existence-uniqueness of solutions for a nonlinear nonautonomous ...

CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 8, Number 1, Spring 2000

EXISTENCE-UNIQUENESS OF SOLUTIONS FOR A NONLINEAR NONAUTONOMOUS SIZESTRUCTURED POPULATION MODEL: AN UPPER-LOWER SOLUTION APPROACH AZMY S. ACKLEH AND KENG DENG ABSTRACT. Existence and uniqueness of weak solutions to a nonlinear nonautonomous size-structured population model are established using the upper-lower solution technique and a comparison principle.

1. Introduction. In this paper we study the following nonlinear and nonlocal first order hyperbolic initial boundary value problem that describes the dynamics of a size-structured population: (1.1) ut +(g(t, x)u)x = − m(t, x, P (u(t, ·)))u, 0 < t < T, b g(t, a)u(t, a) = C(t) + q(t, x)u(t, x) dx, 0 < t < T,

a < x < b,

a

a ≤ x ≤ b,

u(0, x) = u0 (x),

b where P (u(t, ·)) = a η(y)u(t, y) dy. The function u(t, x) in problem (1.1) represents the density of individuals in the size class [x, x + dx) at time t. The parameters q(t, x) and g(t, x) are the time and sizedependent reproduction and growth rates, respectively. The function m(t, x, P ) represents the mortality rate of an individual of size x at time t which depends on the population measure, P , and C(t) represents the inflow of a-size individuals from an external source (e.g., seeds carried by wind). Linear and nonlinear problems similar to (1.1) have been discussed extensively in the literature. Existence-uniqueness results have been Research of the first author was supported by Louisiana Education Quality Support Fund under the grant LEQSF (1996 99)-RD-A-36. 2000 AMS Mathematics Subject Classification. Primary 35A05, 35A35, 35A40, 35L60, 92D99. Key words and phrases. Nonlinear nonautonomous size-structured model, upperlower solution technique, existence-uniqueness. c Copyright 2000 Rocky Mountain Mathematics Consortium

1

2

A.S. ACKLEH AND K. DENG

established using the characteristic method with fixed point argument or the semi-group of linear operators theoretic approach (e.g., [4], [5], [6]). Recently, in [2], [3], we developed a different approach to the study of existence and uniqueness of solutions to linear and nonlinear nonautonomous size-structured models similar to (1.1). This approach is based on the construction of monotone sequences of upper and lower solutions. As is well known, such a monotone approximation heavily relies on a comparison principle. In [1] we proved that a comparison principle works for the case mP ≤ 0. Moreover, we showed via a counter example that, without the restriction mP ≤ 0, the comparison principle fails. To overcome this difficulty, in [3], we introduced a new definition of coupled upper and lower solutions to extend this method to the case mP > 0. However, therein we considered only the case b P (u(t, ·)) = a u(t, x) dx, i.e., η(x) ≡ 1. The goal of this paper then is twofold. On the one hand, we will extend the results of [3] to the case where no restriction on the sign of mP is imposed. On the other hand, we will consider a general function η(x). This allows the theory b to hold, for example, for the total population mass P = a yu(t, y) dy. It is worth mentioning that the monotone sequences developed in [3] do not work for the present case. Hence, in this paper we introduce a new definition of upper and lower solutions. Based on this definition, we are able to establish a comparison result and thus construct monotone sequences of upper and lower solutions which will lead to the existence of solutions by passing to the limit. We organize our paper as follows. In Section 2 we establish a comparison result and show the uniqueness of the weak solution to problem (1.1). In Section 3 we then construct two monotone sequences of upper and lower solutions and show their convergence to the weak solution of (1.1). 2. Comparison and uniqueness results. Throughout the discussion we assume that the parameters in (1.1) satisfy the following: (A1) g(t, x) is continuously differentiable with respect to t and x. Furthermore, g(t, x) > 0 for (t, x) ∈ [0, T ] × [a, b) and g(t, b) = 0 for t ∈ [0, T ]. (A2) q(t, x) (≥ 0) is continuous with respect to t and x for (t, x) ∈ [0, T ] × [a, b].

