S, λ+ S ). ((a) equilibrium points can be algebraically solved with a complicated formula; (b) for any a and c, there are at most 9 equilibrium points.)

34 Numerical bifurcation diagrams Software: Matlab and MatCont (Govaerts, Kuznetsov) similar to Auto, but works under Matlab Bifurcation/continuation of equilibrium, limit cycles, homoclinic orbits of ODEs BP 3 Max(u) H LPC 1 H.5 H BP H e k = 4, m =.5, a =.6, c =.1, bifurcation parameter < e <.4; symmetric Hopf: e =.3, non-symmetric Hopf: e =.387 and e =.1173

35 Numerical solution Software: Matlab coupled population prey 1 predator 1 prey predator k = 4, m =.5, a =.6, c =.1, bifurcation parameter < e <.4; symmetric Hopf: e =.3, non-symmetric Hopf: e =.387 and e =.1173 Here e =.

36 Numerical solution Software: Matlab patch 1 population patch population coupled population coupled population prey 1 predator 1 prey predator prey 1 predator 1 prey predator total prey population total predator population k = 5, m = 9.96, a =, c =.6, bifurcation parameter < e < 1; symmetric Hopf: e = 6.64, no non-symmetric Hopf Here e = 1 and solution appears to be chaotic

37 Attractor? Chaos? An attractor exists as all positive solutions are bounded. When there are non-symmetrical equilibrium points (EQ) or periodic orbits (PO), there is only one stable state (symmetrical PO) in the attractor, but more than unstable states (symmetric EQ, non-symmetric EQ and/or PO). Hence there exist connecting orbits between the two unstable states, which become a heteroclinic loop. This may imply chaotic behavior on a lower dimensional invariant manifold. Conjecture: Chaos occurs on a zero measure set, but the generic dynamics is eventual synchronization (despite possible long transient dynamics) and converges to symmetric PO.

38 Non-symmetric patches ( heterogeneity) where for i = 1,, ) F i (u,v) = u (1 uki u = F 1 (u,v) + a(w u), v = G 1 (u,v) + c(x v), w = F (w,x) a(w u), x = G (w,x) c(x v), m iuv 1 + u, G i(u,v) = m iuv 1 + u e iv. where a,c > are the diffusion rates. 1. synchronization of coupled oscillators. synchronization of equilibrium and oscillator Goldwyn, E.E. and Hastings, A. (8) When can dispersal synchronize populations? Theoretical Population Biology 73:395-4 Goldwyn, E.E. and Hastings, A. (9) Small heterogeneity has large effects on synchronization of ecological oscillators. Bulletin of Mathematical Biology 71:

39 Non-symmetric patches: numerical results Y.Y. Liu (undergraduate research) x = [.95;.5;.;.5]; k = 3; m = ; e1 =.; e =.9 Left: a =.1; c =.; Right: a = 1; c =. coupled population coupled population patch population patch 1 population 4 prey 1 predator prey predator prey 1 predator 1 prey predator total prey population total predator population coupled population coupled population patch population patch 1 population 4 prey 1 predator prey predator prey 1 predator 1 prey predator total prey population total predator population Larger diffusion rates help to achieve synchronization; smaller diffusion rates may not result in synchronization

40 Non-symmetric patches: numerical results Y.Y. Liu (undergraduate research) x = [.95;.5;.;.5]; k = 3; m = ; e1 = 1.8; e =.45 Left: a =.1; c =.; Right: a = 1; c =. coupled population coupled population patch population patch 1 population 4 prey 1 predator prey predator prey 1 predator 1 prey predator total prey population total predator population coupled population coupled population patch population patch 1 population 4 prey 1 predator prey predator prey 1 predator 1 prey predator total prey population total predator population Connecting a prey-only (predator extinction) system to an oscillatory system could make a stable coexistence

41 Non-symmetric patches: numerical results x = [.95;.5;.;.5]; k = 3; m = ; e1 =.; e =.9 Left: a =.1; c =. The limit cycle is non-symmetrical, and not synchronized. patch 1 population patch population prey 1 predator 1 prey predator coupled population 3 1 prey 1 predator 1 prey predator coupled population total prey population total predator population

42 Future Work (for CSUMS) Predator-prey 1. Further work on -patch model (complete bifurcation diagram). Impact or network structure (small number of patches): 3-patch (linear or triangle) 3. If there is a cycle in the network, is there a periodic solution with population running around the cycle? 4. Computer work: writing better Matlab programs for bifurcation diagram calculation

43 References Sze-Bi Hsu; Relaxation oscillator profile of limit cycle in system. Discrete and Continuous Dynamical Systems B, 11, (9) no. 4, Fengqi Yi, Junjie Wei and, Bifurcation and spatiotemporal patterns in a homogeneous diffusive system. Journal of Differential Equations, 46, (9), no. 5, Rui Peng,, Non-existence of Non-constant Positive Steady States of Two Holling Type-II Predator-prey Systems: Strong Interaction Case. Journal of Differential Equations, 47, (9), no. 3,

44 Predator-prey