ADVECTION IN A WATER COLUMN SZE-BI HSU AND YUAN LOU
Abstract. We investigate a nonlocal reaction-diffusion-advection equation which models the growth of a single phytoplankton species in a water column where the species depends solely on light for its metabolism. We study the combined effect of death rate, sinking or buoyant coefficient, water column depth and vertical turbulent diffusion rate on the persistence of a single phytoplankton species. Under a general reproductive rate which is an increasing function of light intensity, we establish the existence of a critical death rate; i.e., the phytoplankton survives if and only if its death rate is less than the critical death rate. The critical death rate is a strictly monotone decreasing function of sinking or buoyant coefficient and water column depth, and it is also a strictly monotone decreasing function of turbulent diffusion rate for buoyant species. In contrast to critical death rate, critical sinking or buoyant velocity, critical water column depth and critical turbulent diffusion rate may or may not exist. For instance, it is shown that if the death rate is suitably small with respect to the water column depth, the phytoplankton can persist for any sinking or buoyant velocity; i.e., there is no critical sinking or buoyant velocity under such situation. We further show that critical water column depth, critical sinking or buoyant velocity and critical turbulent diffusion rate for buoyant species can exist for some intermediate range of phytoplankton death rates and, whenever they exist, are always unique. In strong contrast, we show that there may exist two critical turbulent diffusion rates for sinking species. The phytoplankton forms a thin layer at the surface of the water column for sufficiently large buoyant rate, and it forms a thin layer at the bottom of the water column for sufficiently large sinking rate. Precise characterizations of these thin layers are also given.
1. Introduction
Phytoplankton are microscopic plant-like organisms that drift in the water column of lakes and oceans. They grow abundantly in oceans and lakes around the world, and they are the foundation of the marine food chain. Nutrient and light are the essential resources for the growth of phytoplankton. In phytoplankton communities species compete for nutrient and light in three possible ways. At one extreme, in oligotrophic ecosystems with an ample sup-ply of light, species compete for limiting nutrients [15, 17]. At other extreme, in eutrophic ecosystem with ample nutrient supply, species compete for light [8, 10]. In some aquatic ecosystems the species compete for both nutrients and light which are complementary re-sources for their growth [5, 6, 12, 14, 20]. In the water column the phytoplankton are not only diffusing by the water turbulence but also sinking or buoyant. Most of phytoplankton are heavier than water, they have tendency to sink. On the other hand, some species like some cyanobacteria, green algae, have a lower density than water and they will float and will be called buoyant [8]. In this article we shall restrict our attentions to study the growth of a single species in a water column in eutrophic ecosystem where the species depends solely on light for its metabolism. The model equation is a nonlocal reaction-diffusion-advection equation proposed by Huisman et al. in [8, 9]. We study the combined effect of death rate, vertical turbulent diffusion coefficient, advection (sinking or buoyant) coefficient and water column depth on the survival of the single species (bloom development). Our approach is
1991 Mathematics Subject Classification. 35J55, 35J65, 92D25.
Key words and phrases. phytoplankton, consumption for light, reaction-diffusion-advection, persistence.
different from that in [8]. Under a general reproductive rate which is an increasing function of light intensity, we completely determine the necessary and sufficient conditions for the survival of the phytoplankton species in terms of turbulent diffusion coefficient, advection coefficient, water column depth and death rate of the phytoplankton species.
The rest of the paper is organized as follows: In Section 2, we present the mathematical model proposed in [8, 9] and discuss some previous related works. In Section 3, we state our main results which exclusively focus on the steady states of the model. In Section 4 we establish the existence and uniqueness of positive steady states in terms of the death rate of the phytoplankton species. Sections 5, 6, and 7 are devoted to studying qualitative properties of critical death rate and to determining critical water column depth, critical sinking or buoyant coefficient and critical turbulent diffusion rate, respectively. In Section 8, for large advection coefficients we show that the limiting profile of the steady state solution is a δ-function. Section 9 is the discussion section, where we focus on qualitative properties of critical water column depth, critical advection coefficient and critical turbulent diffusion rate.
