STABILITY OF THE METHOD OF LINES FOR THE MCKENDRICK - VON FOERSTER EQUATION

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1 FUNCTIONAL DIFFERENTIAL EQUATIONS VOLUME 2 213, NO 3 4 PP STABILITY OF THE METHOD OF LINES FOR THE MCKENDRICK - VON FOERSTER EQUATION H. LESZCZYŃSKI AND P. ZWIERKOWSKI Abstract. We analyze te meto of lines for te generalize McKenrick-von Foerster equation. Stability criteria are formulate in l an l 1 norms. Te results of te paper are base on comparison principles corresponing to ifferential equations. Key Wors. meto of lines, comparison, stability AMS(MOS) subject classification. 65M12, 65M2, 92B99 1. Introuction. Suppose tat u(t, x) enotes te ensity of iniviuals of a population wo are of age x at time t. Te initial istribution of te population is given by a function v. In [4] te birt process is escribe by te renewal equation, wereas te eat process is governe by te equation t u(t, x) + x u(t, x) + λ (x, P (t)) u(t, x) =, were P (t) = u(t, x) x is te total number of te members of te population at t. In our moel we assume tat te birt process is given by a fixe function ṽ an te flow of te caracteristics epens on t, x. It means tat te maturation rate of members of te population epens on time an teir age. In particular, as in [4], te age of iniviuals can be measure by teir metrical age. Moreover, te eat process epens on t, x, u an te total number of members of te population. Many biological moels, incluing age-structure populations, can be foun in [3] an [14]. Institute of Matematics, University of Gańsk, Wita Stwosza 57, Gańsk-PL Institute of Matematics, University of Gańsk, Wita Stwosza 57, Gańsk-PL 21

2 22 H. LESZCZYŃSKI AND P. ZWIERKOWSKI 1.1. Formulation of te PDE. Suppose tat E = [, T ] IR +, IR + = [, + ), T >. For given functions c: E IR + IR +, λ: E IR 2 + IR an v : IR + IR +, ṽ : [, T ] IR + consier te ifferential equation (1) u t u (t, x) + c (t, x) (t, x) = u(t, x)λ x wit te initial an bounary conitions ( t, x, u(t, x), (2) u(, x) = v(x) for x IR + ; u(t, ) = ṽ(t) for t [, T ]. ) u(t, y)y Trougout te paper we assume te consistency conition v() = ṽ() an te inequalities c(t, x) M c, (t, x) E, c(t, ) >, t [, T ], wit some real constant M c >. Te iea of te meto of lines (MOL) for PDE s is base on approximations of spatial erivatives by finite ifference quotients. In tis way we get a time epenent, finite or countable, system of orinary ifferential equations, equippe wit an initial conition. Wie classes of numerical proceures can be erive from MOL by te use of numerical metos for ODE s. Te monograp [17] presents MOL for ifferential an ifferential functional equations. Te textbook [18] provies etaile iscussion of numerical proceures in Matlab applie for systems of ODE s obtaine by MOL. Te meto of lines for initial an initial-bounary yperbolic functional ifferential problems is consiere in [7]. In te monograp [16] tere are investigate Rote s metos for parabolic (incluing integro-ifferential) an yperbolic problems, wose generalize solutions are efine as functions minimizing so-calle energy functionals, efine on suitable classes of amissible functions. An approximate minimization of energy functionals leas to Ritz, Galerkin an finite elements metos. Te book contains some results concerning te existence of approximate solutions, convergence an error estimates as well as numerous illustrative examples. Te paper [4] presents results concerning te existence an uniqueness of solutions of non-linear Lotka-McKenrick equations wit birt an eat moules affecte by te size of te wole population. Te results are obtaine by an analysis of integral equations corresponing to te main problem augmente wit a non-local bounary conition. Moreover, stability of an age equilibrium istribution of te population is stuie an some simple examples are given. A S-I-S type moel of a population wit infecte an susceptible iniviuals structure by age is consiere in [6]. Tere is assume tat te life-span

3 STABILITY OF THE METHOD OF LINES 23 of te inabitants is finite, infection spreas bot orizontally an vertically wit age-epenent infection an recovery parameters, wile a vertical transmission coefficient is constant. Te original problem is reformulate as an abstract evolution equation using C -semigroup settings an global existence an asymptotic beavior of solutions are stuie. Solutions of te problem are approximate by numerical solutions of an auxiliary problem. Tere is prove te convergence of te sceme as well as its asymptotic beavior. Te paper [1] reviews two groups of approximation metos for agestructure moels. Te first approac is base on a numerical meto of caracteristics. In tis case tere are use finite ifference metos for te ifferential equations along te caracteristics, or te integral representation of te solutions along te caracteristics. Te secon approac is base on finite ifference metos (for example upwin, Lax-Wenroff) for firstorer yperbolic PDE s. In all te cases tere are introuce respective quaratures (composite rectangular or mipoint, composite Simpson s rules) wic approximate te total number of inabitants or/an te number of offsprings. Consiere numerical proceures are applie for moels wit finite or infinite life-span. Some of reviewe metos are augmente wit illustrative numerical experiments. In [15] tere are given two approaces to an approximation of solutions of te Lotka-McKenrick equation. In te first one te main problem is reuce to a corresponing Volterra integral equation of te secon kin. Next, tere are use various rules (ybri, Lobatto) or Runge-Kutta metos, wic lea to approximate solutions of te renewal equation. Te secon one is base on a construction of finite ifference scemes for a ifferential equation wit an age profile. By an explicit secon-orer Runge-Kutta meto an te trapezoial rule tere is obtaine an implicit numerical proceure wic leas to an approximate solution of te problem. Te presente teory is illustrate by test examples solve numerically or by a symbolic computations package. Apart from age, te structure of a population can be istinguise by some pysiological features of its members suc as size, maturity, weigt, etc. Suc structures are known in te literature as a size-structure. In [2] tere is consiere a moel escribing evelopment of re cells of bloo. Te cells consist of two subpopulations. Te structure of precursors is given by teir maturity level wit a maturation spee epenent on concentration of te erytropoietin ormone. Te number of new precursor cells is ue to erytropoietin concentration. Mature precursor cells become mature erytrocytes wic are structure by teir age. Te eat process of precursors an mature re cells is governe by von Foerster-McKenrick equations. Solutions of iscusse moel are approximate by an implicit finite ifference

