Global attractors, stability, and population persistence

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1 Global attractors, stability, and population persistence Horst R. Thieme (partly joint work with Hal L. Smith*) School of Mathematical and Statistical Sciences Arizona State University partially supported by NSF grant DMS *NSF grant DMS H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

2 Overview A model for bacteria and phages in a chemostat Phage extinction via Laplace transform Global stability of the phage-free equilibrium via global compact attractor How do persistence and compact attractor interact? Global stability of a coexistence equilibrium via Lyapunov functions Global stability of the endemic equilibrium in an epidemic model with spatial spread and non-bilinear incidence H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

3 Literature Hal L. Smith, Horst R. Thieme Dynamical Systems and Population Persistence, Graduate Studies in Mathematics 118 American Mathematical Society, 2011 HLS and HRT Persistence of bacteria and phages in a chemostat (under review) HRT Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators J. Differential Eqns. 250 (2011), H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

4 Bacteria and phages in a chemostat Lenski and B. Levin, 1985 R = D ( R R ) f(r)s, S = ( f(r) D ) S ksp, P (t) = DP(t) ks(t)p(t) + be Dτ ks(t τ)p(t τ). R S I P resource supporting bacterial growth, uninfected susceptible bacteria, phage-infected bacteria, phages (viruses). f(r) is the specific growth rate of bacteria at resource level R. f : R + R + is C 1 and f(0) = 0, f (R) > 0, f( ) <. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

5 The phage-free system Hsu, Hubble, Waltman 1977 R = D ( R R ) f(r)s, S = ( f(r) D ) S. For M = R + S, M = D(R M). Trivial equilibrium: (R,0) Assume f(r ) > D. Then there exists a unique equilibrium ( R, S) with f( R) = D, R + S = R. If S(0) > 0, then R(t) R and S(t) S as t. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

6 The return of the Laplace transform The Laplace transform of f : R + R, ˆf(λ) = Named after Pierre Simon de Laplace ( ) but used before by Euler and Lagrange. 0 e λt f(t)dt. Widder (1941, 1949), real-valued theory Arendt, W., C.J.K. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel 2002 H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

7 phage extinction All non-negative solutions are bounded. P (t) = DP(t) ks(t)p(t) + be Dτ ks(t τ)p(t τ). Laplace transform: ˆP(λ) = e λt P(t)dt, λ > 0. 0 λ ˆP(λ) P(0) = D ˆP(λ) kŝp(λ) + be Dτ k τ e λ(s+τ) S(s)P(s)ds. (λ + D) ˆP(λ) [ be τ(d+λ) ] 1 kŝp(λ) + c, 0 c =P(0) + be Dτ k S(s)P(s)ds. τ H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

8 The return of the Laplace transform D ˆP(λ) [ be Dτ 1 ] + kŝp(λ) + c lim sup S(t) S. t Let ǫ > 0. WoLoG S(t) S + ǫ, t 0; so ŜP ( S + ǫ) ˆP. ( D [ be Dτ 1 ] + k( S + ǫ)) ˆP(λ) c Assume D [ be Dτ 1 ] + k S > 0. For some ǫ > 0, ˆP(λ) c D [ be Dτ 1 ] + k( S + ǫ), λ > 0. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

9 Reproduction number Beppo Levi: 0 P(t)dt = ˆP(0) < and P(t) 0 as t. D [ be Dτ 1 ] + k S > 0 1 > be Dτ k S D + ks =: R. R critical phage reproduction number The condition is sharp: If R > 1, there exists a coexistence equilibrium. Further, bacteria and phages coexist dynamically. If R < 1, is ( R, S,0) globally stable for solutions with S(0) > 0? H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

10 Semiflows and their state spaces The temporal development of a natural or artificial system can conveniently be modeled by a semiflow. A semiflow consists of a state space, X, a time-set, J, and a map, Φ. The state space X comprehends all possible states of the system: the amounts or densities of the system parts; if structure, their structural distribution. Epidemiological system: the amounts or densities of susceptible and infective and possibly exposed and removed individuals. For spatial spread, spatial distributions age-structure: age-distributions H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