2. The mathematical model and previous works
In [8, 9], Huisman et al. proposed and analyzed the following reaction-diffusion-advection equation which describes the population dynamics of a single phytoplankton species in a water column:
(2.1) Pt = DPxx− vPx+ P [g(I(x, t)) − d] , 0 < x < L, t > 0,
with zero flux boundary conditions at x = 0 and x = L
(2.2) DPx(0, t) − vP (0, t) = 0,
DPx(L, t) − vP (L, t) = 0,
and with the initial condition
(2.3) P (x, 0) = P0(x), 0 ≤ x ≤ L,
where P = P (x, t) is the population density of the phytoplankton species; D > 0 is the vertical turbulent diffusion coefficient; v is the sinking velocity (v > 0) or the buoyant velocity (v < 0); L > 0 is the depth of the water column; d > 0 is the death rate; by Lambert-Beer law the light intensity I is given by
(2.4) I = I(x, t) = I0exp(−k0x − k1
Z x
0
P (s, t)ds),
where I0 is the incident light intensity; k0 is the background turbidity, k1 is the absorption
coefficient of phytoplankton. g(I) is the specific growth rate of phytoplankton as a function of light intensity I(x, t). Here we assume all nutrients are in amply supply so that only the light intensity limits the growth rate. We assume that g(I) satisfies
(2.5) g(0) = 0, g0(I) > 0 for I > 0, g(I) ≥ aIγ for I ∈ [0, I
0],
where a > 0 and γ > 0. The simplest example is
(2.6) g(I) = aIγ, 0 < γ ≤ 1.
The typical examples for the reproduction rate saturating for high light intensities are func-tion of Monod type
(2.7) g(I) = mI
Or alternatively by
(2.8) g(I) = m1 − e
−cI
c .
The self-shading model (i.e., k0 = 0) was studied by Shigesada and Okubo in [19]. The
existence, uniqueness and the global stability of the steady state for the infinite long water column (L = ∞) have been established in [13, 19]. More recently, among other things it is shown in [16] that the self-shading model has a unique positive steady state, which is also stable, for any finite water column depth. In particular, this means that the self-shading model has no critical water column depth beyond which the phytoplankton can not persist. This is very different from the case of k0 > 0, where the critical depth exists for some
intermediate range of phytoplankton death rate. See the next and last sections for more detailed discussions on the critical depth.
For the case k0 > 0, it is shown in [8] that the conditions for phytoplankton bloom
development can be characterized by critical water column depth and some critical values of the vertical turbulent diffusion coefficient. In [8] the authors also investigated the phase transition from bloom to no bloom extensively by numerical simulations. They also analyzed in depth the phase transition curve for the case g(I) = aIγ, 0 < γ ≤ 1, by means of reducing
the equation to a Bessel equation. In [7] the authors study both single species and two species competing for light in eutrophic ecosystem with no advections, and the dynamics of single species growth is also completely analyzed in [7]. In this paper, we will use several critical rates to give a complete classification of the phase transition from bloom to no bloom for the general single phytoplankton species model (2.1)-(2.5).
3. Main results Consider the equation
(3.1)
½P
t= DPxx− vPx+ P [g(I(x, t)) − d] , 0 < x < L, t > 0, DPx(0, t) − vP (0, t) = DPx(L, t) − vP (L, t) = 0,
where D > 0, v ∈ (−∞, ∞), g(I) satisfies (2.5), with typical examples (2.6)-(2.8) and I(x, t) takes the form (2.4).
Our first main result concerns the existence and uniqueness of positive steady states of (3.1) in terms of the death rate d. Let λ1(a) denote the principal eigenvalue of
(3.2)
½
− Dϕxx+ vϕx+ a(x)ϕ = λϕ, 0 < x < L, Dϕx(0) = vϕ(0), Dϕx(L) = vϕ(L).
It is well known that λ1(a) is real and can be characterized as
(3.3) λ1(a) = inf ψ∈H1(0,L) RL 0 e(v/D)x(Dψx2+ aψ2)dx RL 0 e(v/D)xψ2dx ,
where H1(0, L) is the closure of C1[0, L] under the norm
kuk = µZ L 0 u2dx ¶1/2 + µZ L 0 u2 xdx ¶1/2 .
For every v ∈ (−∞, +∞), L > 0 and D > 0, set
d∗(v, L, D) := −λ1(−g(I0e−k0x)).
It is easy to show that d∗(v, L, D) is positive. Our following result shows that d∗ is the
critical death rate; i.e., the phytoplankton survives if and only if its death rate is less than
Theorem 1. If 0 < d < d∗(v, L, D), then (3.1) has a unique positive steady state; If d ≥ d∗(v, L, D), then zero is the only non-negative steady state of (3.1).
A natural question is whether there also exist critical water column depth, critical sink-ing/buoyant velocity and critical turbulent diffusion rate. To address these issues, we need to understand the dependence of d∗ on the parameters D, v, L. The following result shows
that d∗ is monotone deceasing in v.
