Analysis of Age-Structured Population Models with an Additional Structure

Transcription

1 Analysis of Age-Structured Population Models with an Additional Structure Horst R. Thieme Department of Mathematics Arizona State University Tempe, AZ 85287, USA Mathematical Population Dynamics Proceedings of the 2 nd International Conference, Rutgers Univ. 989 O. Arino, D.E. Axelrod, M. Kimmel, eds. Lecture Notes in Pure and Applied Mathematics 3, 5-26 Marcel Dekker 99 supported by a Heisenberg scholarship of Deutsche Forschungsgemeinschaft

2 Analysis of Age-Structured Population Models with an Additional Structure Horst R. Thieme Department of Mathematics Arizona State University Tempe, AZ 85287, USA Abstract It is illustrated that age-structured population models with an additional structure nicely fit into the framework of Lipschitz perturbations of non-dense operators which satisfy the resolvent estimates. This makes it not only possible to show that unique solutions to the model equations exist in a well-defined sense, but also that they generate a dynamical system or semiflow which leaves closed convex subsets invariant that satisfy an appropriate subtangential condition.. Introduction We consider the age-structured population model t + a ut, a = Aaut, a + F aut, ; t, a >. ut, = Gut,.; t >. u, a = u a; a. The solution ut, a of we look for takes values in a Banach space E. The Banach space E contains an additional structure of the population, whereas the age-structure has been made explicit in. How the individuals change with respect to the additional population structure which may be the size or/and the spatial distribution of the individuals is described by Aa. Note that the rate of change may depend on the age of the individual. Individual mortality rates and/or supported by a Heisenberg scholarship of Deutsche Forschungsgemeinschaft

3 immigration are described by nonlinear Lipschitz continuous operators F a on E. The population birth rate is given by a nonlinear operator G from L = L [,, E into E. The integrable functions L on [, with values in E are the right space for ut, a considered as a function of age because the L -norm gives the population size. Concrete examples for are age-dependent spatial spread on a bounded open domain Ω R n, n = 2, 3, t + a ut, a, ω = n j,k= ωj d jk a, ω ωk ut, a, ω + F a, ω, ut, ; t, a >, ω Ω ut,, ω = Gω, ut,.; u, a, ω = u a, ω; t >, ω Ω a, ω Ω ν ut, a, ω = ; t, a >, ω Ω or age and size dependent population growth t + a ut, a, σ = σ ga, σut, a, σ + F a, σ, ut, ; t, a, σ >. ut,, σ = Gσ, ut,.; t, σ >. u, a, σ = u a, σ; a, σ >. or combinations of these. Though age-dependent diffusion has been considered by Di Blasio 979, Gurtin 973, Gurtin&McCamy 979, Kunisch et al. 985, Langlais 985, 988, Marcati 98, Marcati&Serafini 979, Webb 982, and others see these references and Webb 985 for more information the general case that the diffusion operator depends both on age and space does not seem to have attracted much interest. Age and size dependent population growth has been dealt with by Tucker&Zimmerman 989. It is the aim of this paper to show how such models fit into the framework of Lipschitz perturbations of non-dense operators which satisfy the resolvent estimates. This theory which has started to be developed in Thieme to appear, b allows the unified analysis of these and many other models in population dynamics. 2

4 Here we restrict ourselves to showing the unique existence of solutions to which includes the nontrivial task of explaining in which precise sense can be satisfied. Since only certain subsets of L the non-negative functions, e.g. may make sense for the model, we also look for conditions under which these subsets are invariant under the solution flow. It readily follows from the theory developed in Thieme to appear, b that T tu a = ut, a provides a semiflow dynamical system, nonlinear semigroup on L. Thieme to appear, b also provides conditions for the regularity of solutions and the stability or instability of equilibria which can be translated for but are not worked out here. Further development of the theory will hopefully provide applicable conditions for the existence of periodic solutions, for asymptotically stable exponential growth in linear population models see Marcati&Serafini, 979, for existence of open dense sets of convergent points see Hirsch 988, Matano 987, Smith&Thieme to appear, submitted, etc. An alternative approach consists in considering as an abstract semilinear bounry value problem see Greiner, 987, 989. But the theory of non-dense operators which bases on the work by Prato&Sinestrari 987 and the theory of integrated semigroups Arendt 987a,b, Kellermann 986, Kellermann&Hieber 989, Neubrander 989, Thieme to appear, a seems to lead more directly to the precise meaning of the operator d + A. 2. Motivation of the approach Let us consider the nonlinear operator H xa = d xa + Aaxa + F ax defined on a suitable subset DH of L which we do not prescribe precisely here except that it contains the bounry condition x = Gx, i.e. DH {x L ; x continuous, x = Gx}. Our method consists in writing both F a and G as bounded perturbations of an extension of the operator d + A. 3