11 Time set Time can be considered as a continuum or in discrete units. time-set J: R + = [0, ) or Z + = N {0} = {0,1,...}. Definition A subset J [0, ) is called a time-set if it has the following properties: 1 0 J and 1 J. 2 If s,t J, then s + t J. 3 If s,t J, and s < t, then t s J. A time-set J is called a closed time-set if J is a closed subset of [0, ). J is a time-set iff Ĵ = J ( J) is a subgroup of (R,+) containing Z. Depending on the model, the time unit can be a year, month, or day. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

12 Semiflow map Φ : J X X. Often Φ itself is called the semiflow. If x X is the initial state of the system (at time 0), then Φ(t,x) is the state at time t. Φ(0,x) = x, x X. Semiflows are characterized by the semiflow property: Φ(t + r,x) = Φ(t,Φ(r,x)), r,t J, x X. Write Φ t (x) = Φ(t,x), Φ t : X X, Φ t Φ r = Φ t+r. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

13 Semiflow for phages Bacteria and phages: Time set: R + state space: X = R + C + [ τ,0] C + [ τ,0], semiflow map: Φ(t,R 0,S 0,P 0 ) = (R(t),S t,p t ), t 0. Translations S t (s) = S(t + s), s [ τ,0]. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

14 Compact attractors Let X be a metric space, J a time-set, and Φ : J X X be a semiflow. A set K X is said to attract a set M X, if K and Φ t (M) K as t. Φ t (M) K: for every open set U, K U X, there exists r J s.t. Φ t (M) U for all t J, t r. K is called an attractor of M, if K is invariant and attracts M. In this situation, we also say that M has K as an attractor. K forward invariant: Φ t (K) K for all t J, K backward invariant: Φ t (K) K for all t J, K invariant: Φ t (K) = K for all t J. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

15 Global compact attractor We avoid the notion of a global attractor because there is no agreement in the literature about this term. Instead we use the following terminology for a non-empty compact invariant set A: If C is a class of sets in X and A attracts every set in C, A is said to attract C. If C is the class of singleton sets in X and A attracts C, A is called a compact attractor of points. If C is the class of bounded (compact) sets in X and A attracts C, A is called an (actually the) compact attractor of bounded (compact) sets. A is a (the) compact attractor of neighborhoods of compact sets if every compact set in X has a neighborhood that is attracted by A. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

16 Global compact attractor in the literature The term global compact attractor has been used in various ways in the literature: compact attractor of points: Ladyzhenskaya, 1991; Matano, Nakamura, 1997 compact attractor of neighborhoods of compact sets: Sell, You, 2002; Magal, Zhao, 2005 compact attractor of bounded sets: Hale 1989; Diekmann, van Gils, Verduyn Lunel, Walter, 1995 exponential attractor (of points) Eden, Foias, Nicolaenko, Temam (1994) Osaki, Tsujikawa, Yagi, Mimura (2002) H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

17 Applicable concepts for global compact attractors Definition (Hale, 1989) Let Φ : J X X be a state-continuous semiflow. Φ is called point-dissipative (or ultimately bounded) if there exists a bounded subset B of X which attracts all points in X. Φ is called asymptotically smooth if every forward invariant bounded closed set has a compact attractor. Φ is called eventually bounded on a set M X if Φ(J r M), J r = J [r, ), is bounded for some r J. If X is a closed subset of R n, Φ is asymptotically smooth. More generally, if Φ r is compact on the metric space X for some r J, Φ is asymptotically smooth. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

18 Applicable conditions for global compact attractors The following result has been inspired by [Magal, Zhao, 2005]. See also [Hale, 1989]. Theorem The following are equivalent for a state-continuous semiflow Φ. (a) Φ is point-dissipative, asymptotically smooth, and eventually bounded on every compact subset K of X. (b) There exists a compact attractor A of neighborhoods of compact sets; A attracts every subset of X on which Φ is eventually bounded. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

