LOCAL EXISTENCE FOR A GENERAL MODEL OF SIZE-DEPENDENT POPULATION DYNAMICS

We shall investigate a size structured population dynamics with aging and birth functions having general forms. The growth rate we deal with depends not only on the size but also on time. We show the existence of a local solution and continuous dependence on the initial data, which shows the uniqueness of the solution as well.


Introduction
We are interested in a size structured population model with the growth rate depending on the individual's size and time. There has been many investigations where the growth rate depends on the size. See, for example, [2; Chap. 10], [3] and the references therein. Recently, A. Calsina and J. Saldaña [1] have studied the case where the growth rate depends on the size as well as the total population at each time. They have the model of plants in forests or plantations in their mind.
We are also motivated by the population model of the forest growth etc. In this case, the growth rate may be influenced by the environment such as light, temperature, and nutrients. These may change with time. It is also reasonable to think that the growth rate varies with the individual's size of plants because the amount of light they capture may depend on it. From these points of view, it is natural to consider the growth rate depending on the size and time.
In this paper we study the following initial boundary value problem with nonlocal boundary condition: Here a ≥ 0, l ∈ (0, ∞] is the maximum size, F and G are given mappings corresponding to birth and aging functions respectively. The function V is the growth rate function depending on the size x and time t and the function C represents the inflow of zero-size individuals from an external source such as seeds carried by the wind or placed in a plantation. The unknown function u(x, t) stands for the density with respect to size x of a population at time t. So the integral x 2 x 1 u(x, t)dx represents the number of individuals with size between x 1 and x 2 at time t. The equation (SDP) is closely related to the age-dependent population dynamics developed by G. Webb [4]. Indeed, from the mathematical point of view, the particular case V (x, t) ≡ 1 is nothing but the age-dependent case.
Our objective is to show the existence of a unique local solution and continuous dependence of the solution on the initial data. The results extend [4, Propositions 2.2 and 2.3] and partially [1, Theorems 1 and 2]. The birth and aging functions treated in [1] are of Gurtin-MacCamy type, i.e., F (u(·, t)) = l 0 β(x, P (t))u(x, t)dx, G(u(·, t))(x) = −m(x, P (t))u(x, t) where P (t) = l 0 u(x, t)dx is the total population at time t, and this is essential for their arguments. We handle more general birth and aging functions which are the same as in [4]. In [1], the growth rate function V depends on the size and the total population P (t), while we deal with V depending on the size and time.
The paper is organized as follows. In Section 2 we state our assumptions and main results (Theorems 2.1 and 2.2). We prepare some lemmas in Section 3 and the proofs of the theorems are established in Sections 4 and 5.

Local existence and uniqueness
In this section we state our main theorems concerning the existence of a unique local solution to (SDP) and the continuous dependence on the initial data. At first, we introduce some notations.
Let L 1 := L 1 (0, l; R n ) be the Banach space of Lebesgue integrable functions from [0, l) to R n with norm f L 1 := l 0 |f (x)|dx for f ∈ L 1 , where | · | denotes the norm of R n . For T > a we set L a,T := C([a, T ]; L 1 ), the Banach space of L 1 -valued continuous functions on [a, T ] with the supremum norm u L a,T := sup a≤t≤T u(t) L 1 for u ∈ L a,T . Note that each element of L a,T is identified with an element of L 1 ((0, l) × (a, T ); R n ) by the relation We assume the following hypotheses.
(F) F : L 1 → R n is locally Lipschitz in the sense that there is an increasing function is Lipschitz continuous with respect to x uniformly for t, i.e., there is a constant L V > 0 such that T ] by the solution of the differential equation Since the function V is Lipschitz continuous as assumed in (V), it is well known that there exists a unique solution Let z a (t) := ϕ(t; a, 0) denote the characteristic through (0, a) in the (x, t)plane. In particular, the curve z 0 (t) is the trajectory in the (x, t)-plane of the newborn individuals at t = 0 and it separates the trajectories of the individuals that were present at the initial time t = 0 from the trajectories of those individuals born after the initial time. For i.e. τ is the initial time of the characteristic through (x, t). And then define τ * a by It is obvious that the solution x(t) = ϕ(t; t 0 , x 0 ) of (2.1) can be extended on [τ * a (t 0 , x 0 ), T ] and ϕ(t; t 0 , x 0 ) satisfies the integral equation Note also that With the characteristics ϕ we define a solution of (SDP) as follows.
Remark 2.1. The above definition is the analogue of the age-dependent case [4, (1.49)]. Note that if u(x, t) satisfies (SDP) in a strong sense, then it is easily seen that u satisfies (2.5).
Our main results are the following two theorems. , and (C) hold and let T > a and r > 0. Let u,û ∈ L a,T be the solutions of (SDP) with initial values u a , u a ∈ L 1 respectively satisfying u L a,T , û L a,T ≤ r. Then we have the following estimate: Remark 2.2. Theorem 2.2 shows the continuous dependence of the solution on the initial data as well as the uniqueness of the solution for the unitial data as long as the solution exists.