EXISTENCE-UNIQUENESS OF SOLUTIONS

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(A3) m(t, x, P ) (≥ 0) is continuous with respect to t and x and continuously differentiable with respect to P for (t, x, P ) ∈ [0, T ] × [a, b] × [0, ∞). In addition, there exists a constant M > 0 such that M + mP (t, x, P ) ≥ 0. (A4) C(t) (≥ 0) is continuous for t ∈ [0, T ]. (A5) η ∈ L∞ (a, b) and η ≥ 0 almost everywhere in (a, b). (A6) u0 ∈ L∞ (a, b) and u0 ≥ 0 almost everywhere in (a, b). Note that g(t, b) = 0 in (A1) is a natural assumption from the biological point of view. It implies that the growth of individuals ceases at size b (see, e.g., [4], [5], [6]). Furthermore, from a mathematical point of view, the following comparison result cannot be established without this assumption. Let DT = (0, T ) × (a, b), and we introduce the following definition of a pair of coupled upper and lower solutions of problem (1.1). Definition 2.1. A pair of functions u(t, x) and v(t, x) are called an upper and a lower solution of (1.1) on DT , respectively, if all the following hold: (i) u, v ∈ L∞ (DT ). (ii) u(0, x) ≥ u0 (x) ≥ v(0, x) almost everywhere in (a, b). (iii) For every t ∈ (0, T ) and every nonnegative ξ ∈ C 1 (DT ), (2.1) a

b

u(t, x)ξ(t, x) dx ≥

b

u(0, x)ξ(0, x) dx b t ξ(s, a) C(s) + q(s, x)u(s, x) dx ds + a

0

a

t

b

+ 0

−

t 0

[ξs (s, x) + g(s, x)ξx (s, x)]u(s, x) dx ds

a b a

ξ(s, x)[m(s, x, P (v(s, ·)))

+ M (P (v(s, ·))−M P (u(s, ·))]u(s, x) dx ds

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(2.2) b v(t, x)ξ(t, x) dx ≤

b

v(0, x)ξ(0, x) dx t ξ(s, a) C(s) + +

a

a

0

q(s, x)v(s, x) dx ds

a

t

b

+ −

b

0

a

0

a

t

[ξs (s, x) + g(s, x)ξx (s, x)]v(s, x) dx ds b

ξ(s, x)[m(s, x, P (u(s, ·)))

+ M P (u(s, ·))−M P (v(s, ·))]v(s, x) dx ds. A function u(t, x) is called a weak solution of (1.1) on DT if u satisfies (2.1) with “≥” replaced by “=” and P (v(s, ·)) by P (u(s, ·)). Such a weak solution definition can be formally derived from multiplying (1.1) by ξ and integrating the resulting equation by parts. Conversely, if a weak solution with enough regularity exists, then one can show that it also satisfies (1.1) in the classical sense. Based on Definition 2.1 the following comparison result can be established. Theorem 2.2. Suppose that (A1) (A6) hold. Let u and v be a nonnegative upper solution and a nonnegative lower solution of (1.1), respectively. Then u ≥ v almost everywhere in DT . Proof. Let w = v − u. Then w satisfies w(0, x) = v(0, x) − u(0, x) ≤ 0 a.e. in (a, b),

(2.3) and (2.4) b

w(t, x)ξ(t, x) dx a

≤

b

a

t

w(0, x)ξ(0, x) dx +

ξ(s, a) 0

b

q(s, x)w(s, x) dx ds a


t

b

[ξs (s, x) + g(s, x)ξx (s, x)]w(s, x) dx ds

+ −

0

a

0

a

t t

b

ξ(s, x)m(s, x, P (u(s, ·)))w(s, x) dx ds

b

b

ξ(s, x)u(s, x)(M +A(s, x))

