We study the long-time behavior of the amount of material within the compartments of a compartmental system for which the flow of material does not have to be instantaneous and may even take an infinite time to occur. Results on the estructure of minimal sets for monotone skew-product semiflows, previously obtained by the authors, are applied to this description

Muñoz Villarragut, Víctor

Novo Martín, Sylvia

Obaya García, Rafael

idUS. Depósito de Investigación Universidad de Sevilla

XX Congreso de Ecuaciones Diferenciales y AplicacionesX Congreso de Matema´tica AplicadaSevilla, 24-28 septiembre 2007(pp. 1–7)Dynamics for a non-linear and non-autonomouscompartmental systemV. Mun˜oz1, S. Novo1, R. Obaya11 Departamento de Matema´tica Aplicada, E.T.S. de Ingenieros Industriales, Universidad de Valladolid,47011 Valladolid. E-mails: vicmun@wmatem.eis.uva.es, sylnov@wmatem.eis.uva.es,rafoba@wmatem.eis.uva.es.Keywords: monotone infinite delay equations, compartmental systems.AbstractWe study the long-time behavior of the amount of material within the compart-ments of a compartmental system for which the flow of material does not have to beinstantaneous and may even take an infinite time to occur. Results on the estruc-ture of minimal sets for monotone skew-product semiflows, previously obtained by theauthors, are applied to this description.1 Some previous resultsIn this section, we focus on some properties of monotone skew-product semiflows de-termined by a family of functional differential equations with infinite delay. They wereproved by Novo et al. [7] and provide infinite delay version of significative results obtainedby Jiang and Zhao [4].Let (Ω, σ,R) be a minimal flow over a compact metric space (Ω, d) and denote σ(t, ω) =ω·t for each ω ∈ Ω and t ∈ R. In Rn, we take the maximum norm ‖v‖ = maxj=1,...,n |vj |and the usual partial order relationv ≤ w ⇐⇒ vj ≤ wj for j = 1, . . . , n ,v < w ⇐⇒ v ≤ w and vj < wj for some j ∈ {1, . . . , n} .We consider the Fre´chet space X = C((−∞, 0],Rn) endowed with the compact-open topol-ogy, i.e. the topology of uniform convergence over compact subsets, which is a metric spacefor the distanced(x, y) =∞∑m=112m‖x− y‖m1 + ‖x− y‖m , x, y ∈ X ,1V. Mun˜oz, S. Novo, R. Obayawhere ‖x‖m = sups∈[−m,0] ‖x(s)‖. The subsetX+ = {x ∈ X | x(s) ≥ 0 for each s ∈ (−∞, 0]}is a positive cone in X, because it is a nonempty closed subset X+ ⊂ X satisfying X+ +X+ ⊂ X+, R+X+ ⊂ X+ and X+ ∩ (−X+) = {0}. Besides, it is normal and has an emptyinterior. As usual, a partial order relation in X is induced, given byx ≤ y ⇐⇒ x(s) ≤ y(s) for each s ∈ (−∞, 0] ,x < y ⇐⇒ x ≤ y and x 6= y .Let BU ⊂ X be the Banach spaceBU = {x ∈ X : x is bounded and uniformly continuous}with the supremum norm ‖x‖∞ = sups∈(−∞,0] ‖x(s)‖. Given r > 0 we will denoteBr = {x ∈ BU : ‖x‖∞ ≤ r} .As usual, given I = (−∞, a] ⊂ R, t ∈ I, and a continuous function z : I → Rn, zt willdenote the element of X defined by zt(s) = z(t+ s) for s ∈ (−∞, 0].We consider the family of non-autonomous infinite delay functional differential equa-tionsz′(t) = F (ω·t, zt) , t ≥ 0 , ω ∈ Ω , (1)ωdefined by a function F : Ω × BU → Rn, (ω, x) 7→ F (ω, x) satisfying the following condi-tions:(H1) F is continuous on Ω×BU and locally Lipschitzian in x for the norm ‖ · ‖∞.(H2) For each r > 0, F (Ω×Br) is a bounded subset of Rn.