Existence and uniqueness of solutions for a diffusion model of host parasite dynamics

Transcription

1 J. Math. Anal. Appl. 279 (23) Existence and uniqueness of solutions for a diffusion model of host parasite dynamics Michel Langlais a and Fabio Augusto Milner b,,1 a UMR CNRS 5466, Mathématiques Appliquées de Bordeaux, case 26, Université Victor Segalen Bordeaux 2, 146 rue Léo Saignat, 3376 Bordeaux, France b Department of Mathematics, Purdue University, 15 N. University St., West Lafayette, IN , USA Received 18 March 22 Submitted by B. Straughan Abstract Milner and Patton (J. Comput. Appl. Math., in press) introduced earlier a new approach to modeling host parasite dynamics through a convection diffusion partial differential equation, which uses the parasite density as a continuous structure variable. A motivation for the model was presented there, as well as results from numerical simulations and comparisons with those from other models. However, no proof of existence or uniqueness of solutions to the new model proposed was included there. In the present work the authors deal with the well posedness of that model and they prove existence and uniqueness of solutions, as well as establishing some asymptotic results. 23 Elsevier Science (USA). All rights reserved. Keywords: Size-structured population models; Diffusion models; Host parasite models; Existence and uniqueness 1. Introduction The goal of this paper is to supply an existence and uniqueness result for an initial boundary value problem arising in modeling the dynamics of a host parasite system and to prove some results about the large-time behavior of the solutions. * Corresponding author. address: milner@math.purdue.edu (F.A. Milner). 1 The work of this author was supported in part by NSF through grant No. INT X/3/$ see front matter 23 Elsevier Science (USA). All rights reserved. doi:1.116/s22-247x(3)2-9

2 464 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) The host parasite model we are interested in includes the parasite load p as a continuous variable and contains both convective and diffusive terms with respect to this variable. It was first described by Milner and Patton [7] and it reads ( t H p a(p) p H ) ( ( ) ) + p v p,t; H(p,t) H + µh (p)h =, p, t>. (1) H = H(p,t) gives the parasite density of the hosts and µ H (p) is the parasite-induced host mortality under a burden of p parasites. Here the convective velocity v represents the difference between parasite recruitment and mortality, and was first modeled nonlinearly in the parasite density variable p [5,6] using a truncated quadratic function earlier proposed by Langlais and Silan [4]. Milner and Patton later proposed a convective velocity linear in p, which gave better results in simulations [7], compared to Bouloux et al. [1]. It is given by v ( p,t; H(p,t) ) [ + ] H(p,t)dp = µ p (t) + ρ C + + p. (2) H(p,t)dp To complete this nonlinear problem one has an initial condition H(p,) = H (p), p, (3) and boundary conditions a(p) p H(p,t)+ v ( p,t; H(p,t) ) H(p,t)=, p=, +, t. (4) In Section 2 we list the assumptions we need to make on the coefficients and initial data, and we state the main result of this paper existence of solutions. In Section 3 we prove some a priori estimates that are necessary for the existence proof, and in Section 4 we give the full proof of the main result. Finally, in Section 5, we establish some results about the asymptotic behavior of solutions. 2. Main result We collect basic assumptions concerning the data. First, { H L (, + ), H (p), p, supp(h ) [,R] for some <R<+. (5) Also { a L (, + ), p a L (, + ), there exists a >, a(p) a, p, and µ p L (,T), T >, µ p (t), t, C >, <ρ, µ H L (,M) for any <M<+, there exist ɛ,ɛ 1,ɛ 2 suchthat µ H (p) ɛ + ɛ 1 p + ɛ 2 p 2, p, (6) (7)