Theorem 2. For any D > 0 and L > 0, d∗(v, L, D) is strictly monotone decreasing for v ∈ (−∞, ∞). Moreover,
lim
v→−∞d∗(v, L, D) = g(I0), v→∞lim d∗(v, L, D) = g(I0e −k0L).
We apply Theorem 2 to study the existence of critical sinking/buoyant velocity. By The-orem 2, for every d ∈ (g(I0e−k0L), g(I0)), there exists a unique v∗ := v∗(d, L, D) such that d = d∗(v∗, L, D). Moreover, v∗ = > 0, if g(I0e−k0L) < d < d∗(0, L, D), = 0, if d = d∗(0, L, D), < 0, if d∗(0, L, D) < d < g(I0).
As a consequence of Theorems 1 and 2 and the definition of v∗, we have
Theorem 3. Given any D > 0 and L > 0.
(a) If 0 < d < g(I0e−k0L), (3.1) has a unique positive steady state, denoted as P (x), for any
v ∈ (−∞, ∞). Moreover, (3.4) Z L 0 P (x) dx > 1 k1 lnI0e −k0L g−1(d) > 0.
(b) If d ∈ (g(I0e−k0L), g(I0)), (3.1) has a unique positive steady state for every v ∈ (−∞, v∗); if v > v∗, zero is the only non-negative steady state of (3.1).
(c) If d > g(I0), zero is the only non-negative steady state of (3.1) for v ∈ (−∞, ∞).
Theorem 3 implies that critical sinking/buoyant velocity may or may not exist, and is unique whenever it exists. If d is suitably small, the phytoplankton can always bloom for any sinking/buoyant velocity; i.e., there is no critical sinking/buoyant velocity for this case. Only when the death rate falls into some intermediate range, there exists a critical sinking/buoyant velocity v∗ such that the phytoplankton can bloom if and only if the sinking/buoyant velocity
is smaller than v∗. For large death rates, the phytoplankton simply can not bloom.
We now turn to the existence of critical water column depth. First, we study how d∗
qualitatively depend on L.
Theorem 4. For any D > 0 and v ∈ (−∞, ∞), d∗(v, L, D) is strictly monotone decreasing for L ∈ (0, ∞). Moreover,
lim
L→0+d∗(v, L, D) = g(I0), L→∞lim d∗(v, L, D) = d∞(v, D),
where d∞(v, D) is a non-negative monotone decreasing function of v ∈ (−∞, ∞), and there exists some v0 > 0 such that d∞(v, D) > 0 for v < v0.
We now apply Theorem 4 to study the existence of critical water column depth. By Theorem 4, given any v ∈ R1 and D > 0, for every d ∈ (d
∞(v, D), g(I0)), there exists a
unique L∗ := L∗(d, v, D) > 0 such that d = d∗(v, L∗, D). As a consequence of Theorems 1
Theorem 5. Given any v ∈ (−∞, ∞) and D > 0.
(a) If 0 < d < d∞(v, D), (3.1) has a unique positive steady state for any L > 0.
(b) If d ∈ (d∞(v, D), g(I0)), (3.1) has a unique positive steady state for every L ∈ (0, L∗); if L > L∗, zero is the only non-negative steady state.
(c) If d > g(I0), zero is the only non-negative steady state of (3.1) for any L > 0.
Theorem 5 also implies that critical water column depth may or may not exist, and is unique whenever it exists. If d is suitably small, there may be no critical water column depth as the phytoplankton can bloom for any water column depth. For some intermediate range of death rates, there exists a critical water column depth L∗ such that the phytoplankton
can persist if and only if the water column depth is less than L∗. We do not know whether d∞(v, D) is positive for every v ∈ (−∞, ∞) and D > 0 and it will be of interest to further
understand d∞(v, D).
Finally, we address the existence of critical turbulent diffusion coefficient. This case is much more subtle as the numerical simulations in [8] suggest that there may exist two critical turbulent diffusion coefficients for sinking species. Similar as before, we first study how the critical death rate d∗ depends on turbulent diffusion coefficient D. It turns out that
the sinking case (v > 0) is indeed more subtle than the buoyant case (v < 0): Theorem 6. For any v ∈ (−∞, ∞) and L > 0,
lim D→∞d∗(v, L, D) = 1 L Z L 0 g(I0e−k0x) dx;
(a) For any v ≤ 0 and L > 0, d∗(v, L, D) is strictly monotone decreasing for D > 0, and
limD→0+d∗(v, L, D) = g(I0).