Properties of characteristic curves
In this section we collect some properties of the characteristic curves defined by (2.1) (or (2.4)) in the previous section. Before that, let us begin with the well-known lemma.
The case (i) is the standard one. The case (ii) is less familiar but we omit the proof since it is quite similar.
Some properties of the characteristic curves ϕ are given as Lemma 3.2. Let ϕ be the characteristic curves defined by the solution of (2.1).
The function τ defined by (2.2) has the following properties.
This contradicts the fact that the initial value problem has a unique solution. Hence x 1 ≥ x 2 , and so τ t (·) is shown to be decreasing. ( x 0 = 0, one can observe that τ t (·) is right or left continuous, respectively, by the same fashion.
(ii) It is easily seen that t → τ (t, x) is increasing. To prove the continuity of τ (t, x), first we observe that for any t ∈ [0, T ] and x ∈ [0, z 0 (t)), is right or left continuous respectively by the same way. Next, we show that τ is continuous on U . Let (t, x) ∈ U and let t n → t and x n → x. We may assume that t n = t. Then there is a subsequence t n k such that t n k ↑ t or t n k ↓ t. We consider the former case. For the latter case, the same fact holds. Take b > 0 such as 0 ≤ x < b < z 0 (t). Then for each y ∈ [0, b], k → τ (t n k , y) is increasing and lim k→∞ τ (t n k , y) = τ (t, y) as shown above. Further, y → τ (t n k , y) (for each sufficiently large k) and y → τ (t, y) are continuous by (i). Hence by Dini's theorem, we have lim k→∞ τ (t n k , y) = τ (t, y) uniformly for y ∈ [0, b]. Therefore, we conclude that lim k→∞ τ (t n k , x n k ) = τ (t, x). Since the limit is common for the subsequences, we establish the continuity of (t, x) → τ (t, x).
The next lemma shows some differentiability properties of the characteristics with respect to the second and third arguments, and they are needed for changes of variables we will use often later.

Lemma 3.4. Let x = ϕ(t; τ, η).
(i) x is differentiable with respect to τ and (ii) x is differentiable with respect to η and Proof. (i) By Lemma 3.2 (ii), the function τ → ϕ(t; τ, η) is differentiable almost everywhere. On the other hand, invoking (3.1) and the Lebesgue bounded convergence theorem, we find that (ii) Similarly to (i), one can show that (ii) holds.
Taking the limit superior on both sides yields lim sup This completes the proof.

Proof of Theorem 2.1
Given r > 0, take u a ∈ L 1 such that u a L 1 ≤ r. Define Obviously, M T is a closed subset of L a,T and so a complete metric space. Define a mapping K on M T as follows: s; τ, 0))ds a.e. x ∈ (0, z a (t)), where τ := τ (t, x) is the one defined by (2.2),F andG are defined by (2.6) and (2.7) respectively in Section 2.
We will seek the fixed point of the mapping K. For that purpose, we will show that K maps M T into itself and that K is contractive for some T > a.
For I 2 and I 4 , use the change of variable η = ϕ(s; τ, 0) = ϕ(s; t, x). By Lemma 3.4 (ii), we obtain By (G) and (V), we have Therefore, we get the following inequality For I 3 , the change of variable ξ = ϕ(a; t, x) leads to Consequently, Choose δ > 0 so small that Then combining (4.2) with (4.5), we have sup a≤t≤T Ku(·, t) L 1 ≤ 2r for T = a + δ.
(ii) (Continuity of t → Ku(·, t)) Let u ∈ M T and t ∈ [a, T ]. We will show only the right-continuity. The left-continuity is proved by exchanging t and t. We will just give some remarks on proving it below.
Noting that B(t) and V (0, t) are continuous in t and bounded on [a, T ], we have J 1 → 0 ast ↓ t by the Lebesgue bounded convergence theorem.
Finally J 5 is estimated as follows. s; t, x)) − G s (ϕ(s,t, x))|dxds By Lemma 3.5, we find that J 5 → 0 ast ↓ t. Consequently, the right-continuity has been shown. To prove the leftcontinuity, lett < t. Then by exchanging t andt, we obtain all the estimates above with t andt exchanged. By the continuity of τ obtained in Lemma 3.3 (ii), we find that all the terms tends to 0 ast ↑ t. Hence the continuity is proved.

Proof of Theorem 2.2
The proof is similar to the proof of the mapping K to be contractive as done in the previous section. We need a little more careful estimation.
As the estimation of I 3 in the previous section, Similarly to the estimate of P 2 + P 3 , by using the change of variable η = ϕ(s; t, x) = ϕ(s; τ, 0), and by (G), we have