Structured Population Models: Measure-Valued Solutions and Difference Approximations

Transcription

1 Structured Population Models: Measure-Valued Solutions and Difference Approximations Azmy S. Ackleh Department of Mathematics University of Louisiana Lafayette, Louisiana 70504

2 Research Collaborators John Cleveland, University of Louisiana. Ben Fitzpatrick, Loyola Marymount University. Kazufumi Ito, North Carolina State University. Horst Thieme, Arizona State University.

3 Outline Present distributed rate model: space of measures as an appropriate space for some population models. Discuss asymptotic behavior of distributed rate model and survival of the fittest. Present a hierarchical continuous-time size-structured model. Discuss related literature and our recent work on this model. Develop a finite difference scheme to approximate the (singular) solution to this model. Present convergence analysis and numerical examples.

4 Distributed Rate Population Model The type of model we consider is based on the simple nonlinear dynamics d h i dt X(t) =X(t) f 1 (X(t)) f 2 (X(t)), Whichinvolvesanonlineargrowthratef 1 and a nonlinear mortality rate f 2 for the population, whose size at time t is denoted by X(t).

5 We begin with an adjustment to our previous equation, given by d h i dt x(t, q) =x(t, q) q 1 f 1 (X(t)) q 2 f 2 (X(t)). Here q =(q 1,q 2 ) Q denotes the subpopulationspecific growth and mortality rate parameters. The function x C 1 (R + ; C(Q)). The nonlinear terms, moreover,areassumedtodependonthetotalpopulation level, X(t) = R Q x(t, q)dq.

6 Lifting up the Model to Space of Measures As will be seen the natural space to examine this model is the space of measures. Thus, we want to extend the (density) model considered on the space C(Q) ½ ẋ(t) =F (x(t)) x(0) = x 0 to the space M = finite signed measures on the measurable space (Q, B(Q)) where B(Q) are the Borel sets on Q. Indeed, if f C(Q), A B(Q), dq is Lebesgue measure and μ f (A) = R A f(q)dq, thenf μ f is an embedding of C(Q), M.

7 Hence we reformulate as follows: if x C 1 (R + ; C(Q)) is asolutionto ½ ẋ(t) =F (x(t)) x(0) = x 0 Then d dt μ(t)(a) = μ x(t)(a) = R A ẋ(t)dq = R A F (x(t))dq and μ x(0) (A) = R A x 0dq = μ 0 (A). So we define ˆF : M M by ˆF (μ)(a) = R A F (x(t))dq, and for any μ 0 M we define the reformulated IVP to be ½ μ = ˆF (μ) μ(0) = μ 0 and the new state space to be C 1 (R + ; M).

8 Hence we seek, a measure-valued function μ : R + M + (non-negative Borel measures), satisfying the differential equation d dt μ(a) = Z A [q 1 f 1 (μ(q)) q 2 f 2 (μ(q))]μ(dq) =F(μ)(A) for all Borel sets A contained in Q. Due to the continuity of f 1 and f 2 and the boundedness of Q, it is clear that F maps non-negative Borel measures M + to signed Borel Measures M. In fact, it is also clear that F(μ) is absolutely continuous with respect to μ.

9 Existence-Uniqueness and Asymptotic Behavior We assume Q is a compact subset of (0, ) (0, ). If q =(q 1,q 2 ), q 1 is a scaled per capita reproduction rate and q 2 a scaled per capita mortality rate and the quotient q 1 q 2 is a scaled reproductive ratio, i.e. it is a measure of the average amount of offspring an individual of characteristic q produces during its lifetime. The actual reproductive ratio at population density X is given by q 1 f 1 (X) q 2 f 2 (X). We define n q1 o R =max ; q Q. q 2

10 Concerning the functions f 1 and f 2, we assume the following. (A1) f 1 is locally Lipschitz continuous on R +,nonnegative, and decreasing on [0, ). (A2) f 2 is locally Lipschitz continuous on +, nonnegative and increasing on [0, ), and strictly positive on (0, ). (A3) f 1 /f 2 is strictly decreasing on (0, ). f 1 (X) (A4) R lim sup X f 2 (X) < 1.