19 Compact attractors and stability Definition A forward invariant subset M of X is called stable if, for any neighborhood U of M there exists some neighborhood V of M such that Φ(J V ) U. A stable subset M of X is called locally asymptotically stable if there exists a neighborhood V of M such that M attracts all points in V. This result can be found in Hale (1989) and in Magal and Zhao (2005). Theorem Let the semiflow Φ be continuous and A be a compact forward invariant subset of X that attracts all compact subsets of a neighborhood of itself. Then A is locally asymptotically stable. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

20 Total trajectories Recall Ĵ = J ( J). Let Φ be a semiflow on X. φ : Ĵ X is a total Φ-trajectory if Φ(t,φ(s)) = φ(t + s), t J,s Ĵ. In ODEs, a total trajectory is a solution that exists for all times. Special case: E X is an equilibrium if Φ(t,E) = E for all t J. Theorem A X is invariant if and only if for every x A there is a total Φ-trajectory φ : Ĵ A with φ(0) = x. Corollary Let A be an invariant subset of X and E A be an equilibrium of Φ. Then A = {E} if and only if every total trajectory with range in A is constant with value E. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

21 Attractors and persistence J = Z + or J = R +, X is metric space, persistence function ρ : X R +, continuous. Φ is uniformly weakly ρ-persistent, if there exists η > 0 s.t. lim supρ(φ t (x)) > η, whenever ρ(x) > 0, t and is uniformly ρ-persistent if we can replace lim sup by lim inf above. The following set is closed and forward invariant (though possibly empty), X 0 = {x X; t J : ρ(φ(t,x)) = 0}. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

22 Partition of the attractor: assumptions Assume that Φ has a compact attractor, A, of neighborhoods of compact sets, Φ is state-continuous. Assume that X 0, ρ and ρ Φ are continuous, Φ is uniformly weakly ρ-persistent, and (H1) there exists no total Φ-trajectory φ with range in A such that ρ(φ( r)) > 0, ρ(φ(0)) = 0, and ρ(φ(s)) > 0 with r,s J, H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

23 Partition of the attractor: results The attractor A is the disjoint union A = A 0 C A 1 of three invariant sets A 0, C, and A 1. A 0 and A 1 are compact and: (a) A 0 = A X 0 is the compact attractor of compact subsets of X 0 ; in fact, every compact subset K of X 0 has a neighborhood in X 0 that is attracted by A 0. (b) inf ρ(a 1 ) > 0 and A 1 is the compact attractor of neighborhoods of compact sets in X \ X 0. A 1 is stable. (c) (results on total trajectories, characterization of C) We call A 1 the (ρ-)persistence attractor of Φ and A 0 the (ρ-)extinction attractor of Φ. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

24 Back to the phages Trivial equilibrium: (R,0,0). Assume f(r ) > D: phage-free equilibrium E S = ( R, S,0). Choose ρ(r 0,S 0,P 0 ) = S 0 (0) for R 0 R +, S 0,P 0 C + [ τ,0]. Show: The ρ-persistence attractor equals {E S }. Let R,S,P be a bounded solution that is defined for all times, inf t R S > 0: Ad hoc estimates: S(t) S for all t R modified Laplace-transform argument: if R < 1 then P 0. contradiction arguments: S S, R R. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

25 Global stability of the phage-free equilibrium Theorem Let R < 1. Then the phage-free equilibrium E S = ( R, S,0) is stable. Further, if c > ǫ > 0, (R(t),S(t),P(t)) E S as t uniformly for all non-negative solutions with R(0) c, S(0) ǫ, and S(s) + P(s) c, τ s c. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

26 Global stability of endemic equilibria Volterra type Lyapunov functions: x x ln(x/x ) Beretta, Takeuchi and coworkers (2001) Korobeinikov and coworkers ( ) M. Li and coworkers ( ) McCluskey ( ) Sallet and coworkers (2006, 2007) Webb and coworkers (2006, 2010) H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