(H3) For each r > 0, F : Ω × Br → Rn is continuous when we take the restriction of thecompact-open topology to Br, i.e. if ωm → ω and xm d→ x as m→∞ with x ∈ Br,then limm→∞ F (ωm, xm) = F (ω, x).(H4) If x, y ∈ BU with x ≤ y and xj(0) = yj(0) holds for some j ∈ {1, . . . , n}, thenFj(ω, x) ≤ Fj(ω, y) for each ω ∈ Ω.(H5) If x, y ∈ BU with x ≤ y and xi(0) < yi(0) holds for some i ∈ {1, . . . , n}, thenzi(t, ω, x) < zi(t, ω, y) for each t ≥ 0 and ω ∈ Ω.(H6) There is an r > 0 such that all the trajectories with initial data in Br are uniformlystable in Br and relatively compact for the product metric topology.From Hypothesis (H1), the standard theory of infinite delay functional differentialequations (see Hino et al. [1]) assures that for each x ∈ BU and each ω ∈ Ω the system (1)ωlocally admits a unique solution z(t, ω, x) with initial value x, i.e. z(s, ω, x) = x(s) for eachs ∈ (−∞, 0]. Therefore, the family (1)ω induces a local skew-product semiflowτ : R+ × Ω×BU −→ Ω×BU(t, ω, x) 7→ (ω·t, u(t, ω, x)) , (2)2Dynamics for a non-linear and non-autonomous compartmental systemwhere u(t, ω, x) ∈ BU and u(t, ω, x)(s) = z(t+ s, ω, x) for s ∈ (−∞, 0].Hypothesis (H4) implies that the semiflow is monotone, i.e. u(t, ω, x) ≤ u(t, ω, y)provided that x ≤ y and whenever they are defined.Let z(t, ω0, x0) be a bounded solution of equation (1)ω0 . From Hypotheses (H1-H3)the omega-limit set of the trajectory of the point (ω0, x0)O(ω0, x0) = {(ω, x) ∈ Ω×BU : ∃ tm ↑ ∞ with ω0·tm → ω , u(tm, ω0, x0) d→ x}is a positively τ -invariant compact subset for the product metric topology, which admitsa flow extension.Under Hypotheses (H1–H6), the following result was proved in [7].Theorem 1.1. Assume that Hypotheses (H1–H6) hold and let (ω0, x0) ∈ Ω×Br be suchthat O(ω0, x0) ⊂ Ω×Br. Then O(ω0, x0) = {(ω, c(ω)) : ω ∈ Ω} is a copy of the base andlimt→∞ d(u(t, ω0, x0), c(ω0·t)) = 0 ,where c : Ω→ BU is a continuous map.2 Application to compartmental systemsCompartmental systems have been widely used as a mathematical model for the studyof the dynamical behavior of many processes in biological and physical sciences whichdepend on local mass balance conditions (see Jacquez and Simon [2, 3] for a review ofcompartmental systems with or without delay, Gyo¨ri [6], Gyo¨ri and Eller [5] and Wu andFreedman [8]).Firstly, we introduce the model with which we are going to deal as well as somenotation. Let us suppose that we have a system formed by n compartments C1, . . . , Cnamong which there is a flow of material through some pipes; we denote by Pi,j the pipetaking material from the ith comparment to the jth one for i, j ∈ {1, . . . , n} with i 6= j.We call C0 the environment surrounding the system and, for each i ∈ {1, . . . , n}, zi willdenote the amount of material within the compartment Ci. Let g˜i,j : R × R → R+ bethe so-called transport function determining the volume of material flowing from Cj to Cigiven in terms of the time t and the value of zj in t for i ∈ {0, . . . , n}, j ∈ {1, . . . , n}. Foreach i ∈ {1, . . . , n}, we will assume that there exists an incoming flow of material I˜i fromthe environment into the compartment Ci which only depends on time.Thus, considering a non-instantaneous flow of material among the compartments leadsus to a model governed by the following system of infinite delay differential equations:z′i(t) = −g˜0,i(t, zi(t))−n∑j=1j 6=ig˜j,i(t, zi(t)) +n∑j=1j 6=i∫ t−∞g˜i,j(τ, zj(τ))hi,j(t− τ)dτ + I˜i(t), (3)i ∈ {1, . . . , n}, where hi,j ∈ L1([0,+∞),R+) is the transit time density