4 24 H. LESZCZYŃSKI AND P. ZWIERKOWSKI sceme. Te paper [2] presents convergence of te sceme in l, l 1. Tere are presente results of numerical experiments base on some experimental ata. Tere are very few papers in te literature ealing wit semiiscretization of integro-ifferential equations of matematical biology. In [9] tere is consiere a simplifie version of te Gurtin-MacCamy problem from [4]. Namely, te life-span is finite, te mortality coefficient epens only on te size of te population, an te number of offspring is given by a prescribe function. Tere is applie te Rote s meto, terefore a semiiscretization is conucte wit respect to te time variable. Tis proceure leas to a finite system of orinary ifferential equations (wit respect to te space variable) equippe wit an initial conition. Te Caucy problem is solvable an a recurrent formula for te solution can be written since a starting function is te initial istribution of te population. Tere is prove te bouneness of solutions of te semiiscrete meto an its convergence to te exact solution of te original problem. A trapezoial rule is use to approximate te integrals appearing in te solution of te Caucy problem. Tis approac leas to approximate solutions of te main problem. A convergence teorem is prove uner a restrictive assumption tat σ/ 2 is boune, were σ, are iscretization parameters for space an time, respectively. Results of presente numerical experiments attest tat tis assumption can be weakene. Te paper [19] eals wit a moel of an age-structure population augmente wit an aitional pysiological feature wic canges ue to some velocity epenent on te age. Terefore te ensity istribution of inabitants is a function of time t, age a an an aitional feature g. Te population mortality epens on te variables a an g, wereas te birt process is governe by a renewal equation wit a fertility function wic epens on t, a, g. Te time an age variables are iscretize wit te same iscretization parameter an te main problem is transforme to a system of orinary ifferential equations wit an inepenent variable g. Te convergence of presente semi-iscrete meto is prove. A generalize version of te moel wit te velocity of g epenent on g an a is briefly presente. Moels of a population ynamics wit an age (size-structure) structure can be equippe wit a spatial iffusion penomena. Ten te istribution of iniviuals epens on time, teir age, location, an te total number of inabitants is a function of time an location. Te paper [5] presents two kins of iffusion penomena, namely it istinguises iffusion as a ranom walk from iffusion as te movement to avoi crowing pressure. Tere are consiere tree forms of te birt laws, starting from te simplest (te birt rate epens only on te number of inabitants) to te most realistic (te

5 STABILITY OF THE METHOD OF LINES 25 birt rate is a prouct of te number of inabitants, age an some ecreasing exponential factor epenent on age). Tese consierations lea to six moels wic are consiere subject to two types of bounary conitions: witout any movement across bounaries or wit inospitable bounaries. In [1] tere is consiere a generalization of te moel from [5] as a multiimensional space variable is introuce. Te autors analyze separable solutions of te moel wit some simplifying assumptions concerning a eat-rate an a fertility function. Te existence, uniqueness an asymptotic properties of solutions are stuie. Tere are impose extinction or survival conitions for te consiere population. Te nonlinear case is stuie by means of an auxiliary linear problem. A numerical proceure for age-structure population moels wit iffusion is stuie in [8]. A problem is iscretize in time by a finite ifference meto along caracteristics, wereas te spatial variable is iscretize by a mixe finite element meto. A maturation-proliferation moel of te erytroi prouction is presente in [11]. Due to te biological mecanism of te evelopment of te re bloo cells precursors, te moel can o witout any bounary conition. Given an initial istribution, a solution of te moel can be foun by te meto of caracteristics. Te moel was successfully applie in a clinical terapy. A generalization of te moel from [11] is consiere in [13]. In a problem of [13] past states of population are taken into consieration by means of a functional epenence, represente by Hale type operators. Terefore, it is possible to inclue in te moel elays, eviations an integrals. Te problem requires some aitional assumptions concerning te flow of te caracteristics since te bounary conition is not given. Te main result of [13] is an existence teorem wose proof is base on te meto of bicaracteristics an an iterative proceure. A finite ifference sceme for te problem from [13], forwar in time an backwar wit respect to te spatial variable, is consiere in [2]. Its convergence of in l 1, l is prove by means of recurrence inequalities an comparison principles Semiiscretization. Suppose tat > is a iscretization parameter. Denote x (m) = m an E = [, T ] {,, 2,... }. Given any function u: E IR an (t, x (m) ) E, we write u (m) (t) = u(t, x (m) ), in particular c (m) (t) = c(t, x (m) ) for (t, x (m) ) E. Introuce te ifference operator δ u (m) (t) = u(m) (t) u (m 1) (t).