11 Under these assumptions, we see that for any q = (q 1,q 2 )thedifferential equation d h i dt Z(t) =Z(t) q 1 f 1 (Z(t)) q 2 f 2 (Z(t)) has a unique solution. If q = (q 1,q 2 ) Q and q 1 f q 2 lim 1 (X) X 0 f 2 > 1, the (X) above equation has a unique positive stable equilibrium. We call this equilibrium the carrying capacity of populations of characteristic q. This equilibrium is denoted by K(q). If q 1 f q 2 lim 1 (X) X 0 f 2 1, we set (X) K(q) = 0. Notice that K(q) only depends on q 1 q 2. Therefore we can define K = K(q ), q Q := {q Q; q 1 /q 2 = R}.

12 If R lim X 0 f 1 (X) f 2 (X) > 1, K istheuniquepositivesolution of Rf 1 (K ) f 2 (K )=0;otherwiseK =0. The monotonicity properties of f 1 and f 2 imply that K K(q) forallq Q which is related to the fact that R is the highest (scaled) reproductive ratio.

13 Theorem Let μ 0 be a non-negative Borel measure on Q. Then there exists a unique continuously Differentiable bounded function μ : R + M + such that Z d dt μ(a) = [q 1 f 1 (μ(q)) q 2 f 2 (μ(q))]μ(dq) =F(μ)(A) A for all Borel sets A contained in Q. Moreover μ(t) max{μ 0 (Q),K } t 0 and lim sup t μ(t)(q) K.

14 Of primary interest here is the asymptotic behavior, as t, of the solution μ(t)(dq). The following is an immediate consequence of the previous theorem. Corollary Let the assumptions (A1) to (A4) be f satisfied and R lim 1 (X) X 0 f 2 (X) 1. Then μ(t)(q) 0ast.

15 (A5) R lim X 0 f 1 (X) f 2 (X) > 1. We will show that μ(t)(q) K as t under certain assumptions concerning the initial data. Proposition Let Q = {q =(q 1,q 2 ) Q; q 1 q 2 = R, δ 0 > 0andU 0 be a relatively open subset of Q which contains Q.Letμ(t) be a solution such that for every δ > 0thereexistssomeq Q with μ(0)(b δ (q )) > 0. Then μ(t)(q \ U 0 ) 0as t.

16 Proposition Let (A1) to (A5) be satisfied. Let μ(t) be a solution such that for every δ > 0there exists some q Q with μ(0)(b δ (q )) > 0. Then lim t μ(t)(q) =K,whereK is the unique solution K of R f 1(K) f 2 (K) =1.

17 Finally we add the assumption that (A6) There exists a unique q Q such that q 1 = R = q2 max{q 1 /q 2 ; q Q}. Assumption (A6) states that there is a unique fittest subpopulation. It implies that μ(t) K δ q in the weak sense, i.e., ultimately the population trait characterized by q will be concentrated at q. Theorem Assume that (A1)-(A6) hold. Let μ(t)be a solution such that for every δ > 0 the initial datum satisfies μ(0)(b δ (q )) > 0. Then for every f(q), Z f(q)μ(t)(dq) f(q )K, t Q

18 Casting the problem in the space of measure of course allows the simultaneous treatment of both discrete and continuous cases. In particular, in the new setting one can choose the initial condition to be a linear combination of Dirac measures centered at a finite collection of points in the parameter space Q. These results can be modified to apply to competitive exclusion results for epidemic models similar to those considered by Bremmerman and Thieme (1991), Ackleh and Allen (2003,2005). One may wonder why such a general model favors a survival of the fittest asymptotic behavior. One of the reasons for this behavior is that the growth in our model is subpopulation specific (selection).

19 If for example, one consider the following generalization of our model (which allows for mutation): Z d dt x(t, q) =f 1(X(t)) ˆq 1 p(q, ˆq)x(t, ˆq)dˆq Q q 2 f 2 (X(t))x(t, q) Here p(q, ˆq)dq represents the probability that anindividual of type ˆq reproducing an individual of type q. If we let p(q, ˆq) =δ q (ˆq) then we obtain the selection (density) model discussed earlier.