5 To this end we consider the Banach space X = E L and the closed subspace X = {} L. We set A, x = x, d x + A x, without explaining the precise meaning and domain of definition now, and Fx = Gx, F x. The intention of this construction becomes clear by noticing that H is the part of the operator H = A + F in X. Apparently the operator A is not densely defined, but it will turn out that A satisfies the resolvent estimates of semigroup theory. Our procedure consists in applying the theory of Lipschitz perturbations of non-densely defined operators which satisfy the resolvent estimates Thieme, to appear, b. 3. A semigroup on X and an integrated semigroup on X First of all the definition of the operator d + A in L see and of the operator A in X have to be made precise. We assume that there exists an evolutionary system Ut, s with the following properties: Assumption 3.. i The family Ut, s, t s, consists of bounded linear operators. ii Ut, sus, r = Ut, r if t s r. iii Us, sx = x if x E, s. iv Ut, sx is a continuous mapping of t for t s and x E. v There is a constant c > such that Ua + s, a c, s, a. vi Ut, sx is a Bochner measurable function of s, s t. 4

6 We tentatively define the operator d + A in and the operator A: d + Aaxa = lim h Ua, a hxa h xa h provided that this limit makes sense and is integrable. DA = {, x; x D d + A } for a.a a > 3. A, x = x, d + A x. 3.2 The definition of A takes for granted that any element x D d continuous at a =. The definitions of d + A and A are motivated by the fact that + A is { T txa = Ua, a txa t if a t 3.3 if a < t defines a strongly continuous semigroup on L and are modeled in analogy to the generator A of T. Actually A will turn out to be the restriction of d + A to those elements in D d + A with x =. It is convenient to identify T with a strongly continuous semigroup on X by setting T t, x =, T tx. Then A can be identified with the part of A in X. Let us formally determine λ A. Let i.e. u be a solution of λ A, u = η, ũ λua + u a Aaua = ũa, u = η. Analogy with non-autonomous differential equations suggests to try ua = e λa Ua, η + e λs Ua, a sũa sds. 3.4 Recalling the relation between the resolvent of the generator and the Laplace integral of a semigroup, we realize that the integral on the right hand side of this equation is the resolvent of T. As e λa Ua, η = λ a 5 e λt Ua, ηdt,

7 we have, formally, with Stη, ũ = λ A η, ũ = λ e λt Stη, ũdt 3.5, a Ht aua, η + Here H denotes the Heaviside function. considerations rigorous. T sũads. 3.6 We now start to make our formal Lemma 3.2. The family S of operators defined in 3.6 is an integrated semigroup which is locally Lipschitz continuous in the uniform operator topology. An integrated semigroup is a family of bounded linear operators St, t, satisfying the relations StSr = Sr + τ Sτdτ, S =. This definition is motivated by the fact that any strongly continuous semigroup T gives rise to an integrated semigroup S via S t = T sds. Proof: It is sufficient to check the functional equation for x = because, for ξ =, S is nothing else than the integrated semigroup originating from integrating T. Then the second component of SrStξ, is = minr,a minr,a = = r Ua, a sstξ, a sds Ua, a sht + s aua s, ξds minr,a Ht + s adsua, ξ [Ht + s a Hs a]dsua, ξ But this is the second component of r [St + s Ss]ξ, ds. The local Lipschitz continuity needs checking only for x = because, for ξ =, S is just the integrated semigroups originating from T. Now Ht a Hs a Ua, ξ Ua, ξ s 6

8 sup Ua, ξ t s. s a t For integrated semigroups a generator A can be defined in analogy to semigroups: u DA, Au = v d Stu = u + Stv, t. dt It has been shown by Thieme to appear, a that relation 3.5 holds between the resolvent of the generator A and the Laplace transform of an integrated semigroup S. Note the analogy with semigroups. Hence we can rigorously derive for the generator A of S that λ A η, ũ =, a e λa Ua, η + e λs Ua, a sũa sds. 3.8 We notice that this expression makes formal sense for λ = and so we guess that the generator A of S can be characterized as follows: ξ, x DA, Aξ, x = η, y iff ξ =, xa = Ua, η Ua, sysds. 3.9 It follows that any x with, x DA is continuous and x = η. Before we prove 3.9 we notice the following. Lemma 3.3. Stξ, x is continuously differentiable in t iff ξ =, and S t, x =, T tx. With this lemma in mind we translate the definition of the generator of an integrated semigroup to S. First we notice that DA X. Further, x DA, A, x = η, y iff Ua, a txa t = xa + Ht aua, η + mint,a 7 Ua, a sya sds 3.