6 26 H. LESZCZYŃSKI AND P. ZWIERKOWSKI (3) Consier te meto of lines t u(m) (t) + c (m) (t) δ u (m) (t) ( ) = u (m) (t) λ t, x (m), u (m), u (m ) (t), m 1, m =1 (4) u (m) () = v(x (m) ) for m 1, u () (t) = ṽ(t) for t [, T ]. By L 1 (IR + ) enote te class of all real-value Lebesgue integrable functions on IR + wit te stanar norm L 1. Te supremum or maximum norms are enote by. For convenience, in te space l 1, of all summable sequences φ = (φ (m) ) m, we efine te seminorm φ 1 = φ (m). Definition 1. Let f L 1 (IR + ). Te function f is of class L 1 M iff tere is a ecreasing function F L 1 (IR + ) suc tat f(x) F (x) for x IR +. Observe tat if f L 1 M an f = (f(x (m) )) m, ten f 1 f L 1 <. It is easy to verify tat if v an ṽ, ten te solutions of (3), (4) wit continuous c, λ are nonnegative. 2. Auxiliary lemmas. Since te matrix obtaine by MOL epens on 1, we apply te ieas of [12] to erive acceptable estimates for solutions of (3), (4) in te supremum norms. Te same tecnique is use to prove stability in te supremum norm. Estimates for solutions of (3), (4) an for stability in l 1 are also erive by te metos base on [12]. By B (m) we enote te only solution of te problem (5) wic is equal to t B(m) (t) = c(m) (t) B (m) (t), B (m) () = 1, m, ( exp ) c (m) (s) s. Given a continuous function r : [, T ] IR an constants p, P IR suc tat p, consier te system (6) t Z(m) = c(m) (t) B (m) (t) Z(m 1) (t) B (m 1) (t) + P Z(m) (t) + r(t)e Q(t) B (m) (t), m 1,

7 wit te initial conition STABILITY OF THE METHOD OF LINES 27 (7) Z (m) () = p, m 1 an Z () (t) = pe Q(t) B () (t), t [, T ], were Q(t) = tp + 1 p r(s) s. Lemma 1. Suppose tat: 1. c: E IR +, r : [, T ] IR + are continuous; 2. real constants p, P satisfy p >, P. Ten tere exists a unique solution of (6), (7) suc tat Z (m) (t) for m, t [, T ], an (Z/B)(t) pe Q(t), t [, T ], were Z/B = (Z () /B (), Z (1) /B (1),...). Proof. Define Z (m) (t) = pe Q(t) B (m) (t), m. Ten Z (m), m, is te solution of system (6), (7) an Z (m) (t) for m, t [, T ]. Te assertion is a consequence of te efinitions of Z (m) an Q. Given real constants G 1, G 2, γ, γ an a function V L 1 (IR + ) consier te system (8) t Z(m) (t) = c(m) (t) B (m) (t) Z(m 1) (t) B (m 1) (t) + G 1Z (m) (t) + G 2 V (m) B (m) Z (m ) (t) (t) B (m ) (t) + γ V (m) B (m) (t), wit te initial conition m =1 m 1, (9) Z (m) () = γv (m), m 1 an Z () (t) = γv () B () (t), t [, T ]. If we multiply (8) by 1/B (m) (t), ten te system transforms to ( ) Z (m) (t) = c(m) (t) Z (m 1) (t) t B (m) (t) B (m 1) (t) Z(m) (t) Z (m) (t) + G B (m) 1 (t) B (m) (t) + G 2 V (m) Z (m ) (t) (1) B (m ) (t) + γ V (m), m 1, m =1

8 28 H. LESZCZYŃSKI AND P. ZWIERKOWSKI (11) Z (m) () = γv (m), m 1, Z () (t)/b () (t) = γv (), t [, T ]. (12) (13) Consier te ODE system Z () (t) t B () (t) = G 1 Z () (t) B () (t) + G 2V (1) W(t) + γ V (1), t W(t) = M Z () (t) c B () (t) + (G 1 + G 2 V 1 ) W(t) + γ V 1, wit te initial conition (14) Z () () = γv (), W() = γ V 1. Te system (12)-(14) as a unique solution, wic we enote by Z (), W. Consier an auxiliary system (15) Z (m) (t) t B (m) (t) = M c wit te initial conition ( ) Z (m 1) (t) B (m 1) (t) Z(m) (t) Z (m) (t) + G B (m) 1 (t) B (m) (t) + G 2 V (m) W (t) + γ V (m), m 1, (16) Z (m) () = γv (m), m 1, corresponing to (1), (11), were Z () Z (). Te problem (15), (16) as a unique solution, wic we enote by Z (m), m. Moreover, one can ceck tat Z (m) (t) W (t) =, t [, T ]. B (m) (t) We formulate a lemma on monotone properties of te sequence (Z (m) ) m. Lemma 2. Suppose tat: 1. te constants G 1, G 2, γ, γ are nonnegative; 2. V L 1 (IR + ) is nonnegative an ecreasing; 3. Z (m) : [, T ] IR +, m 1, satisfy (15), (16) an Z () (t) = Z () (t), t [, T ], were Z () is te solution of (12)-(14). Ten Z (m+1) (t)/b (m+1) (t) Z (m) (t)/b (m) (t), m, t [, T ]. Proof. Te proof is by inuction on m. Let m =. Ten ( ) ( ) ( Z (1) (t) t B (1) (t) Z() (t) Z (1) (t) = B () (t) B (1) (t) Z() (t) G B () 1 M ) c + β(t), (t)