3 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) and finally, either µ p (t) ρ, t T, or ɛ 2 >. (8) The boundary condition at p =+ will be contained in the functional spaces to which the solution belongs. By constructionone has v(,t,h)=. We shall consider the system ( t H p a(p) p H ) ( ( ) ) + p v p,t; H(p,t) H + µh (p)h =, p, t>, (9) v ( p,t; H(p,t) ) [ + ] 1 = µ p (t) + ρ C + + H(p,t)dp p, (1) H(p,t)dp with the initial condition at t = : H(p,) = H (p), p, (11) and the boundary condition on p = : a() p H(,t)=, t. (12) Definition. A solution of (9) (12) on (, ) (, T) is a nonnegative function H such that: H lies in L ((,T); L 1 (, )); H is exponentially decaying to at p =+ ; more precisely, for any fixed T> there exists a couple (α, β) with α>, β>such that H(x,t) exp(αt βp), p, t T ; H lies in L 2 (,T; H 1 (, )) and satisfies (9) (11) in a weak sense; lim t + H(,t)= H a.e. Theorem 1. Assume conditions (5) (8) hold. Then, given any positive and finite number T, problem (9) (12) has at least one solution. This solution is unique when ɛ > or when µ H and µ p (t) ρ, <t<t. Outline of the proof. We shall use the theorem of Schauder. Set X(T ) = { H : (, + ) (,T) R, H L (,T; L 1 (, ) )}. Given any nonnegative element H in X(T),let [ + ] v 1 (p, t) = µ p (t) + ρ C + + H (p, t) dp p. H (p, t) dp We shall prove below that the boundary value problem ( t H p a(p) p H ) ( + p v (p, t)h ) + µ H (p)h =, p, <t<t, (13) H(p,) = H (p), p, (14) a() p H(,t)=, <t<t, (15)

4 466 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) has a unique nonnegative solution H in X(T), according to the previous definition. Introducing a mapping Φ : X(T) X(T), Φ(H ) = H, any nonnegative fixed point for Φ is a solution of the original problem. 3. A priori estimates In this section we supply a priori estimates for an auxiliary linear initial and boundary value problem posed on the bounded domain (,n) (,T)of the p t space with <R<n, supp(h ) [,R]. This system reads wherein t H p ( a(p) p H ) + p ( w(p,t)h ) + µh (p)h =, p n, <t<t, (16) w(p,t) = [ µ p (t) + ρr(t) ] p, r L (,T), r(t) r < 1, t T. (17) To complete this problem one has an initial condition H(p,) = H (p), p n, (18) and boundary conditions { a() p H(,t)=, <t<t, (19) H(n,t)=, <t<t. According to the previous definition, (16) (19) has a unique suitable solution [2]. In order to quickly derive a priori estimates that do not depend on n, one assumes some smoothness properties on the coefficients; namely for some δ in (, 1): H C 2+δ ([, + )), supp(h ) (,R), µ p,r C δ ([, T )), (A) µ H C δ ([, )), a C 2+δ ([, )). When (A) holds, this auxiliary linear boundary value problem (16) (19) has a unique classical solution H C 2+δ,1+δ/2 ([,n] [,T]) [2]. The maximum principle yields H(p,t), p n, <t<t. (2) Upon integrating (16) over (,n)one gets for <t<t d dt n n + H(p,t)dp+ [ a(p) p H(p,t)+ w(p,t)h(p,t) ] p=n p= µ H (p)h (p, t) dp =.

5 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) Now, because of the second boundary condition (19), H achieves a minimum value along p = n so that a(n) p H(n,t) ; this together with (19) implies n t H(p,t)dp+ n µ H (p)h (p, s) dp ds Let us now look for a supersolution independent of n. Lemma 1. Let R H (p) dp, <t T. (21) z(p, t) = Ke αt βp2, where α>, <β<1, K= H () L (,+ ) eβr2. Then, for β sufficiently small and α sufficiently large, z is a supersolution for (16) (19). Proof. A straightforward computation gives ( t z p a(p) p z ) ( ) + p w(p,t)z + µh (p)z = [ A (p, t) + A 1 (t)p + A 2 (t)p 2 + µ H (p) ] z, with A (p, t) = α + 2a(p)β + ρr(t) µ p (t), A 1 (p, t) = 2β p a(p), A 2 (t) = 2β[2a(p)β + ρr(t) µ p (t)]. Keeping in mind that T is a fixed positive number, one has that A 2 (t) > for t T, when µ p (t) ρ and β is small enough, i.e., for <β 1 ( ) inf 2a µ p(t) ρ max r(t) t T t T it is bounded from below by a positive constant δ 2 (β). Then, for α large enough A (p, t) + A 1 (t)p + A 2 (t)p 2 + µ H (p) is also strictly positive and thus a larger constant α results in a positive right-hand side for µ H (p). Next, when µ H (p) ɛ 2 p 2, a sufficiently small β gives A 2 (t) + ɛ 2 δ 2 (β) >, t T. Once again, a large enough constant α gives a nonnegative right-hand side. Finally, z satisfies the boundary condition at p = and, from the choice of K above, one has that z(p, ) H (p). Thus, z is a supersolution. 4. Proof of Theorem 1 Let us begin with showing that the mapping Φ defined above makes sense. Lemma 2. For any nonnegative H X(T ) there is a unique nonnegative H X(T), asolutionof(13) (15).