20 Selection-Mutation Model We obtain the following DE for the selection-mutation model Z d dt μ(t)(a) =f 1(μ(t)(Q)) ˆq 1 Π(ˆq)(A)μ(t)(dˆq) Z Q f 2 (μ(t)(q)) q 2 μ(t)(dq) = ˆF (μ)(a) A where Π(ˆq)(A) = R A P (q, ˆq)dq. Note that if P (q, ˆq) = δ q (ˆq), then the above model reduces to the selection (measurevalued) model.

21 The Hierarchical Structured Model We consider the following hierarchical size-structured population model: u t +(g(x, Q(x, t))u) x + m(x, Q(x, t))u =0 g(x 0,Q(x 0,t))u(x 0,t)=C(t)+ u(x, 0) = u 0 (x). Here u(x, t) is the density of individuals having size x at time t, andfor0 α < 1, Q(x, t) =α Z x Z x1 x 0 x 0 w(ξ)u(ξ,t)dξ + β(x, Q(x, t))u(x, t)dx Z x1 x w(ξ)u(ξ,t)dξ,

22 α =0 When such models have been used to describe the dynamics of tree populations when light is limiting

23 Review of Related Literature The case α =1 Z x1 Q(x, t)=q(t)= w(ξ)u(ξ,t)dξ, This problem has been studied by several authors. Calsina and Saldana (1995) and Ackleh and Ito (1997) studied this problem. Diekmann, Gyllenberg, Metz, Thieme, and their co-workers (1998, 2001). x 0

24 The case 0 α < 1, and g Q 0 Cushing (1994) and Cushing and Henson (1996) Ackleh and Deng (2005) studied this problem for the age structured case (i.e., ). g(x, Q) =1 Calsina and Saldana (1997), Kraev (2001), Blayneh (2002) and Ackleh et al. (2004) studied the case g Q (x, Q) 0.

25 What happens if g Q 0 is not satisfied? To formally explain this, assume for simplicity that α =0, w =1, g = g(q), β = β(q), and m = m(q). Consider the IBVP u t + gu x = ( dg + m)u dx = (g Q Q x + m)u = mu + g Q u 2

26 Integrating the equation over (x, x 1 )weget: Q t + g(q)q x + M(Q) =0, d dt Q(x 0,t)=C(t)+B(Q(x 0,t)) M(Q(x 0,t)), Q(x, 0) = Q 0 (x). where M(Q) = Z Q 0 m(s)ds and B(Q) = Z Q 0 β(s)ds This is a local quasilinear hyperbolic equation in Q and for a<bwe have Q 0 (a) = Z x1 a u 0 (x)dx Z x1 b u 0 (x)dx = Q 0 (b) Hence, if the condition g Q 0 is not imposed then the characteristic curves may intersect causing a discontinuity in Q which corresponds to a Dirac delta measure in u since Q x = u.

27 Vanishing Viscosity Method For the sake of simplicity we first assume: m(x,q) =m(q), β(x, Q) =β(q), w(x) =1, and α =0. In particular, for ² > 0weconsiderthefollowing viscous problem u t +(g(x, Q) u) x + m(q)u = ²u xx g(x 0,Q(x 0,t)) u(x 0,t) ²u x (x 0,t) = C(t)+ R x 1 (1) x 0 β(q(y, t))u(y, t)dy u x (x 1,t)=0, where Q(x, t) = Z x1 x u(x, t) dx.

28 Integrating the previous PDE on (x, x 1 ), we obtain where Q t + g(x, Q)Q x + M(Q) =²Q xx d dt Q(x 0,t)=C(t)+B(Q(x 0,t)) M(Q(x 0,t)) Q(x 1,t)=0, M(Q) = Z Q 0 m(s) ds, B(Q) = Z Q 0 β(s) ds. This problem is a parabolic equation with a nonhomogeneous but local boundary value at x = x 0. Clearly the first boundary equation has a unique solution Q(x 0,t)whichsatisfies Q(x 0,t) e ωt Q(x 0, 0) + Z t 0 e ω(t s) C(s) dx K 1, for some positive constant K 1. t [0,T],