9 were STABILITY OF THE METHOD OF LINES 29 Z () (t) β(t) = G 1 B () (t) + G 2V (1) W (t) + γ V (1) Z () (t) t B () (t) an β(t) =. Since Z (1) () Z () () = γ(v (1) V () ), te solution of te above ifferential equation satisfies Z (1) (t)/b (1) (t) Z () (t)/b () (t), t [, T ]. Suppose tat for some m > te inequality Z (m) (t)/b (m) (t) Z (m 1) (t)/b (m 1) (t), t [, T ], ols. Subtracting (15) for m + 1 an m, we get ( ) ( ) ( Z (m+1) (t) t B (m+1) (t) Z(m) (t) Z (m+1) (t) = B (m) (t) B (m+1) (t) Z(m) (t) G B (m) 1 M ) c (t) ( ) + γ (V (m+1) V (m) Z (m) (t) ) + B (m) (t) Z(m 1) (t) B (m 1) (t) + (V (m+1) V (m) )G 2 W (t) an Z (m+1) () Z (m) () = γ(v (m+1) V (m) ). To complete te proof one as to ceck tat te inequality Z (m+1) (t)/b (m+1) (t) Z (m) (t)/b (m) (t), t [, T ], wic follows from te above ifferential equation, is satisfie. Due to Lemma 2, te solution of (12)-(16), i.e. W, Z (m), m, satisfies te inequality Z (m+1) (t)/b (m+1) (t) Z (m) (t)/b (m) (t), t [, T ], m. Terefore, Z (m), m, is an upper solution of (1), (11), an tus Z (m), m, is an upper solution of (8), (9). Te existence of solutions of (8), (9) is prove by means of an iterative proceure, wic starts from Z (m), m, W. Moreover, uniqueness an estmates for solutions of (8), (9) in 1 will be erive. Lemma 3. Suppose tat: 1. c: E IR + is continuous, boune by a constant M c an satisfies te Lipscitz conition: c(t, x) c(t, x) L c x x for (t, x), (t, x) E, wit a constant L c ; M c

10 21 H. LESZCZYŃSKI AND P. ZWIERKOWSKI 2. constants G 1, G 2, γ, γ are nonnegative; 3. V L 1 (IR + ) is nonnegative an ecreasing. Ten tere exists a unique solution of (8), (9) suc tat for t [, T ], were (Z/B)(t) 1 [ γ V 1 + t ( γ V 1 + γ c () V ())] e Γt Z/B = (Z () /B (), Z (1) /B (1),...), Γ = L c + G 1 + G 2 V 1. Proof. Te proof is ivie into tree steps. 1. Existence. Te existence is base on an iterative proceure starting from an upper solution of (8), (9), namely Z (m), m, an te associate function W, wic are te unique solution of (12)-(16). Suppose tat for some k tere are given Z (m) k : [, T ] IR +, m, W k : [, T ] IR +, were Define (17) t Z(m) k+1 (t) = c(m) (t) W k (t) = Z (m) k (t) B (m) (t). B (m) (t) Z(m 1) k (t) B (m 1) (t) + G 1Z (m) k+1 (t) + G 2 V (m) B (m) (t)w k (t) + γ V (m) B (m) (t), m 1, wit te initial conition Z (m) k+1 () = γv (m), m 1, an (18) t Z() k+1 (t) = G 1Z () k+1 (t) + G 2V (1) W k (t) + γ V (1), Z () () = γv (). Since Z (m), m, is an upper solution of (8), (9) an te r..s. of (17), (18) are nonecreasing wit respect to Z k, W k, we obtain Z (m) k+1 (t) Z(m) k (t), m, W k+1 (t) W k (t), k, t [, T ]. Moreover, we ave Z (m) k (t) for m, k, t [, T ]. Terefore, by a monotonicity of Z (m) k wit respect to k, a solution of (8), (9) is given by Z (m) (t) = lim k + Z(m) k (t), m, t [, T ].

11 STABILITY OF THE METHOD OF LINES Uniqueness. Suppose Z (m), Z (m) : [, T ] IR +, m, are solutions of (8), (9). Denote Z (m) (t) = Z (m) (t) Z (m) (t). Subtracting te integral representation of (8) wit Z (m) an Z (m) we obtain Z (m) (t) an c (m) (s) B (m) (s) Z(m 1) (s) t s + G B (m 1) 1 Z (m) (s) s (s) + G 2 V (m) B (m) Z (m ) (s) (s) B (m ) s, m 1, (s) m =1 Z () (t) =, t [, T ]. Let us multiply te above inequalities by 1/B (m) (t). Ten, for m 1 we obtain Z (m) (t) B (m) (t) + G 1 c (m) (s) + G 2 V (m) B (m) (s) Z (m 1) (s) s B (m) (t) B (m 1) (s) B (m) (s) Z (m) (s) s B (m) (t) B (m) (s) B (m) (s) B (m) (t) m =1 Z (m ) (s) B (m ) (s) Taking into account tat B (m) (s)/b (m) (t) 1, s t T, an summing over m 1 te bot sies of te above inequalities multiplie by, we get ( ) Z () (s) t Mc W (t) M c s + B () (s) + G 1 + G 2 V 1 W (s) s, were Te Gronwall lemma leas to W (t) = Z (m) (t) B (m) (t). W (t) =, t [, T ], since Z () (t) = for t [, T ]. Hence Z (m) Z (m), m. 3. Te estimate. Suppose tat Z (m), m, is te nonnegative solution of (8), (9), or equivalently (1), (11). Multiplying te equations (1) by an summing its bot sies over m 1, we obtain Z (m) (t) ( ) Z = c (m) (m 1) (t) (t) t B (m) (t) B (m 1) (t) Z(m) (t) Z (m) (t) + G B (m) 1 (t) B (m) (t) + G 2 V (m) Z (m ) (t) (19) B (m ) (t) + γ V (m). m =1 s.