6 468 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) Proof. For n R let H n be the nonnegative solution of (16) (19) with w = v ;set + r 1 (t) = C + + H (p, t) dp < 1. (22) H (p, t) dp Using the maximum principle one gets that the sequence (H n ) n R is nondecreasing in n. It is uniformly bounded in n in L ((,n) (,T)) by a function depending only on r (),(,T ) and lying in L 1 ((, + ) (,T)); see Lemma 1. From this and the Lebesgue s dominated convergence theorem one may conclude that as n + the sequence of functions { Hn (p, t) = Hn (x, t), p n, t T,, n p, t T, is strongly convergent in L q ((, ) (,T)) for any q 1toafunctionH. By a monotonicity argument it satisfies the estimate (21) with the upper limit n replaced by +. Last, the function z supplied in Lemma 1 is still a supersolution. Because one has chosen homogeneous Dirichlet boundary conditions on p = n, Hn(,t) lies in the first order Sobolev space H 1 (, ); upon integrating by parts one gets that the sequence (Hn) n R is bounded there. Actually, for <t<t 1 d 2 dt + n H 2 n t n t (p, t) dp + n µ H H 2 n (p, s) dp ds 1 2 a(p) p H n (p, s) 2 dp ds t n µ p (s)hn 2 (p, s) dp ds. As a consequence, the limit H is a solution of (9) and (11) in the desired weak sense and a t p H(p,s) 2 dp ds + 1 µ p () t 2,(,T ) t µ H (p)h 2 (p, s) dp ds H 2 (p, s) dp ds. (23) Last, for any finite M the sequence ( t Hn) n R is bounded in L 2 (,T;[H 1 (,M)] ), [H 1 (,M)] being the topological dual space of H 1 (,M). A standard compactness argument yields (Hn) n R is relatively compact in C ([,T]; L 2 (,M)) and the limiting function satisfies the initial condition (14). Uniqueness is proved upon integrating by parts, as in the derivation of (23). The next step consists in finding an invariant closed convex domain for Φ in X(T ).This domain should lie in the nonnegative cone of X(T). Again let H X(T ) be nonnegative and let r be given by (22). As pointed out in the proof of Lemma 2, H = Φ(H ) satisfies (21) with the upper limit n replaced by +. Having in mind (21), one has

7 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) H(p,t)dp+ t We can summarize these results into µ H (p)h (p, s) dp ds = Lemma 3. Let us define a subset C(T ) of X(T ) as { H X(T): p, <t<t H (p, t), H (p) dp, <t<t. (24) and H (p, t) dp } H (p) dp. Then C(T ) is a closed convex subset of X(T ) positively invariant by Φ. There exists β, <β < 1, α>, and K> such that for any H C(T ) the solution H = Φ(H ) satisfies H(p,t) K exp(αt β p), p, <t<t. The solution H = Φ(H ) satisfies the estimates (23) (24). The last step is Lemma 4. The mapping Φ : C(T ) C(T ) is completely continuous. Proof. Continuity relies on a well-known continuity dependence argument concerning the solutions to linear parabolic equations with respect to their coefficients, together with the exponential decay of solutions at p =+.Let(H k ) k be a sequence of functions in C(T ) converging to some limit H in X(T).Then(r k ) k will converge to r in L (,T) and therefore the sequence (H k = Φ(H k )) k will converge to H in any space occurring in the definition of solutions. Now, let (H λ ) λ Λ be a subset of C(T ), bounded in X(T ). From Lemma 3 the corresponding solutions (H λ = Φ(H λ )) λ Λ satisfy (23) (24) as well as an uniform exponential decay for t T and λ Λ at p =+. Arguing as we did in the proof of Lemma 2, for any finite M>theset(H λ ) λ Λ is relatively compact in C ([,T]; L 2 (,M)). It follows that, given any δ>, there exists M(δ) with H λ (p, t) δ, t T, p M(δ), and a sequence (H k(δ) ) k strongly converging in C ([,T]; L 2 (, M(δ))) to some limit H δ as k(δ) +. Using a diagonal argument one can construct a sequence (H k ) k strongly converging in C ([,T]; L 2 (, + )) to some limit H. Up to a subsequence it is also converging almost everywhere on (,T) (, + ). Invoking again the Lebesgue s dominated convergence theorem, one gets the convergence of the sequence (H k ) k in X(T). Thus the range of Φ is relatively compact in C(T ), as needed.