(b) For any v > 0 and L > 0, limD→0+d∗(v, L, D) = g(I0e−k0L); Moreover, given any L > 0,
there exists some v1 > 0 such that for every 0 < v < v1,
(3.5) sup
0<D<∞
d∗(v, L, D) > lim
D→∞d∗(v, L, D) > limD→0+d∗(v, L, D); In particular, for L > 0 and 0 < v < v1, d∗(v, L, D) is not monotone in D.
We do not know whether (3.5) holds for general v > 0 and L > 0. By Theorem 6, given any v ≤ 0 and L > 0, for every d ∈ (1
L
RL
0 g(I0e−k0x), g(I0)), there exists a unique
D∗ := D∗(d, v, L) > 0 such that d = d∗(v, L, D∗). By Theorems 1 and 6 and the definition of D∗, we have
Theorem 7. Given any v ≤ 0 and L > 0. (a) If 0 < d < 1
L
RL
0 g(I0e−k0x), (3.1) has a unique positive steady state for any D > 0.
(b) If d ∈ (1
L
RL
0 g(I0e−k0x), g(I0)), (3.1) has a unique positive steady state for every D ∈
(0, D∗); if D > D∗, zero is the only non-negative steady state.
(c) If d > g(I0), zero is the only non-negative steady state of (3.1).
Similar to other critical rates, critical turbulent diffusion rate depth may or may not exist for buoyant species and whenever it exists, it is unique. However, the story is quite different for sinking species. Let v1 be as given in Theorem 6 such that (3.5) holds for 0 < v < v1.
Set
d = sup
0<D<∞
d∗(v, D, L), d = inf
0<D<∞d∗(v, D, L).
By Theorem 6, we see that d ∈ (0, g(I0e−k0L)] and d > L1
RL
0 g(I0e−k0x). The following
result shows that, in strong contrast to buoyant species, there may exist two or more critical turbulent diffusion rate for sinking species:
Theorem 8. Given L > 0 and 0 < v < v1.
(a) If 0 < d < d, (3.1) has a unique positive steady state for any D > 0. (b) If d ∈ (1
L
RL
0 g(I0e−k0x), d), there exist 0 < Dmin < D ≤ D < Dmax such that (3.1) has
no positive steady state for any D ∈ (0, Dmin) ∪ (Dmax, ∞), and (3.1) has a unique positive steady state for any D ∈ (Dmin, D) ∪ (D, Dmax).
(c) If d > d, zero is the only non-negative steady state of (3.1).
From these results we can conclude that critical death rate always exists and is unique. In contrast, there are either zero or one critical water column depth, zero or one critical sinking/buoyant velocity, and zero or one critical turbulent diffusion rate for buoyant species. Interestingly, there may exist two critical turbulent diffusion rates for sinking species which was first shown numerically in [8]. These theoretical findings may shed some new insight into the combined effects of death rate, water column depth, sinking/buoyant velocity and turbulent diffusion rate in the persistence of single phytoplankton species.
The rest of this section concerns qualitative properties of the unique positive steady state
P (x; v) of (2.1)-(2.2) when the advection coefficient v varies, assuming that other parameters D, d, L, k0, k1 are all fixed. For the simplicity of notation and the clarity of the presentation,
we perform the following scaling for the equation (2.1)-(2.2). Let
(3.6) ˜ x = x L, ˜t = D L2t, ˜k0 = k0L, ˜k1 = k1L, ˜d = L2 Dd, ˜v = v DL, ˜
P (˜x, ˜t) = P (x, t), ˜I(˜x, ˜t) = I(x, t) = I0e−˜k0˜xexp(−˜k1
Z x˜ 0 ˜ P (s, ˜t)ds), ˜g( ˜I)(˜x, ˜t) = L2 Dg(I(x, t)).
Then equation (2.1)-(2.2) becomes (3.7) ˜ P˜t = ˜P˜x˜x− ˜v ˜Px˜+ ³ ˜g( ˜I) − ˜d ´ ˜ P , 0 < ˜x < 1, ˜ Px˜(0, ˜t) − ˜v ˜P (0, ˜t) = 0, P˜x˜(1, ˜t) − ˜v ˜P (1, ˜t) = 0.
If we drop the ∼ sign, equation (3.7) becomes (3.8)
½P
t= Pxx− vPx+ (g(I) − d) P, 0 < x < 1, t > 0, Px(0, t) − vP (0, t) = 0, Px(1, t) − vP (1, t) = 0,
where I is still given by (2.4).