27 Infectious disease in a structured population S(t,x) susceptible individuals with structural characteristic x at time t I(t,x) infective individuals... Ω compact subset of R n, Ω is the closure of its interior. t S(t,x) =Λ(x) µ(x)s(t,x) f ( x,y,s(t,x),i(t,y) ) dy, t I(t,x) = f ( x,y,s(t,x),i(t,y) ) dy ν(x)i(t,x), Ω t 0,x Ω. Removal rates µ ν. Ω Disease-free equilibrium: S (x) = Λ(x) µ(x), I 0 H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

28 The incidence function f : Ω 2 R 2 + R is continuous; for all x,y Ω, if I 0, f(x,y,s,i) is an increasing function of S 0, with the increase being strict if x = y and I > 0, and, if S 0, f(x,y,s,i) is an increasing function of I 0, f(x,y,0,i) = 0 = f(x,y,s,0) for all x,y Ω, S,I 0. Finally f is locally Lipschitz continuous in S and I. Theorem Unique non-negative solutions exist for non-negative initial data. The solutions induce a continuous semiflow that has a compact attractor of bounded sets. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

29 More assumptions, main result f has the following form f(x,y,s,i) = f 1 (x,s,i)f 2 (x,y,i), S,I 0. f(x,y,s,i) I decreases in I > 0. I f(x,y,s,0) is continuous on Ω 2 (0, ) and I f(x,x,s,0) > f(x,x,s,i), I > 0,S > 0. I Theorem Either the disease-free equilibrium is stable and attracts all solutions, uniformly for initial data in bounded sets, or there exists a stable endemic equilibrium which attracts all solutions with I(0,x) > 0 for some x Ω. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

30 Persistence Lemma Assume that an endemic equilibrium ( S,Ĩ) exist. Then there exists a compact invariant set A 1 which is stable and attracts all points (S 0,I 0 ) with I 0 (x) > 0 for some x Ω. Further inf Ω I > 0 and inf Ω S > 0 for all (S,I) A 1. Aim: Show A 1 = {( S,Ĩ)}. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

31 LaSalle invariance principle for compact attractors Theorem Let A be a compact invariant subset of X and E A be an equilibrium. Let L : A R be a Lyapunov-function: L is continuous and L(Φ(t,x)) decreases in t J for all x A. Finally assume that if φ : Ĵ A is a total trajectory and L φ is constant, then φ E. Then A = {E}. This principle has also been used by Magal, McCluskey, Webb (2010). H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

32 Lyapunov function We assume that an endemic equilibrium S,Ĩ : Ω R + exists. Then S(x) > 0 and Ĩ(x) > 0 for all x Ω. For strictly positive functions S,I : Ω R +, we define V (S,I)(x) = S(x) S(x) f ( x,x,s,ĩ(x)) f ( x,x, S(x),Ĩ(x)) f ( x,x,s,ĩ(x)) ds I(x) + I(x) Ĩ(x)ln Ĩ(x), W(S,I)(x) = V (S, I)(x) w(x)dx. Ω Generalization of the Volterra Lyapunov function: M.Li and Shuai (2010). Finite-dimensional version, graph theoretic methods H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

33 rescued by Krein and Rutman Orbital derivative V (x) Ω k(x,y)(g(x) g(y))dy with g : Ω R and k : Ω 2 R +, k(x,x) > 0. Wanted: w : Ω (0, ) with w(x)dx k(x,y)(g(x) g(y))dy = 0. Find w as Ω Ω w(x) = w(x) γ(x), γ(x) = k(x, y)dy, Ω w(y) = w(x) k(x,y) dx =: (L w)(y). γ(x) Ω L compact linear positive operator with spectral radius 1. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

34 Checking the invariance principle Orbital derivative Ẇ T(S) 0 where T(S) = 0 only if S = S. If W(S(t),I(t)) does not depend on t, S(t) = S. From the differential equations: I(t) = Ĩ. Lemma Every strictly positive solution S,I that is defined for all times and is bounded satisfies S(t,x) = S(x) and I(t,x) = Ĩ(x) for all t R and x Ω. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35

35 Summary Laplace transform yields extinction results. Compact global attractors help avoiding linearized stability analysis. Solve eigenvalue problems to construct Lyapunov functions. H. R. Thieme (ASU) Tsing-Hua Univ, May / 35