12 212 H. LESZCZYŃSKI AND P. ZWIERKOWSKI Define te function W (t) = Z (m) (t) B (m) (t). Ten, by (19) an te inequality ( ) Z c (m) (m 1) (t) (t) B (m 1) (t) Z(m) (t) B (m) (t) γ c() V () + L c W (t), we ave t W (t) γ c() V () + γ V 1 + (L c + G 1 + G 2 V 1 )W (t). Moreover, W () = γ V 1, wic follows from te initial conition (9). Hence, by te Gronwall lemma, we get W (t) [ γ V 1 + t ( γ V 1 + γ c () V ())] e Γt, were Γ = L c + G 1 + G 2 V 1. Te assertion follows from te efinition of te function W. We inten to estimate solutions of system (3), (4) wit respect to an 1 by means of Lemmas 1, 3. Lemma 4. Suppose tat: 1. c(t, x) M c, < c(t, ), (t, x) E, an c(t, x) c(t, x) L c x x for (t, x), (t, x) E; 2. λ(t, x, p, q) M λ for (t, x), (t, x) E, p, q IR + ; 3. V : IR + IR + is ecreasing, V L 1 (IR + ) an v(x) V (x), x IR + ; 4. ṽ : [, T ] IR + is continuous an ṽ V () ; 5. u (m) : [, T ] IR +, m, satisfy (3), (4). Ten u(t) V () e M λt, u(t) 1 e (M λ+l c )t ( V 1 + tv () c () ), t [, T ]. Proof. Let u (m) : [, T ] IR +, m, satisfy (3), (4). Denote ( ) λ (m) (t) = λ t, x (m), u (m) (t), u (m ) (t). m =1 Multiplying (3) by B (m), were B (m) is te solution of (5), we obtain t [u(m) (t)b (m) (t)] = c(m) (t) B (m) (t)u (m 1) (t) + u (m) (t)b (m) (t)λ (m) (t),

13 STABILITY OF THE METHOD OF LINES 213 an u (m) ()B (m) () = v (m), m 1, u () (t)b () (t) = ṽ(t)b () (t), t [, T ]. To erive te first estimate in te assertion let Z (m) : [, T ] IR +, m, be te only solution of problem (6), (7) wit P = M λ, p = V () max{ v, ṽ }, r(t) =, t [, T ]. We ave te following inequalities u (m) () Z (m) (), m 1, u () (t)b () (t) Z () (t), t [, T ]. Since λ (m) (t) M λ, one can verify by inuction on m tat u (m) (t)b (m) (t) Z (m) (t), m, t [, T ]. Terefore, accoring to Lemma 1, we ave te estimate u(t) max{ v, ṽ }e M λt, t [, T ]. Suppose tat Z (m) : [, T ] IR +, m, is te only solution of (8), (9) wit Since ṽ V (), we ave G 1 = M λ, G 2 =, γ = 1, γ =. u (m) () Z (m) (), m 1, u () (t)b () (t) Z () (t), t [, T ]. Moreover, as λ (m) (t) M λ, te estimate u (m) (t)b (m) (t) Z (m) (t), m, t [, T ], can be cecke by inuction on m. Ten, by Lemma 3, we obtain Te proof is complete. u(t) 1 e (M λ+l c )t ( V 1 + tv () c () ), t [, T ]. 3. Stability. Consier te meto of lines wit te perturbe rigtan sie (2) tū(m) (t) + c (m) (t) δ ū (m) (t) = ( = ū (m) (t) λ t, x (m), ū (m) (t), ) ū (m ) (t) + m =1