Let P (x; v) denote the unique positive steady state of (3.8). By Theorem 3, if 0 < d <
g(I0e−k0), P (x; v) exists for any v ∈ (−∞, ∞). The following result describes the asymptotic
profiles of P (x; v) for large positive v.
Theorem 9. Suppose that 0 < d < g(I0e−k0).
(a) If v ≥ 2pg(I0) − d, then P (x; v) is strictly increasing in [0, 1];
(b) As v → ∞, P (x; v) → 0 uniformly in any compact subset of [0, 1), P (1; v)/v → κ∗, P (·; v) → κ∗δ(1), where κ∗ > 0 is uniquely determined by
(3.9) Z 1 0 g(I0e−k0−k1κ ∗z ) dz = d. Moreover, (3.10) lim v→∞ ° °P (x; v) − P (1; v)e−v(1−x)°° L∞(0,1) = 0
and (3.11) lim v→∞ ° ° ° °veP (x; v)−v(1−x) − κ ∗ ° ° ° ° L∞(0,1) = 0.
Remark 3.1. δ(1) denotes the Dirac measure at x = 1, and P (·; v) → κ∗δ(1) as v → ∞ means that for any continuous function f in [0, 1],
lim
v→∞
Z 1
0
f (x)P (x; v) dx = κ∗f (1).
Similarly, the asymptotic profiles of P (x; v) for large negative v can be characterized as follows:
Theorem 10. Suppose that 0 < d < g(I0).
(a) If v ≤ 0, then P (x; v) is strictly decreasing in [0, 1];
(b) As v → −∞, P (x; v) → 0 uniformly in any compact subset of (0, 1], P (0; v)/v → κ∗, P (·; v) → −κ∗δ(0), where κ∗ < 0 is uniquely determined by
(3.12) Z 1 0 g(I0ek1κ∗(1−z)) dz = d. Moreover, (3.13) lim v→−∞kP (x; v) − P (0; v)e vxk L∞(0,1) = 0 and (3.14) lim v→−∞ ° ° ° °P (x; v)vevx − κ∗ ° ° ° ° L∞(0,1) = 0.
By Theorem 10, the buoyant species is always monotone decreasingly distributed in the water column, and the phytoplankton forms a thin layer at the surface of the water column when the buoyant coefficient is sufficiently large. On the other end, by Theorem 9, P (x; v) is monotone increasing in the water column when the sinking velocity is suitably large, and the phytoplankton forms a thin layer at the bottom of the water column.
4. Proof of Theorem 1 Consider the steady state equation
(4.1) ½ DPxx− vPx+ P [g(I) − d] = 0, 0 < x < L, DPx(0) − vP (0) = 0, DPx(L) − vP (L) = 0, where (4.2) I = I(x) = I0e−k0xexp(−k1 Z x 0 P (s)ds).
The proof of Theorem 1 is similar to that of case v = 0, which was studied in [7], with some modifications. For the sake of completeness we give the proof here in details.
Lemma 4.1. (4.1) has no positive solution when d 6∈ (0, d∗). Proof. We note that the first equation in (4.1) can be rewritten as
(4.3) −DPxx+ vPx+ (−g(I))P = −dP.
If (d, P ) is a positive solution of (4.1), from (4.2), (4.3) and the comparison principle of the principal eigenvalue,
That is, d < d∗. Multiplying the first equation of (4.1) by e−(v/D)x, integrating the result in
(0, L), and applying the boundary condition in (4.1), we obtain Z L
0
e−(v/D)xP [g(I) − d] dx = 0,
which implies that d > 0. Therefore, (4.1) has no positive solution when d 6∈ (0, d∗). ¤
Lemma 4.2. Given any η > 0, there exists some positive constant C(η) such that every
positive solution P of (4.1) with η < d < d∗ satisfies kP kL∞(0,L) ≤ C(η).
Proof. We argue by contradiction. If not, suppose that there exists a sequence dn ∈ (η, d∗), n = 1, 2, ..., and positive solution Pn of (4.1) with d = dn such that kPnkL∞(0,L) → ∞ as
n → ∞. Passing to a subsequence if necessary we may assume that dn → d ∈ [η, d∗]. Set
˜
Pn= Pn/kPnkL∞(0,L). Then ˜Pn satisfies k ˜PnkL∞ = 1 and
(4.4) ( D ˜Pn,xx− v ˜Pn,x+ ˜Pn[g(In) − dn] = 0, 0 < x < L, D ˜Pn,x(0) − v ˜Pn(0) = 0, D ˜Pn,x(L) − v ˜Pn(L) = 0, where (4.5) In(x) = I0e−k0xexp(−k1 Z x 0 Pn(s)ds).