function, whichhas integral 1 on [0,+∞) and satisfies ∫∞0 s hi,j(s)ds < +∞.3V. Mun˜oz, S. Novo, R. Obaya2.1 The system on the hullFor simplicity, we define g˜i,0 : R × R → R, (t, v) 7→ I˜i(t) for i ∈ {1, . . . , n} and denoteg˜ = (g˜i,j)i,j : R × R → Rm, where m = n(n + 1). For each t ∈ R, let g˜t : R × R →Rm, (s, v) 7→ g˜(t+ s, v) as usual.Hereafter, the following property will be assumed:(C1) For all i, j ∈ {0, . . . , n}, g˜i,j ∈ C(R2,R+) and is uniformly continuous and boundedon all the sets of the form R× {v0} for v0 ∈ R.Under Hypothesis (C1), it is easy to check that the family {g˜t : t ∈ R} is equicon-tinuous and that the sets {g˜t(s, v) : t ∈ R} have compact closure in Rm for all s, v ∈ R.Consequently, since R2 is separable and Rm is complete, the set Ω = cls{g˜t : t ∈ R}turns out to be compact and metrizable when the compact-open topology is considered onC(R2,Rm) (see [1]). Henceforth, Ω will be considered to be endowed with this topology.Now, we define a flow on Ω byσ : R× Ω −→ Ω(t, (ωi,j)i,j) 7→ (ωi,j · t)i,j ,where ωi,j · t(s, v) = ωi,j(t + s, v) for all i, j and all t, s, v ∈ R. Again, we will use thenotation σ(t, ω) = ω · t, t ∈ R, ω ∈ Ω. It is clear that the flow σ is continuous.Let us introduce some more notation. Let g : Ω × R → Rm, (ω, v) 7→ ω(0, v). Thismap is continuous on Ω. We denote g = (gi,j)i,j . It is easy to check that, for all ω =(ωi,j)i,j ∈ Ω and all i ∈ {1, . . . , n}, ωi,0 is a function dependent only on t; thus, we candefine Ii = ωi,0, i ∈ {1, . . . , n}. Let F : Ω×BU → Rn be the map defined byFi(ω, x) = −g0,i(ω, xi(0))−n∑j=1j 6=igj,i(ω, xi(0)) +n∑j=1j 6=i∫ 0−∞gi,j(ω · τ, xj(τ))hi,j(−τ)dτ + Ii(ω),for (ω, x) ∈ Ω×BU and i ∈ {1, . . . , n}.We have a family of infinite delay differential equations corresponding to (1)ωz′(t) = F (ω·t, zt) , t ≥ 0 , ω ∈ Ω. (1)ωIf ω = g˜ is taken in (1)ω, we recover equation (3).2.2 Conditions for the systemMore properties of the map g˜ will be needed in order to apply the aforementioned resultsof [7]. The following hypotheses will be assumed throughout the reminder of this section.(C2) g˜i,j is locally Lipschitz in its second variable with Lipschitz constant independentfrom t.(C3) g˜i,j(t, v) ≤ g˜i,j(t, w) for all t, v, w ∈ R and all i, j whenever v ≤ w.(C4) g˜i,j(t, 0) = 0 for all t ∈ R and all i, j.4Dynamics for a non-linear and non-autonomous compartmental system(C5) g˜ = (g˜i,j)i,j is a recurrent function, i.e. the flow σ is minimal on Ω.This last property holds for instance if g˜, i.e. all the transport and input functions arealmost periodic or almost automorphic in t.Now, we can define the total mass of the system (1)ω asM : Ω×BU −→ R(ω, x) 7→n∑i=1xi(0) +n∑i=1n∑j=1j 6=i∫ 0−∞(∫ 0sgj,i(ω · τ, xi(τ))dτ)hi,j(−s)ds.Under assumptions (C1–C5), σ is a minimal flow on Ω and hypotheses (H1–H5) are sat-isfied. In particular, the skew-product semiflow (2) can be defined.Proposition 2.1. The total mass M is a continuous function on all the sets of the formΩ×Br with r > 0. Moreover, we have thatddtM(τt(ω, x)) =n∑i=1(Ii(ω · t)− g0,i(ω · t, zi(t, ω, x))) for t ≥ 0.Proposition 2.2. Let (ω, x) ∈ Ω× BU . If there exists r > 0 such that zt(ω, x) ∈ Br forall t ≥ 0, then, for all ε > 0, there exists δ > 0 such that‖z(t+ s, ω, x)− z(t, ω · s, y))‖ < εfor all t ≥ 0 whenever y ∈ Br and d(zs(ω, x), y) < δ.Corollary 2.3. If there exists a bounded solution of equation (1)ω, then all solutions of(1)ω are bounded as