14 214 H. LESZCZYŃSKI AND P. ZWIERKOWSKI +ξ (m) (t), m 1, an te perturbe te initial an bounary conitions (21) ū (m) () = v(x (m) )+ ˆξ (m) for m 1; u () (t) = ṽ(t)+ ξ(t) for t [, T ]. Te main result of te paper, i.e. stability of MOL in, 1, is base on Lemmas 1, 3, respectively. Teorem 1. Assume tat: 1. c: E IR + is continuous, c(t, x) M c for a constant M c >, < c(t, ) an c(t, x) c(t, x) L c x x for (t, x), (t, x) E; 2. λ: E IR 2 + IR is continuous, boune by a constant M λ > an λ(t, x, p, q) λ(t, x, p, q) L λ p p + ˆL λ q q for (t, x) E, p, p, q, q IR + ; 3. ū, u are solutions of (2), (21) an (3), (4), respectively, for wic tere is a ecreasing function V : IR + IR + suc tat V L 1 (IR + ) an ū (m) (t), u (m) (t) V (m) for t [, T ], m ; 4. tere are C, C > an α as suc tat ˆξ (m) α CV (m), ξ (m) (t) α C V (m), m 1, ξ(t) α CV () for t [, T ]. Ten ū (m) (t) u (m) (t), ū (m) (t) u (m) (t) 1 as on [, T ]. Remark 1. Te assumptions on te perturbations in (2), (21) mean tat ˆξ 1, ˆξ as, wereas ξ(t) 1, ξ(t), ξ(t) uniformly on [, T ] as. Proof. Denote u (m) (t) = ū (m) (t) u (m) (t) an λ (m) (t) = λ (m) (t) λ (m) (t), were ( ) λ (m) (t) = λ t, x (m), ū (m) (t), ū (m ) (t). m =1 Subtracting te bots sies of (4), (21) an (3), (2), respectively, we obtain u () (t) = ξ(t) an u (m) () = ˆξ (m), (22) t u(m) (t) + c(m) (t) u (m) (t) = c(m) (t) u (m 1) (t) + u (m) (t) λ (m) (t) + u (m) (t) λ (m) (t) + ξ (m) (t)

15 STABILITY OF THE METHOD OF LINES 215 for m 1. Let B (m) be te only solution of (5). Multiplying (22) by B (m) (t), we get u () (t)b () (t) = ξ(t)b () (t) an t [ u(m) (t)b (m) (t)] = c(m) (t) B (m) (t) u (m 1) (t) + B (m) (t) u (m) (t) λ (m) (t) + B (m) (t)u (m) (t) λ (m) (t) + B (m) (t)ξ (m) (t), m 1, wit te initial conition u (m) ()B (m) () = ˆξ (m), m 1, or equivalently u (m) (t)b (m) (t) = ˆξ (m) c (m) (s) B (m) (s) u (m 1) (s) s B (m) (s) u (m) (s) λ (m) (s) s B (m) (s)u (m) (s) λ (m) (s) s + Hence, we obtain u () (t) B () (t) α CV () B () (t) an u (m) (t) B (m) (t) α CV (m) + + (23) + (M λ + L λ V () ) + ˆL λ V (m) B (m) (s) c (m) (s) B (m) (s) u (m 1) (s) s+ B (m) (s) u (m) (s) s B (m) (s)ξ (m) (s) s. u (m ) (s) s + α C V (m) B (m) (s) s m =1 for m 1. Consier te comparison integral system Z () (t) = α CV () B () (t), Z (m) (t) = α CV (m) + + (M λ + L λ V () ) c (m) (s) + ˆL λ V (m) B (m) (s) Z (m) (s) s m =1 for m 1, wic can be written as follows B (m) (s) Z(m 1) (s) B (m 1) (s) s Z (m ) (s) B (m ) (s) s + α C V (m) B (m) (s) s

16 216 H. LESZCZYŃSKI AND P. ZWIERKOWSKI t Z(m) (t) = (24) = c(m) (t) B (m) (t) Z(m 1) (t) B (m 1) (t) + (M λ + L λ V () )Z (m) (t) + ˆL λ V (m) B (m) Z (m ) (t) (t) B (m ) (t) + α C V (m) B (m) (t), m =1 wit te initial an bounary conitions (25) Z (m) () = α CV (m), m 1, Z () (t) = α CV () B () (t), t [, T ]. Denote Z/B = (Z /B, Z 1 /B 1,...). Te estimate for (Z/B)(t) 1 is erive by means of Lemma 3. Consier te system (8), (9) wit G 1 = M λ + L λ V (), G 2 = ˆL λ, γ = α C, γ = α C. By Lemma 3 we obtain (Z/B)(t) 1 α ω(t), t [, T ], were ω(t) = [ C V 1 + t ( C V 1 + C c () V ())] e Γt, an Γ = L c + M λ + L λ V () + ˆL λ V 1. Define Q(t) = t(m λ + L λ V () ) + ˆL λ ω(s) + C s, t [, T ]. C Te estimates (Z/B)(t) 1 α ω(t), 1 e Q(t), t [, T ], an V (m) V (), m, applie to (24), (25) yiel t Z(m) (t) c(m) (t) B (m) (t) Z(m 1) (t) B (m 1) (t) + (M λ + L λ V () )Z (m) (t) + α V () (ˆLλ ω(t) + C ) e Q(t) B (m) (t), m 1, wit te initial an bounary inequalities Z (m) () α CV (), m 1, Z () (t) α CV () B () (t)e Q(t), t [, T ]. Te solution of te above system of inequalities is estimate by te solution of (6), (7) wit P = M λ + L λ V (), p = α CV (), r(t) = α V () (ˆL λ ω(t) + C ).