9 Choosing t > a we obtain 3.9. As this is possible for any a, 3.9 is a necessary condition for 3. to hold. It is also sufficient for 3. to hold, if t > a. Let now t a. Then, by using 3.9 twice, t Ua, a txa t = Ua, a t Ua t, η Ua t, sysds = Ua, η t = xa + But this is 3. for t a. Ua, sysds = xa + a t Ua, a sya sds Ua, sysds Proposition 3.4. ξ, x DA if and only if ξ =, x is continuous, and xa = Ua, x Ua, sysds for some y L. Moreover A, x = x, y. Actually it is possible to describe A explicitly. Corollary 3.5. A, x = x, y with a Let x L and, x DA. Then x is continuous and ya = lim Ua, a hxa h xa for a.a. a >. h h b Let x, y L, x continuous and ya = lim Ua, a hxa h xa for a.a. a >. h h Then, x DA and A, x = x, y. 8

10 Proof: Let A, x = x, y. By proposition 3.4, = Ua, a h Hence h Ua h, x h xa = Ua, a hxa h Ua h, sysds a h Ua, a hxa h xa ya h Ua, sysds. a h a h Ua, sysds Ua, sys ya ds Ua, s ys ya ds + h a h h a h Ua, sya ya ds c h a h ys ya ds + h a h Ua, sya ya ds. with a suitable constant c which we obtain from the uniform boundedness theorem. Convergence to a.e. now follows from theorem and its corollary 2 in Hille&Phillips 957 and from the strong continuity of U. Let now y be the limit in the statement of the corollary. Then a Ua, sysds = lim Ua, s hxs h Ua, sxs ds h h h a h a = lim Ua, sxsds Ua, sxsds h h h a h = lim Ua, sxsds + Ua, sxsds h h a h = xa + Ua, x for a.a. a >. The last conclusion holds though we have not assumed that Ua, s is strongly continuous in s. To make it precise one has actually to take the L -norm of the left and the right hand side of this equation and to use the strong continuity of Ua, s in a. 9

11 Now we can give a precise meaning to the operator d + A see 3.2: Definition 3.6. Let x, y L. Then x D d A xa iff x is continuous and xa = Ua, x Ua, sysds. + A and ya = d + The justification of our tentative definition 3. follows from corollary 3.5. Corollary 3.7. Let x, y L. Then x D d A xa iff x is continuous and ya = lim Ua, a hxa h xa h h + A and for a.a. a >. d ya = + 4. Integrated solutions of inhomogeneous linear problems and the variation of constants formula With the procedure of section 3 the inhomogeneous linear equation t + a ut, a = Aaut, a + ft, a; t, a >. ut, = gt; t >. 4. u, a = u a; a. can be rigorously formulated as with vt = v + A vsds + hsds. vt =, ut,, v =, u, ht = gt, ft,. is uniquely solved by the variation of constants formula vt = S tv + d dt = T tx + lim λ St shsds T t sλλ A hsds.

12 See Kellermann&Hieber 989 and Thieme to appear, b. By using our characterization of A in proposition 3.4 becomes equivalent to us, ads = Ua, gsds + Ua, a ut, a + u a + i.e. to fs, a ds, us, ds = gsds ut, a = Ua, gt d dt u, a = u a Ua, a ut, a + Ua, a ft, a for a.a. a. Another way of giving 4. a precise meaning is the following: us, ds = gsds ut, a u a = d + A us, ads + fsds 4.2 with the operator d + A being defined by definition 3.6 and equivalently described by corollary 3.7. Translating yields { Ua, a tx a t + Ua, a sft s, a sds if a > t ut, a = a Ua, a sft s, a sds + Ua, gt a if a < t for a.a. a. This corresponds to the formula one obtains by integrating 4. along characteristic lines. 5. Solutions to the semilinear problem and invariance of closed convex sets Following section 4 we can rewrite as vt = v + A vsds + Fvsds. 5.

13 with vt =, ut,, v =, u, F, u = Gu, F u. 5.2 Formulations of 5. which are closer to can be translated from 4.2: us, ads = ut, a u a = Gusds d + A us, ads + F aus, ds 5.3 with the operator d + A being defined by definition 3.6 and equivalently described by corollary 3.7, or from. Solutions to 5. are equivalent to solutions to the equation vt = T tv + lim λ T t sλλ A Fvsds is useful in many respects, e.g. for proving the existence and uniqueness of solutions to 5., in particular if the equations only make sense in a subset C of L such that one has to guarantee that the solutions stay in C. In the following we assume that C is a closed convex subset of L. We make the following Assumptions 5.. a Let the evolutionary system Ut, s associated with the operator A satisfy the assumptions 3.. b Let the operators F a : C E satisfy and F au c + ua for u C F au F aũ c ua ũa for u, ũ C with some constant c >. c Let the operator G : C E satisfy Gu Gũ c u ũ for u, ũ C. 2