well.Thus, Hypothesis (H6) is satisfied and Theorem 1.1 may be applied to our equation.In particular, when a bounded solution exists, and hence all the solutions are bounded,we deduce that they are asymptotically of the same type as the transport functions,i.e. asymptotically almost periodic (resp. almost automorphic) if g˜ is almost periodic(resp. almost automorphic). This result is a generalization of the asymptotic constancyof solutions for the autonomous case.2.3 Long-time behavior of the solutionsIn order to state precisely some subsequent results, two definitions on the way pipesconnect compartments will be needed. Let I = {1, . . . , n}.Definition 2.4. Given i, j ∈ I, a pipe Pi,j is said to carry material (from compartmentCj to compartment Ci) if, for all t, v ∈ R, v = 0 whenever g˜i,j(t, v) = 0.Definition 2.5. Let τ : P(I) → P(I), J 7→ ∪i∈J{j ∈ I \ {i} : Pj,i carries material}. Asubset J of I is said to be irreducible if τ(J) ⊂ J and no proper subsets of J have thatproperty.Note that τ(I) ⊂ I, so there is always some irreducible subset of I. Our next resultgives a useful property of the irreducible sets.5V. Mun˜oz, S. Novo, R. ObayaProposition 2.6. If a subset J of I is irreducible, then, for all i, j ∈ J with i 6= j, thereexist s ∈ N and i1, . . . , is ∈ J such that Pi1,i, Pi2,i1 , . . . , Pis,is−1 and Pj,is carry material.Let J1, . . . , Jk be all the irreducible subsets of I and let J0 = I \ ∪kl=1Jl. These setsreflect the geometry of the compartmental system in a good enough way as to describethe long-time behavior of the solutions, as we will see below.Let K be a minimal subset of Ω × BU for the semiflow (2). In virtue of Theorem1.1, K will be of the form K = {(ω, x(ω)) : ω ∈ Ω} where x is a continuous map fromΩ into BU . It is clear that, in such circumstances, x gives the solution of (1)ω in thatz(t, ω, x(ω)) = x(ω·t)(0) for all ω ∈ Ω and all t ∈ R.Let us check that, on any minimal subset of Ω × BU for the semiflow (2), each com-partment is either empty all the time or it is never empty.Proposition 2.7. If there exist i ∈ J, ω ∈ Ω and t ∈ R such that xi(ω)(t) = 0, thenxi ≡ 0.All of the subsequent results give qualitative information about the long-time behaviorof the solutions. Let us see that, provided that we are working on a minimal set, if thesystem is closed, i.e. it has no inflow from the environment and has no outflow either,then the total mass is constant, all compartments out of an irreducible subset are emptyand, in an irreducible subset, either all compartments are empty or all are never empty.Theorem 2.8. Suppose that the system is closed, i.e. g˜0,i ≡ 0, I˜i ≡ 0 for all i ∈ I. Foreach c ≥ 0, there exists a minimal subset K of Ω× BU such that M |K ≡ c. Conversely,if K is a minimal subset of Ω × BU , then there exists c ≥ 0 such that M |K ≡ c. IfK = {(ω, x(ω)) : ω ∈ Ω} is a minimal subset of Ω×BU , then(i) xi ≡ 0 for all i ∈ J0.(ii) If, for some l ∈ {1, . . . , k}, there exists jl ∈ Jl such that xjl ≡ 0, then xj ≡ 0 for allj ∈ Jl.In our next result, it is seen that, provided that we are working on a minimal set,if there is no inflow from the environment, then the total mass is non-increasing, allcompartments out of an irreducible subset are empty and, in any irreducible subset withsome outflow of material, all compartments are empty. Besides, if an irreducible subsethas no outflow of material, then its compartments are all empty or all never empty.Theorem 2.9. If I˜i ≡ 0 for all i ∈ I, then(i) M is non-increasing.