17 Hence, by Lemma (1), we obtain One can verify tat STABILITY OF THE METHOD OF LINES 217 (Z/B) α CV () e Q(t), t [, T ]. u (m) (t) B (m) (t) Z (m) (t), m, t [, T ], were Z (m), u (m) are solutions of (24), (25) an (??), respectively. Terefore u(t) 1 (Z/B)(t) 1, u(t) (Z/B)(t) on [, T ]. Hence u(t) 1, u(t) as on [, T ], wic completes te proof. 4. Examples. 1. Consier te ifferential equation (26) t u(t, x) + c x u(t, x) = Λu(t, x), c, Λ IR, c >, wit te initial an bounary conitions (27) u(, x) = v(x), x IR +, u(t, ) = ṽ(t), t [, T ], were v : IR + IR +, ṽ : [, T ] IR + are given functions suc tat v() = ṽ(). Te solution of te above problem is u: [, T ] IR + IR +, v(x ct)e Λt, x ct, (28) u(t, x) = ṽ(t x c )eλ x c, x < ct. If v L 1 (IR + ), ten te total number of iniviuals is given by u(t, x) x = c ṽ(s)e Λ(t s) s + e Λt v L 1. For a fixe iscretization parameter >, consier te meto of lines for (26), (27): (29) t u(m) (t) + c u(m) (t) u (m 1) (t) = Λu (m) (t), m 1, (3) u (m) () = v(x (m) ), m 1, u () (t) = ṽ(t), t [, T ].

18 218 H. LESZCZYŃSKI AND P. ZWIERKOWSKI Te unique solution of (29), (3) is given by u () (t) = ṽ(t), u (m) (t) = e (Λ c )t m i=1 v (i) ( c ) m i t m i c ( c m (m i)! + e(λ ) )t I (m) (t), were I (m) (t) = 1 m 1... ṽ(t m )e (Λ c )t m t m t m 1... t 1 for m 1, t [, T ]. Suppose v L 1 M, i.e. v L1 (IR + ), tere is a ecreasing function V : IR + IR + suc tat V L 1 (IR + ) an v(x) V (x), x IR +. We also assume ṽ is continuous an ṽ V (). Ten if we apply Lemma 4 wit L c =, M λ = Λ, we obtain te estimates u(t) V () e Λ t, u(t) 1 ( V 1 + tcv () )e Λ t, t [, T ]. 2. Let Λ IR. Consier te ifferential equation (31) t u(t, x) + x u(t, x) = Λu(t, x)z(t), were z(t) = u(t, x) x, wit te initial an bounary conitions (32) u(, x) = v(x), x IR +, u(t, ) = ṽ(t), t [, T ], v : IR + IR +, ṽ : [, T ] IR + are given functions suc tat v() = ṽ() an v L 1 (IR + ). Applying te meto of caracteristics we fin u: [, T ] IR + IR +, wic is te solution of (31), (32) (33) u(t, x) = v(x t) exp ( Λ z(s) s), x t, ṽ(t x) exp ( Λ t x z(s) s), x < t. Te total number of inabitants z satisfies te integral equation ( ) ( ) z(t) = ṽ(t x) exp Λ z(s) s x + v L 1 exp Λ z(s) s. t x Te above integral relation is equivalent to te Riccati equation z (t) ṽ(t) = Λz 2 (t), z() = v L 1. If ṽ = const, ten te solution of tis initial value problem can be foun analytically. Oterwise, te solution can be approximate by a numerical

19 STABILITY OF THE METHOD OF LINES 219 proceure. We estimate te existence omain of z. Consier te comparison problem ẑ (t) ṽ + Λẑ 2 (t), ẑ() = v L 1, wit respect to te above Riccati equation. Te function z is efine at least on te interval [, T ], were ( ) π/2 arctan v L 1 Λ/ ṽ T, Λ ṽ provie tat Λ >, ṽ =. Moreover, z(t) ẑ(t) on [, T ] an tere is ˆM > suc tat ẑ(t) ˆM for t [, T ]. Suppose v L 1 M. Consier te meto of lines for (31), (32) wit a iscretization parameter > : (34) t u(m) (t) + u(m) (t) u (m 1) (t) = Λu (m) (t)z(t), m 1, (35) u (m) () = v(x (m) ), m 1, u () (t) = ṽ(t), t [, T ], were z(t) = u (m) (t), t [, T ]. If we sum te bot sies of (34) for m 1, ten we obtain (36) z (t) ṽ(t) = Λz 2 (t), z() = v 1. Assume tat z satisfies (36). Ten te solution of (34), (35) is given by te following recurrent proceure: u () (t) = ṽ(t) an ( ( u (m) (t) = v (m) exp ) ) Λz(s) 1 s + 1 ( τ t t ) u (m 1) (τ) exp + Λ z(s) s τ for m 1, t [, T ]. If we take c = 1, L c =, M λ = Λ ˆM in Lemma 4, ten we obtain te estimates u(t) V () e Λ ˆMt, u(t) 1 ( V 1 + tv () )e Λ ˆMt, t [, T ]. 3. Numerical examples We present two numerical examples. Te experiments are performe by Matlab version 7.1 an its ODEs solver oe45, wic generates a mes on τ