14 b. The subsequent result is a consequence of theorem 2.4 by Thieme to appear, Proposition 5.2. Under the assumptions 5. there exists a unique solution to 5.3 with values in C for any u C iff the following subtangential condition is satisfied for any u C : h dist T tu + ShF, u ; C, h. 5.5 Here T and S are the semigroup and integrated semigroup associated with the evolutionary system U via 3.3 and then takes the form h h dist T tu + T sf u ds + Ht U, Gu ; C, h. Actually it seems to be more useful to replace h T sf u ds by ht hf u in this formula. Using 3.3 we obtain the following form of the subtangential condition. Theorem 5.3. Under the assumptions 5. there exists a unique solution to 5.4 with values in C for any u C iff the following subtangential condition is satisfied for any u C : h dist a Ha hua, a h u a + hf au + Hh aua, Gu ; C for h. The following special case seems to be of particular interest: Let C be a closed convex set in E, C. We consider the closed convex subset C in L defined by C = {u L ; ua C for a.a. a > }. Corollary 5.4. Let the assumptions 5. be satisfied, u C. Then there exists a unique solution to 5.4 with values in C if the following holds: 3

15 a Ut, sc C, t s. b F satisfies the subtangential condition c GC C. h dist u + hf u, C, h, u C. It follows from section 3 in Thieme to appear, b that the definition T tu = ut defines a dynamical system strongly continuous nonlinear semigroup on L. Section 3 also provides regularity results. Section 4 in Thieme to appear, b provides conditions for the stability and instability of equilibria which can be translated to this model. 4

16 References Arendt, W. 987a: Resolvent positive operators and integrated semigroups. Proc. London Math. Soc. 3 54, Arendt, W. 987b: Vector valued Laplace transforms and Cauchy problems. Israel J. Math. 59, Da Prato, G.; Sinestrari, E. 987: Differential operators with non-dense domain. Ann. Sc. Norm. Pisa 4, Di Blasio, G. 979: Nonlinear age-dependent diffusion. J. Math. Biol. 8, Greiner, G. 987: Perturbing the bounry conditions of a generator. Houston J. Math. 3, Greiner, G. 989: Linearized stability for hyperbolic evolution equations with semilinear bounry conditions. Semigroup Forum 38, Gurtin, M.E. 973: A system of equations for age dependent population diffusion. J. Theor. Biol. 4, Gurtin, M.E.; MacCamy, R.C. 982: Product solutions and asymptotic behavior in age dependent population diffusion. Math. Biosc. 62, Hille, E.; Phillips, R.S. 957: Functional Analysis and Semigroups. AMS Hirsch, M. 988: Stability and convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383, -53 Kellermann, H. 986: Integrated Semigroups. Thesis. Tübingen Kellermann, H.; Hieber, M. 989: Integrated semigroups. J. Funct. Anal. 84, 6-8 Kunisch, K.; Schappacher, W.; Webb, G.F. 985: Nonlinear age-dependent population dynamics with random diffusion. Comp.&Maths. with Appls., Langlais, M. 985: A nonlinear problem in age dependent population diffusion. SIAM J. Math. Anal. 6, Langlais, M. 988: Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion. J. Math. Biol. 26, Marcati, P. 98: Asymptotic behavior in age dependent population diffusion model. SIAM J. Math. Anal. 2, Marcati, P.; Serafini, R. 979: Asymptotic behavior in age dependent population dynamics with spatial spread. Boll. Un. Mat. Ital. 6-B, Matano, H. 987: Strong comparison principle in nonlinear parabolic equations. Nonlinear Parabolic Equations: Qualitative Properties of Solutions. Pitman Research Notes in Mathematics 49. Longman Scientific & Technical Neubrander, F. 988: Integrated semigroups and their applications to the abstract Cauchy problem. Pac. J. Math. 35, -55 Pazy, A. 983: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer 5

17 Smith, H.L.; Thieme, H.R. to appear: Quasiconvergence and stability for strongly order preserving semiflows. SIAM J. Math. Anal. Smith, H.L.; Thieme, H.R. submitted: Convergence for strongly order preserving semiflows. Thieme, H.R. to appear, a: Integrated Semigroups and integrated solutions to abstract Cauchy problems. JMAA Thieme, H.R. to appear, b: Semiflows generated by Lipschitz perturbations of non-densely defined operators. Diff. and Int. Equations Tucker, S.L.; Zimmerman, S.O. 988: A nonlinear model of population dynamics containing an arbitray number of continuous structure variables. SIAM J. Appl. Math. 48, Webb, G.F. 982: A recovery-relapse epidemic model with spatial diffusion. J. Math. Biol. 4, Webb, G.F. 985: Theory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker. New York 6