8 47 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) At this point we have proved the existence of at least one fixed point for Φ, i.e., at least one solution for our original problem. Let us now focus on uniqueness. Lemma 5. Problem (9) (12) has a unique solution when ɛ 1 >. Proof. Let H i,i= 1, 2, be to solutions with r i,i= 1, 2, being defined as in (24); set H = H 1 H 2 and r = r 1 r 2. Then, one gets from (9), for p, t>, [ t H p a(p) p H ( µ p (t) + ρr 1 (t) ) ph ] + µ H (p)h ( ( = p ρ r1 (t) r 2 (t) ) ) ph 2. (25) By definition, any solution decays exponentially to at p =+ at a rate β>. Let γ> be such that { } 2ɛ 1 γ<min β 1,β 2, µ p,(,t ) and let θ(p)= e γp. Multiplying both sides of (25) by θ(p)h(p,t) and integrating by parts, one has 1 d 2 dt + γ θ(p)h 2 (p, t) dp + a(p)θ(p) p H(p,s) 2 dp a(p)θ(p)h(p,t) p H(p,t)dp + 1 [ µp (t) + ρr 1 (t) ] [ γpθ(p)h 2 (p, t) θ(p)h 2 (p, t) ] dp 2 + µ H (p)θ(p)h 2 (p, t) dp = ρ [ r 1 (t) r 2 (t) ] ph 2 (p, t)θ(p) [ γh(p,t)+ p H(p,t) ] dp. (26) Since ph 2 (p, t)θ(p) L (,T; L 2 (, )), there exists c 1 L (,T) such that ph 2 (p, t)θ(p) [ γh(p,t)+ p H(p,t) ] dp [( 1/2 c 1 (t) θ(p)h 2 (p, t) dp) + ( θ(p) ( H(p,t) ) ) 1/2 2 dp ]. (27)

9 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) Next note that r 1 (t) r 2 (t) e (γ /2)p θ 1/2 (p)h (p, t) dp [ 1/2 γ 1/2 θ(p)h 2 (p, t) dp]. (28) It follows from (26) (28) together with the assumptions µ H (p) ɛ 1 p, ɛ 1 >, and a(p) a > forp that for a suitable constant c 2 >, 1 d 2 dt θ(p)h 2 (p, t) dp + ɛ 2 pθ(p)h 2 (p, t) dp c 2 θ(p)h 2 (p, t) dp. Gronwall s lemma now implies θ(p)h(p,t) forp and<t<t, which completes the proof of uniqueness. We can also show uniqueness of the solution in case there is no host mortality. Lemma 6. Problem (9) (12) has a unique solution when µ H (p). Proof. From (24) one immediately concludes that H(p,t)dp = H (p) dp for t. Hence the convective velocity in (1) is known and problem (9) (12) is linear, with a unique solution. 5. Asymptotic behavior of solutions We shall establish in this section some results about the large-time behavior of solutions of (9) (12). Lemma 7. Assume conditions (5) (8) hold and assume ɛ >. ThenH(,t) in L 1 (, ) as t. Proof. It follows from (24) that the function t H(p,t)dp is nonincreasing. Moreover, for ɛ > we know that H(p,t)dp decays exponentially to at a rate ɛ,which completes the proof. In some special cases, when there is no parasite-induced mortality and some of the other coefficients in the model are constant, we can establish the exact asymptotic behavior of solutions. Lemma 8. Assume conditions (5) (8) hold and assume also that for p a(p) a >, µ p (p) µ p ρ, and µ H (p).

10 472 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) Let β = µ [ p ρr() and k = e βp2 dp 2a ] 1 H (p) dp. (29) Then, as t, H(p,t) k e βp2 in L 2 (, ) and uniformly on [,M] for any M>. Proof. First, we show that H L ((, ) (, )) by exhibiting a supersolution. Let z(p) = K e βp2, with β given by (29) and K = H L (, )e βr2 (R defined by (5)). Now recall that, as was pointed out in the proof of uniqueness, when µ H (p) wehave r(t) r(). Then, a straightforward calculation yields ( t z p (a p z) + p [ µp + ρr()]pz ) =, t, p, a p z(,t)=, t, z(p, ) H (p), p. This shows z is a supersolution and for p,t wehave H(p,t) K e βp2. Next, we let H(p,t)= e βp2 H(p,t). Then, for p,t, H is nonnegative and bounded, and it is a solution of e βp2 t H p (a e βp2 p H)=, t, p, a p H(,t)=, t, (3) H(p,) = e βp2 H (p), p. From the approximating sequence used in Section 3, one can check that H satisfies e (1/2)βp2 p H L 2 (,T; L 2 (, )) for any T>. Hence, there exists τ>such that e βp2 p H(p,τ) dp <. Multiplying both sides of (3) by t H and integrating by parts over (τ, T ) (, ), we see that T τ e βp2 t H(p,t) 2 dp dt = 1 2 a e βp2 p H(p,T) 2 dp e βp2 p H(p,τ) 2 dp. (31) As a conclusion, the semi-orbits { H(,t), t } and {H(,t), t } are bounded in H 1 (,M)and relatively compact in C ([,M]) for each finite M>. Now, let us fix M >, and let (t n ) n be a sequence such that, as n, t n and H(,t n ) H M ( ) in C([,M]), and weakly in H 1 (,M). Upon iteratively extracting infinitely many subsequences, one sees that there exists a nonnegative continuous function H H 1 [, ) such that H(,t n ) H uniformly on each compact interval [,M],