+ 0

t

a

b

+ a

η(y)w(s, y) dy dx ds a

b

ξ(s, x)M v(s, x) 0

5

η(y)w(s, y) dy dx ds, a

where A(t, x) = mP (t, x, θ(t)) with θ(t) between P (u(t, ·)) and P (v(t, ·)). Following [1], we let ξ(t, x) = eλt ζ(t, x) where ζ ∈ C 1 (DT ) and λ (> 0) is chosen so that λ − m ≥ 0 on DT × [P , P ] where P = min{inf [0,T ] P (u(t, ·)), inf [0,T ] P (v(t, ·))} and P = max{sup[0,T ] P (u(t, ·)), sup[0,T ] P (v(t, ·))}. Then we find (2.5) eλt

b

w(t, x)ζ(t, x) dx

a

≤

b

t

w(0, x)ζ(0, x) dx + a

t

b

+ 0

a

0

a

t

b

+ t

b

+ 0

a

t + 0

b

eλs ζ(s, a)

b

q(s, x)w(s, x) dx ds

0

a

w(s, x)eλs [ζs (s, x) + g(s, x)ζx (s, x)] dx ds eλs ζ(s, x)(λ − m)w(s, x) dx ds eλs ζ(s, x)u(s, x)(M +A) eλs ζ(s, x)M v(s, x)

a

b

b


η(y)w(s, y) dy dx ds. a

We now set up a backward problem as follows:

(2.6)

ζs + gζx = 0, 0 < s < t, a < x < b, ζ(s, b) = 0, 0 < s < t, ζ(t, x) = ϕ(x), a ≤ x ≤ b.

Here ϕ(x) ∈ C0∞ (a, b), 0 ≤ ϕ ≤ 1.

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The existence of ζ ∈ C 1 (DT ) follows from the fact that by the variable change τ = t − s, (2.6) can be written into 0 < τ < t, ζτ − gζx = 0, ζ(τ, b) = 0, 0 < τ < t, ζ(0, x) = ϕ(x), a ≤ x ≤ b,

(2.7)

a < x < b,

and thus (2.7) can be solved by the characteristic method. Note that the initial and boundary values for ζ imply that 0 ≤ ζ ≤ 1 on DT . Substituting such a ζ in (2.5) yields

b

(2.8) a

w(t, x)ϕ(x) dx ≤

b

t

+

w(0, x) dx + ν a

0

b

w(s, x)+ dx ds,

a

where w(t, x)+ = max{w(t, x), 0} and ν = maxDT {q(t, x) + (λ − m) + (b − a)[u(t, x)(M + A(t, x)) + M v(t, x)]η(x)}. From the condition on initial data in (2.3), we then have

b

a

w(t, x)ϕ(x) dx ≤ ν

t 0

b

w(s, x)+ dx ds.

a

Since this inequality holds for every ϕ ∈ C0∞ (a, b) with 0 ≤ ϕ ≤ 1, we can choose a sequence {ϕn } on (a, b) converging to χ=

1 if w(t, x) > 0, 0 otherwise.

Consequently, we find that

b a

+

w(t, x) dx ≤ ν

t 0

b

w(s, x)+ dx ds,

a

which by Gronwall’s inequality leads to

b

w(t, x)+ dx = 0.

a

Thus, the proof is completed.


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Remark 1. From the proof of Theorem 2.2, it easily follows that for any function w ∈ L∞ (DT ) if w(0, x) ≤ 0 almost everywhere in (a, b) and the following inequality holds for every nonnegative ξ ∈ C 1 (DT ), b w(t, x)ξ(t, x) dx a

≤

b

w(0, x)ξ(0, x) dx a

(2.9)

t

ξ(s, a)

+ 0

t

b

+ −

0

a

0

a

t

b

q(s, x)w(s, x) dx ds a

[ξs (s, x) + g(s, x)ξx (s, x)]w(s, x) dx ds b

ξ(s, x)F (s, x)w(s, x) dx ds

with F ∈ L∞ (DT ), then w(t, x) ≤ 0 almost everywhere in DT . Such a result will be used in Section 3. The following uniqueness result easily follows from Theorem 2.2. Corollary 2.3. If u(t, x) is a nonnegative weak solution of problem (1.1) and if P (u(t, ·)) ∈ C[0, T ], then u is unique. Proof. Suppose that u1 (t, x) and u2 (t, x) are two nonnegative solutions of (1.1). Clearly, if P (u1 (t, ·)) ≡ P (u2 (t, ·)) for 0 ≤ t ≤ T , u1 (t, x) ≡ u2 (t, x). For that reason, we may assume that P (u1 (t, ·)) = P (u2 (t, ·)) for 0 ≤ t ≤ t0 and P (u1 (t, ·)) > P (u2 (t, ·)) for t0 < t ≤ t1 , 0 ≤ t0 < t1 ≤ T . Then from assumption (A3), we find that − m(t, x, P (u1 (t, ·))) = − [m(t, x, P (u1 (t, ·))) + M P (u1 (t, ·))] + M P (u1 (t, ·)) ≤ − m(t, x, P (u2 (t, ·))) − M P (u2 (t, ·)) + M P (u1 (t, ·)) and − m(t, x, P (u2 (t, ·))) ≥ − m(t, x, P (u1 (t, ·))) − M P (u1 (t, ·)) + M P (u2 (t, ·)).