20 22 H. LESZCZYŃSKI AND P. ZWIERKOWSKI te interval [, T ], wose iameter is enote by t. We coose = 1 3 an perform te computation for m =, 1,..., N. Let u be te solution of te PDE from examples 1, 2, an ū is an approximate solution of te respective meto of lines eiter for example 1 or 2. Define te error of tis approximation u = max u(t, t,m x(m) ) ū (m) (t), were (t, x (m) ) belongs to te mes. Let were z(t) = u(t, x) x an (37) z N (t) = z N = max z(t) z N (t), t N ū (m) (t). Te quantity z N measures te influence of N on te istance between z(t) an te quarature (37). Our first experiment refers to (26)-(27) an te corresponing meto of lines (29)-(3). Suppose tat T = 1, Λ = 1 1, c = 2 an Ten z(t) = c Λ ṽ(t) = 1 + sin(πt), t [, 1], v(x) = 1 + sin2 x 1 + x 2, x IR +. ( 1 e Λt ) + c π 2 + Λ 2 ( πe Λt Λ sin(πt) π cos(πt) ) + e Λt v L 1, were v L 1 = π(3 4 e 2 ). In our experiments we obtaine u =.126 an t = Figure 1 presents te error z N as a function of N, wereas Figure 2 sows te plots of ū (m) for m =, 15, 3,..., 75. Consier te example (31)-(32) an te associate meto of lines (34)- (36). We begin wit some special case. Suppose tat T = 1, Λ = 1 1 an ṽ(t) = 1, t [, 1], v(x) = 1 + sin2 x, 1 + x 2 x IR +. Ten v L 1 = π 4 (3 e 2 ). Problem (36) wit te above ata as te solution z(t) = ( Λ 1 tan t ) Λ + D,

21 STABILITY OF THE METHOD OF LINES 221 ) were D = arctan ( v 1 Λ. From (33) we obtain cos D v(x t) cos(t Λ + D), x t, ( u(t, x) = cos (t x) ) Λ + D cos(x, x < t. Λ + D) In tis experiment we get u =.39 an t = Figure 3 sows te error z N as a function of N. Figure 4 sows te plots of ū (m), m =, 15, 3,..., 75. We moify te secon example as follows. We take ṽ(t) = 1 + sin(πt), t [, 1]. Ten we ave to solve (36) by a numerical proceure an compare tis approximation wit z N, obtaine by (37). Te errors z N are presente by Figure 5. Te plot of ū (m), m =, 15, 3,..., 75 is given by Figure 6.

22 222 H. LESZCZYŃSKI AND P. ZWIERKOWSKI REFERENCES [1] L. M. Abia, O. Angulo, J. C. López-Marcos, Age-structure population moels an teir numerical solution, Ecological Moelling, 188 (25), [2] A. S. Ackle, K. Deng, K. Ito, J. Tiboeaux, A structure erytropoiesis moel wit nonlinear cell maturation velocity an ormone ecay rate, Matematical Biosciences, 24 (26), [3] W. S. C. Gurney, R. M. Nisbet, Ecological Dynamics, Oxfor University Press, New York, [4] M. E. Gurtin R. C. MacCamy, Non-linear Age-epenen Population Dynamics, Arcive for Rational Mecanics an Analysis, 54 (1974), [5] M. E. Gurtin, R. C. MacCamy, Diffusion moels for age-structure populations, Matematical Biosciences 54 (1981), [6] M. Iannelli, M. Kim, E. Park, Asymptotic beavior for an SIS epiemic moel an its approximation, Journal Nonlinear Analysis: Teory, Metos & Applications, 35 (1999) [7] Z. Kamont, Hyperbolic Functional Differential Inequalities an Applications, Kluwer Acaemic Publisers, [8] M. Kim, E. Park, Mixe Approximation of a Population Diffusion Equation, Computers & Matematics wit Applications., 3 (1995), [9] T. V. Kostova, Numerical solutions of a yperbolic ifferential-integral equation, Computers & Matematics wit Applications, 15 (1988), [1] M. Langlais, Large time beavior in a nonlinear age-epenent population ynamics problem wit spatial iffusion, Journal of Matematical Biology, 26 (1988), [11] A. Lasota, M. C. Mackey, M. Ważewska-Czyżewska, Maximizing cances of survival, Journal of Matematical Biology, 13 (1981), [12] H. Leszczyński, Comparison ODE teorems relate to te meto of lines, Journal of Applie Analysis, 17 (211), [13] H. Leszczyński, P. Zwierkowski, Iterative Metos for Generalize von Foerster Equations wit Functional Depenence, Journal of Inequalities an Applications, vol. 27, Article ID 12324, 14 pages, 27, [oi:1.1155/27/12324]. [14] J. D. Murray, Matematical Biology, I. An Introuction, Springer-Verlag, Heielberg, 22. [15] G. Pelovska, M. Iannelli, Numerical metos for te Lotka-McKenrick s equation, Journal of Computational an Applie Matematics, 197 (26), [16] K. Rektorys, Te Meto of Discretization in Time an Partial Differential Equations, D. Reiel Publising Company, Dorrect, [17] W. E. Sciesser, Te Numerical Meto of Lines, Acaemic Press, San Diego, [18] W. E. Sciesser, G. W. Grifflts, A Compenium of Partial Differential Equation Moels: Meto of Lines Analysis wit Matlab, Cambrige University Press, 29. [19] J. M. Tcuence, A Partially iscretize age-epenent population moel wit an aitional structure, Applications an Applie Matematics 1 (26), [2] P. Zwierkowski, Convergence of a finite ifference sceme for von Foerster equation wit functional epenence, Demonstratio Matematica 95 (212), # 3,

23 STABILITY OF THE METHOD OF LINES z N N Fig. 1. Computation error for z u 1.5 t x Fig. 2. Meto of lines.

24 224 H. LESZCZYŃSKI AND P. ZWIERKOWSKI.5.4 z N N Fig. 3. Computation error for z u t x Fig. 4. Meto of lines.

25 STABILITY OF THE METHOD OF LINES z N N Fig. 5. Computation error for z u t x Fig. 6. Meto of lines.