(ii) If K = {(ω, x(ω)) : ω ∈ Ω} is a minimal subset of Ω×BU , then xi ≡ 0 for all i ∈ J0.(iii) If K = {(ω, x(ω)) : ω ∈ Ω} is a minimal subset of Ω × BU and, for some l ∈{1, . . . , k}, there exists jl ∈ Jl such that g0,jl(ω, v) > 0 for all ω ∈ Ω and all v ∈ Rwith v > 0, then xj ≡ 0 for all j ∈ Jl.6Dynamics for a non-linear and non-autonomous compartmental system(iv) Suppose that, for some l ∈ {1, . . . , k}, g˜0,jl ≡ 0 for all jl ∈ Jl. Define the mass of JlasMl : Ω×BU −→ R(ω, x) 7→∑i∈Jlxi(0) +∑i∈Jl∑j∈Jlj 6=i∫ 0−∞(∫ 0sgj,i(ω · τ, xi(τ))dτ)hi,j(−s)ds.Then, for each c ≥ 0, there exists a minimal subset K = {(ω, x(ω)) : ω ∈ Ω}of Ω × BU such that xi ≡ 0 for all i ∈ I \ Jl and Ml|K ≡ c. Conversely, ifK = {(ω, x(ω)) : ω ∈ Ω} is a minimal subset of Ω × BU such that xi ≡ 0 for alli ∈ I \ Jl, then there exists c ≥ 0 such that Ml|K ≡ c. Moreover, if there existsjl ∈ Jl such that xjl ≡ 0, then xj ≡ 0 for all j ∈ Jl.Finally, in a non-closed system, i.e. a system which may have any inflow and anyoutflow of material, if there exists a bounded solution in an irreducible set which has someinflow, then all compartments of that irreducible set are never empty and there must besome outflow from the irreducible set. In particular, we have the following result.Theorem 2.10. Let us suppose that there exists a bounded solution of equation (1)ω. LetK = {(ω, x(ω)) : ω ∈ Ω} be a minimal subset of Ω×BU . If, for some l ∈ {1, . . . , k}, thereexist jl ∈ Jl and ω0 ∈ Ω such that Ijl(ω0) > 0, then(i) xj(ω)(t) > 0 for all j ∈ Jl, all t ∈ [0,+∞) and all ω ∈ Ω.(ii) There exist j ∈ Jl, ω1 ∈ Ω and v1 ∈ (0,+∞) such that g0,j(ω1, v1) > 0.In the general frame of non-closed systems, some results on the existence of boundedsolutions of (1)ω have been studied; specifically, the linear case is particularly interestingand some results in this direction have been reached.AcknowledgmentsThe authors were partly supported by Junta de Castilla y Leo´n under project VA024/03,and C.I.C.Y.T. under project MTM2005-02144.Bibliography[1] Y. Hino, S. Murakami, T. Naiko, Functional Differential Equations with Infinite Delay, Lecture Notesin Math. 1473, Springer-Verlag, Berlin, Heidelberg, 1991.[2] J.A. Jacquez, C.P. Simon, Qualitative theory of compartmental systems, SIAM Review 35 No.1(1993), 43–49.[3] J.A. Jacquez, C.P. Simon. Qualitative theory of compartmental systems with lags, Math. Biosci.180 (2002), 329–362.[4] J. Jiang, X.-Q. Zhao, Convergence in monotone and uniformly stable skew-product semiflows withapplications, J. Reine Angew. Math 589 (2005), 21–55.[5] I. Gyo¨ri, J. Eller, Compartmental systems with pipes, Math. Biosci. 53 (1981) 223–247.[6] I. Gyo¨ri, Connections between compartmental systems with pipes and integro-differential equations,Math. Modelling 7 (1986) 1215–1238.[7] S. Novo, R. Obaya, A.M. Sanz, Stability and extensibility results for abstract skew-product semiflows,J. Differential Equations 235 No. 2 (2007), 623-646.[8] J. Wu, H.I. Freedman, Monotone semiflows generated by neutral functional differential equationswith application to compartmental systems, Can. J. Math. 43(5) (1991), 1098-1120.7

Dynamics for a non-linear and non-autonomous compartmental system

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Dynamics for a non-linear and non-autonomous compartmental system

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