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Hence, u1 and u2 are a lower and an upper solution of (1.1) on Dt1 , respectively. By Theorem 2.2 we have that u1 (t, x) ≤ u2 (t, x) almost everywhere in Dt1 and hence P (u1 (t, ·)) ≤ P (u2 (t, ·)) for 0 ≤ t ≤ t1 , which is a contradiction. 3. Monotone sequences and existence-uniqueness of solutions. We begin this section by constructing a pair of nonnegative lower and upper solutions of (1.1). Let α0 (t, x) = 0. Choose a constant γ large enough such that max q(t, x)/ min g(t, a) ≤ γ/2. DT

[0,T ]

Fix this γ and choose δ large enough such that max u0 ∞ , max C(t) min g(t, a) ≤ (δ/2) exp(−γb). [0,T ]

[0,T ]

Now choose σ large enough such that

σ ≥ 2M δη∞ exp(−γa)/γ + γ max g(t, x) + max |gx (t, x)| . DT

DT

Let β 0 (t, x) = δ exp(σt) exp(−γx). Then it can be easily shown that α0 and β 0 are a pair of coupled lower and upper solutions of (1.1) on [0, T0 ]×[a, b] with T0 = min{T, (ln 2/σ)}. We then define two sequences k ∞ {αk }∞ k=0 and {β }k=0 as follows. For k = 1, 2, . . . , αtk + (g(t, x)αk )x = −Dk−1 (t, x)αk , 0 < t < T0 , ˆ k−1 (t), (3.1) 0 < t < T0 , g(t, a)αk (t, a) = R k α (0, x) = u0 (x), a ≤ x ≤ b,

a < x < b,

where Dk−1 (t, x) = m(t, x, P (β k−1(t, ·)))+ M P (β k−1 (t, ·))− M P (αk−1 (t, ·)) b ˆ k−1 (t) ≡ C(t) + R q(t, x)αk−1 (t, x) dx, a

9


and βtk + (g(t, x)β k )x = −E k−1 (t, x)β k , ˆ k−1 (t), (3.2) g(t, a)β k (t, a) = B k

β (0, x) = u0 (x),

0 < t < T0 ,

a < x < b,

0 < t < T0 , a ≤ x ≤ b,

where E k−1 (t, x) = m(t, x, P (αk−1(t, ·)))+ M P (αk−1 (t, ·))− M P (β k−1 (t, ·)) b ˆ k−1 (t) ≡ C(t) + B q(t, x)β k−1 (t, x) dx. a

The existence of solutions to problems (3.1) and (3.2) follows from ˆ k−1 are given functions. In particular, using ˆ k−1 and B the fact that R the method of characteristics [7], [8], [9], [11], one can find explicitly solutions to these problems as follows. Consider the equation for the characteristic curves given by  d   t(s) = 1 ds (3.3)   d x(s) = g(t(s), x(s)). ds The solution αk of (3.1) along a characteristic curve (t(s), x(s)) satisfies the following equation d k α (s) = − (gx (t(s), x(s)) + Dk−1 (t(s), x(s)))αk (s). ds Clearly, for any initial point (t(s0 ), x(s0 )), the existence of a unique solution of (3.3) follows from the assumption (A1). Parametrizing the characteristic curves with the variable t, then a characteristic curve passing through (tˆ, x ˆ) is given by (t, X(t; tˆ, x ˆ)) where X satisfies d X(t; tˆ, x ˆ) = g(t, X(t; ˆt, x ˆ)) dt and X(tˆ; tˆ, x ˆ) = x ˆ. From (A1) it follows that the function X is strictly increasing. Hence, a unique inverse function τ (x; tˆ, x ˆ) exists. Now define G(x) = τ (x; 0, a) where (G(x), x) represents the characteristic