29 Define Q(x, t) =Q(x, t) x 1 x x 1 x 0 Q(x 0,t). Then we have ½ Qt + g(x, Q)Q x + M(Q) =² Q xx x 1 x Q(x 1,t)= Q(x 0,t)=0. x 1 x 0 d dt Q(x 0,t) Let H = L 2 (x 0,x 1 )andv = H0(x 1 0,x 1 ). Applying standard techniques for parabolic systems which are based on the Gelfand triple V H = H V we see that for ²>0the PDE has a unique solution Q ² C(0,T; V ) L 2 (0,T; H 2 (x 0,x 1 )) H 1 (0,T; H). Hence, PDE with nonhomogeneous boundary has a unique solution Q ² (0,T; H 1 ) L 2 (0,T; H 2 (x 0,x 1 )) H 1 (0,T; H).

30 Apriori Estimates and Existence Wewillprovethatlim ² 0 + Q ² and lim ² 0 + u ² exist, and thatlim ² 0 + u ² defines a measurevalued solution to (1). Multiplying by the test function φ =max(0,q ² (x, t) K 2 )weobtain Q ² (x, t) max( Q ² (x, 0), Q ² (x 0,t) ) K 2, where K 2 is a positive constant independent of ². Using the fact that u ² (x, t) 0and R x1 x 0 u ² (x, t) dx e R ωt x 1 x 0 u 0 (x) dx + R t 0 eω(t s) C(s) ds K 3.

31 Multiplying by Q, integrating over (x 0,x 1 ), and using the above estimates we get R d x1 Q ² (x,t) 2 dt x 0 2 dx + ² R x 1 x 0 ( Q ² ) x 2 dx K 4, for some positive constant K 4 (independent of ²> 0). Upon integrating the above inequality over (0,T) we see that ² Z T 0 Z x1 x 0 ( Q ² ) x 2 dxdt is uniformly bounded in ²>0.

32 Existence of Vanishing Viscosity Solution Therefore Q ² is bounded in L ((0,T) (x 0,x 1 )) L ((0,T); BV (x 0,x 1 )). Furthermore, it follows that d dt Q ² is bounded in L 2 (0,T; V ). Now, using Aubin s lemma to pass to the limit ² 0 +,wegetq ² has a strong convergent subsequence in L 2 ((0,T) (x 0,x 1 )) where the limit Q L ((0,T) (x 0,x 1 )) L (0,T; BV (x 0,x 1 )) satisfies R t 0 R x1 x 0 ( Qφ s G(x, Q)φ x +( G x (x, Q)+M(Q)) φ) dx ds + R x 1 x 0 (Q(x, t)φ(x, t) Q(x, 0)φ(x, 0)) dx =0, d Q(x dt 0,t)=C(t)+B(Q(x 0,t)) M(Q(x 0,t)), for all φ C 1 ((0,T) (x 0,x 1 )) satisfying φ(x 0,t)=φ(x 1,t)=0.

33 Moreover u(,t)= Q x (,t) is a measure-valued function and satisfies R t 0 R x1 x 0 ( uψ s g(x, Q)uψ x + m(q)uψ) dx ds + R x 1 x 0 (u(x, t)ψ(x, t) u 0 (x)ψ(x, 0)) dx + R t 0 (C(s)+B(Q(x 0,s))ψ(x 0,s) ds =0, for all ψ C 1 ((0,T) (x 0,x 1 )), where g(x, Q)u =(G(x, Q)) x G x (x, Q), m(q)u = (M(Q)) x L (0,T; C ).