11 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) M>M, and weakly in H 1 (, ). From Lebesgue s dominated convergence theorem it follows that H(,t n ) e βp2 H ( ) in L 1 (, ) and e βp2 H (p) dp = Last, from the a priori estimates in (31), that is e (1/2)βp2 t H L 2( τ, ; L 2 (, ) ) and e (1/2)βp2 p H L ( τ, ; L 2 (, ) ), H (p) dp. (32) and the large time behavior result of Langlais and Phillips [3], one may conclude that H is a nonnegative solution of the steady state problem associated to (3), that is H (p) k is the nonnegative constant given in (29) by (32). This convergence result being true for any M > and any convergent subsequence, the proof of Lemma 8 is thus complete. From (24) it is readily seen that the mapping t r(t) is nondecreasing; thus let lim r(t)= t r, r < 1. (33) It is obvious that µ p ρr when µ p (t) µ p ρ. It is interesting to note that this is still true if µ p <ρ. Lemma 9. Assume conditions (5) (8) hold and assume also that for p a(p) a and µ p (t) µ p <ρ. (34) Then, µ p ρr. Proof. Let us assume that µ p <ρr() and µ p <ρr. Then, for β small enough and K = H L (, )e βr2, z(p) = Ke βp2 is a supersolution. In fact, t z p (a p z) + ([ µ p + ρr(t) ]) p (pz) + µ H (p)z = [ ] [ ( ) 2a β + ρr(t) µ p z + 2β 2a β + ρr(t) µ p p 2 + µ H (p) ] z. Noting that (8) and (36) imply µ H (p) ɛ 2 p 2 for some ɛ 2 >, one has for p 2β ( 2a β + ρr(t) µ p ) p 2 + ɛ 2 p 2, provided β is positive and small enough. It then follows that 2a β + ρr(t) µ p. It suffices now to check the initial condition at t = and the boundary condition at p = toprovethatz is indeed a steady supersolution. In order to do this, multiply both sides of (9) by H and integrate by parts over (,T) (, ) to see that

12 474 M. Langlais, F.A. Milner / J. Math. Anal. Appl. 279 (23) so that 1 d 2 dt d dt + H 2 (p, t) dp + a p H(p,t) 2 dp ( µ H (p) + 1 ( ) ) ρr(t) µp H 2 (p, t) dp =, 2 H 2 (p, t) dp + (ρr µ p ) and Gronwall s lemma then yields the relation H 2 (p, t) dp e (ρr µ p )t H 2 (p, t) dp, H 2 (p) dp, t. Hence, as t,wehaveh(,t) inl 2 (, ) and a.e. in (, ). The supersolution allows to invoke Lebesgue s dominated convergence theorem to obtain, as t, that H(,t) inl 1 (, ). This implies that r =, contradicting the assumption µ p <ρr. References [1] C. Bouloux, M. Langlais, P. Silan, A marine host parasite model with direct biological cycle and age structure, Ecological Modelling 17 (1998) [2] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, Providence, [3] M. Langlais, D. Phillips, Stabilization of solutions of nonlinear evolution equations, Nonlinear Anal. 9 (1985) [4] M. Langlais, P. Silan, Theoretical and mathematical approach of some regulation mechanism in a marine host parasite system, J. Biol. Systems 3 (1995) [5] F.A. Milner, C.A. Patton, A new approach to mathematical modeling of host parasite systems, Comput. Math. Appl. 37 (1999) [6] F.A. Milner, C.A. Patton, Existence of solutions for a host parasite model, J. Comput. Appl. Math. 137 (21) [7] F.A. Milner, C.A. Patton, A diffusion model for host parasite interaction, J. Comput. Appl. Math., in press.