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curve passing through (0, a) and dividing the (t, x)-plane into two parts. Then for any point (t, x) with t ≤ G(x), the solution αk (t, x) is determined through the initial condition by (3.4) αk (t, x) = u0 (X(0; t, x)) t · exp − (gx (s, X(s; t, x)) + Dk−1 (s, X(s; t, x))) ds , 0

and for any point (t, x) with t > G(x) the solution is determined via the boundary condition by (3.5) αk (t, x) = Rk−1 (τ (a; t, x)) t k−1 · exp − (gx (s, X(s; t, x)) + D (s, X(s; t, x))) ds , τ (a;t,x)

ˆ k−1 (t)/g(t, a)). Note that a representation similar where Rk−1 (t) = (R to (3.4) (3.5) can be obtained for the solution β k by interchanging αk with β k and replacing Dk−1 with E k−1 and Rk−1 with B k−1 where ˆ k−1 /g(t, a)). B k−1 = (B k ∞ Next we show that the sequences {αk }∞ k=0 and {β }k=0 are monotone. 0 1 To this end we first let w = α − α . Then w satisfies (2.9) with F (t, x) = D0 (t, x). Hence by Remark 1 w ≤ 0, which implies α0 ≤ α1 . Similarly, it can be seen that β 0 ≥ β 1 . From this, in view of (A3) it easily follows that α1 and β 1 are a lower and an upper solution, respectively, and hence α1 ≤ β 1 .

Assume that for some k > 1, αk and β k are a lower and an upper solution of (1.1), respectively. By similar reasoning, we can show that αk ≤ αk+1 ≤ β k+1 ≤ β k and that αk+1 and β k+1 are also a lower and an upper solution of (1.1), respectively. Thus, by induction, we obtain two monotone sequences that satisfy α0 ≤ α1 ≤ · · · ≤ αk ≤ β k ≤ · · · ≤ β 1 ≤ β 0

a.e. in DT0

for each k = 0, 1, 2, . . . . Hence it follows from the monotonicity of k ∞ the sequences {αk }∞ k=0 and {β }k=0 that there exist functions α and β k k such that α → α and β → β pointwise in DT0 . Clearly α ≤ β almost everywhere in DT0 .


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Upon establishing the monotonicity of our sequences, we can prove the following convergence result. Theorem 3.1. Suppose that (A1) (A6) hold. Then the sequences k ∞ {αk }∞ k=0 and {β }k=0 converge uniformly along characteristic curves to a limit function u. Moreover, the function u is the unique solution of problem (1.1) on [0, T0 ] × [a, b]. From the solution Proof. Consider first the sequence {αk }∞ k=0 . representation for αk given in (3.4) (3.5), the fact that α0 ≤ αk ≤ β 0 , and the monotonicity of the sequence {αk }, we obtain by arguing as in [10, p. 189] that, along the characteristic curve passing through (0, x0 ), the solution αk (t, X(t; 0, x0 )) t k−1 (gx (s, X(s; 0, x0 )) + D (s, X(s; 0, x0 ))) ds = u0 (x0 ) exp − 0

converges to α(t, X(t; 0, x0 ))

t = u0 (x0 ) exp − (gx (s, X(s; 0, x0 )) + D(s, X(s; 0, x0 ))) ds 0

uniformly and monotonically for 0 ≤ t ≤ T0 , where D(t, x) = m(t, x, P (β(t, ·))) + M P (β(t, ·)) − M P (α(t, ·)). On the other hand, since Rk (t) is monotone and uniformly bounded for 0 ≤ t ≤ T0 , along the characteristic curve passing through (t0 , a), the solution αk (t, X(t; t0 , a)) =R

k−1

t k−1 (t0 ) exp − (gx (s, X(s; t0 , a)) + D (s, X(s; t0 , a))) ds t0

converges to α(t, X(t; t0 , a)

t (gx (s, X(s; t0 , a)) + D(s, X(s; t0 , a))) ds = R(t0 ) exp − t0

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uniformly and monotonically for t0 ≤ t ≤ T0 , where R(t) =

b 1 q(t, x)α(t, x) dx . C(t) + g(t, a) a

Consequently, the limit α has the following implicit representation. For any point (t, x) with t ≤ G(x),

(3.6)

α(t, x) = u0 (X(0; t, x)) t (gx (s, X(s; t, x)) + D(s, X(s; t, x))) ds · exp − 0

and, for any point (t, x) with t > G(x), (3.7) α(t, x) = R(τ (a; t, x)) t · exp −

τ (a;t,x)

(gx (s, X(s; t, x)) + D(s, X(s; t, x))) ds .