34 The Finite-Difference Method We let x 0 =0andx 1 = 1. We consider a semi-implicit method as follows. Let x = 1 N and t = T M.Forx j = j x,j =0, 1, 2,,N and t k = k t, t =0, 1, 2,,M, we will denote by u k j and Qk j the difference approximation of u(t k,x j )andq(t k,x j ) and we define g k j = g(x j,q k j ), β k j = β(x j,q k j ), m k j = m(x j,q k j ) and C k = C(t k )

35 We assume that g(x, Q) > 0andg(1,Q)=0. Thesemiimplicit scheme is given by with u k+1 j u k j t + gk j uk+1 j g k j 1 uk+1 j 1 x g0u k k+1 0 = C k + P N i=1 βk i uk+1 i Q k j = α jx w i u k i x + i=1 + m k j uk+1 j =0, 1 j N x NX i=j+1 w i u k i x

36 Convergence Analysis If we define c k j =1+ t x gk j + tm k j, 1 j N then (2.1) is equivalently written as the following system oflinear equation for ~u k+1 =[u k+1 0,u k+1 1,...,u k+1 N ]T R N+1 A k ~u k+1 = f ~ k (3.1)

37 where ~f k =[C k,u k 1,u k 2,...,u k N] T and A k = If t x g0 k xβ1 k xβ2 k xβn k x gk 0 c k t x gk 1 c k t x gk N 1 c k N t Ĉ then (3.1) has a unique solution..

38 Lemma 1. u k j 0providedthatu0 j 0. Lemma 2. Assume m(x, Q) β(x, Q) ω 1 and ω 1 t< 1. We have the estimate ³ ku k 1 k 1 Q max = 1 ω 1 t k ku0 k 1 + P ³ k 1 i=1 1 ω 1 t k+1 i C i t. and 0 Q k j Q max.

39 Lemma 3. There exists an L > 0, independent of x, t, such that for any m>p NX Qm j Q p j t j=1 x L(m p) Define a family of functions {Q x, t } by Q x, t (x, t) =Q k j for x [x j 1,x j ), t [t k 1,t k ) Then, the set of functions {Q x, t } is compact in the topology of L 1 ((0, 1) (0,T)) and we have the following theorem.

40 Theorem 4. There exists a subsequence {Q xi, t i } {Q x, t } which converges to a BV ([0, 1] [0,T]) function Q(x, t) in the sense that for all t>0 and Z T 0 Z 1 0 Z 1 0 Q xi, t i (x, t) Q(x, t) dx 0, Q xi, t i (x, t) Q(x, t) dx dt 0, as i. Furthermore the limit function satisfies kqk BV ([0,1] [0,T ]) c( u 0 1, C ).

41 Theorem 5. The sequence Q x, t (x, t) constructed via our difference scheme converges in L ((0,T); L 1 (0, 1)) to the unique entropy solution Q(x, t). Corollary 6. Let u x, t (x, t) =u k j for x [x j 1,x j ), t [t k 1,t k ). Then, u x, t u in weak star topology of C for every t [0,T].

42 Numerical Results Example 1. Let α = 0 and choose the parameters g, m and β as follows: g(x, Q) =5(1 x)(q +0.01) exp( 2Q), m(x, Q) =(Q +1)exp(x) β(x, Q) =0.2x exp( 0.2Q) C(t) =1+sin(2πt).

43

44 Graph of Q(x,t) Graph of U(x,t)

45 Indefinite Growth Rate If g(x, Q) isindefinite but g(0,q) > 0andg(1,Q)=0,weusethe following approximation; u k+1 j u k j t if g k j > 0andg k j 1 < 0. + gk j u k+1 j 0 x + m k j u k+1 j =0, 1 j N u k+1 j u k j t if g k j 0andg k j gk j+1u k+1 j+1 gk j u k+1 j x + m k j u k+1 j =0, 1 j N u k+1 j u k j t if g k j 0andg k j+1 > gk j u k+1 j x + m k j u k+1 j =0, 1 j N

46 Example 2. Let α = 0 and choose the parameters g, m andβ as follows: g(x, Q) =2(1 x)(1 xq), m(x, Q) =0.01(Q +1)exp(x) β(x, Q) =0.2x exp( 0.2Q) C(t) =0.

47

48 Concluding Remarks We have developed a finite difference approximation to the singular solutions of the model. We have established convergence of the approximating environments Q x, t strongly in L ((0,T); L 1 (0, 1)) to the unique solution Q and showed that the approximating measures u x, t converge in the weak* topology to u the measure-valued solution of the model. While uniqueness of the entropy solution Q has been established. We have not established uniqueness of u. Investigate the case Q(x, t) = R 1 0 k(x, y)u(y, t)dy.