A similar convergence argument can be established for β k , and we find that the limit β can be represented as follows. For any point (t, x) with t ≤ G(x), (3.8)

t β(t, x) = u0 (X(0; t, x)) exp − (gx (s, X(s; t, x))+E(s, X(s; t, x))) ds 0

and for any point (t, x) with t > G(x), (3.9) β(t, x) = B(τ (a; t, x)) t · exp −

τ (a;t,x)

(gx (s, X(s; t, x)) + E(s, X(s; t, x))) ds ,

where E(t, x) = m(t, x, P (α(t, ·))) + M P (α(t, ·)) − M P (β(t, ·)) b 1 B(t) = q(t, x)β(t, x) dx . C(t) + g(t, a) a


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We now show that α = β. To this end, let w = β − α. Since β ≥ α, w(t, x) ≥ 0 and w(0, x) = 0. In view of (2.4), by choosing ξ(t, x) ≡ 1, we have that (3.10) b t b w(t, x) dx = q(s, x)w(s, x) dx ds a

0

−

a

t 0

b

a

t

m(s, x, P (β(s, ·)))w(s, x) dx ds

b

β(s, x)(M +A(s, x))

+ 0

a

t

b

+ 0

≤ C0

a

t

b


b

M α(s, x)

b


w(s, x) dx ds, 0

a

where C0 = maxDT [q(t, x)+η∞ β(t, x)(M +A(t, x))+M η∞ α(t, x)]. 0 Hence it follows from Gronwall’s inequality that w(t, x) = 0 almost everywhere in DT0 , i.e., α = β. Defining this common limit by u, we find that u satisfies the following. For any point (t, x) with t ≤ G(x), (3.11) u(t, x) = u0 (X(0; t, x)) t (gx (s, X(s; t, x)) + m(s, X(s; t, x), P (u(s, ·)))) ds · exp − 0

and for any point (t, x) with t > G(x), (3.12) u(t, x) = R(τ (a; t, x)) t · exp − (gx (s, X(s; t, x))+m(s, X(s; t, x), P (u(s, ·)))) ds . τ (a;t,x)

Using arguments such as those in [6], one can establish that P (u(t, ·)) is continuous. Hence, by Corollary 2.3, u(t, x) is the unique weak solution of problem (1.1). Remark 2. Mathematically all results in Sections 2 and 3 hold if we assume instead of (A3) that M − mP (t, x, P ) ≥ 0 and define another

14


pair of coupled upper and lower solutions by replacing Definition 2.1(iii) with the following inequalities: (3.13) b u(t, x)ξ(t, x) dx a

≥

b

t

u(0, x)ξ(0, x) dx + a

0

t

−

[ξs + g(s, x)ξx ]u(s, x) dx ds

a

t 0

a

b

+ 0

b ξ(s, a) C(s) + q(s, x)u(s, x) dx ds

b

a

ξ(s, x)[m(s, x, P (u(s, ·))) − M P (u(s, ·)) + M P (v(s, ·))]u(s, x) dx ds

(3.14) b v(t, x)ξ(t, x) dx a

≤

b

t

v(0, x)ξ(0, x) dx + a

t

0

−

0

[ξs + g(s, x)ξx ]v(s, x) dx ds

a

t

a

a

b

+ 0

b ξ(s, a) C(s) + q(s, x)v(s, x) dx ds

b

ξ(s, x)[m(s, x, P (v(s, ·))) − M P (v(s, ·)) + M P (u(s, ·))]v(s, x) dx ds.

Biologically, however, it is more relevant to consider the case M + mP (t, x, P ) ≥ 0, for this allows the mortality to increase dramatically without a bound as the population becomes unbounded, i.e., P → ∞. Acknowledgment. The authors thank the referee for many valuable comments.


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