Integrating the first equation of (4.4) from 0 to x we have
D ˜Pn,x(x) − v ˜P (x) +
Z x
0
˜
Pn[g(In) − dn] = 0.
As g(In) and ˜Pn are uniformly bounded, ˜Pn,x is uniformly bounded. By (4.4), ˜Pn,xx is
uniformly bounded. Passing to a sequence if necessary we may assume that ˜Pn → ˜P in C1[0, L], ˜P ≥ 0, k ˜P k
L∞ = 1. As 0 ≤ g(In) ≤ g(I0) in [0, L], we may assume that g(In) → q(x)
weakly in L2(0, L) for some function q satisfying 0 ≤ q ≤ g(I
0). Hence, ˜P is a weak solution
of (4.6)
(
D ˜Pxx− v ˜Px+ ˜P [q(x) − d] = 0, 0 < x < L, D ˜Px(0) − v ˜P (0) = 0, D ˜Px(L) − v ˜P (L) = 0.
As ˜P ≥ 0, ˜P 6≡ 0 and q ∈ L∞(0, L), by the strong maximum principle we have ˜P > 0 in
(0, L). As ˜Pn → ˜P > 0 in (0, L) and kPnkL∞(0,L) → ∞, (4.7) In(x) = I0e−k0xexp(−k1kPnkL∞([0,L]) Z x 0 ˜ Pn(s)ds) → 0
for every x ∈ (0, L) as n → ∞. This implies that q ≡ 0. Integrating (4.6) in (0, L), we
obtain d = 0, which is a contradiction. ¤
Proof of Theorem 1. By a standard bifurcation argument of Crandall and Rabinowitz [3] and Rabinowitz [18], (4.1) has an unbounded connected branch of positive solutions, denote by Γ = {(d, P ) ⊂ R1 × C1([0, 1])}, which bifurcations from the trivial branch {(d, 0)} at
(d∗(v, L, D), 0). Since (4.1) has no positive solution when d 6∈ (0, d∗) (Lemma 4.1) and all
positive solutions of (4.1) are uniformly bounded when d is positive and bounded away from zero (Lemma 4.2), we see that Γ can only become unbounded as d → 0+. As Γ is connected, (4.1) has at least one positive solution for every d ∈ (0, d∗).
It remains to show the uniqueness. Let U(x) = e−(v/D)xP (x). Then (4.1) becomes
(4.8)
½DU
xx+ vUx+ [g(I) − d] U = 0, 0 < x < L, Ux(0) = 0, Ux(L) = 0,
where (4.9) I = I(x) = I0e−k0xexp(−k1 Z x 0 e(v/D)sU(s)ds). (4.8) can be rewritten as (4.10) ( D(e(v/D)xUx)x+ [g(I) − d] Ue(v/D)x= 0, 0 < x < L, Ux(0) = 0, Ux(L) = 0.
The proof of the uniqueness of positive solution of (4.1) basically follows from the argument in [7] applying to (4.10). Suppose that (4.8) has two positive solutions U1 6≡ U2. If U1 ≤ U2
then we deduce −d = λ1[−g(I0e−k0xexp(−k1 Z x 0 e(v/D)sU1(s)ds))] < λ1[−g(I0e−k0xexp(−k1 Z x 0 e(v/D)sU 2(s)ds))] = −d,
a contradiction. Therefore U1 − U2 changes sign in (0, L). We claim that U1(0) 6= U2(0).
Otherwise, for i = 1, 2, we denote Vi(x) =
Rx
0 Ui(s)e(v/D)sds, Wi(x) = Ui0(x)e(v/D)x, and find
that (Ui, Vi, Wi) are solution of the initial value problem
U0 = W e−(v/D)x, V0 = e(v/D)xU,
DW0 = −[g(I0e−k0xexp(−k1V )) − d]e(v/D)xU,
(U(0), V (0), W (0)) = (U(0), 0, 0).
By the uniqueness of ODE, we conclude that (U1, V1, W1) = (U2, V2, W2), a contradiction.
Therefore U1(0) 6= U2(0).
For definiteness we assume U1(0) < U2(0). Since U1 − U2 changes sign in (0, L), there
exists x0 > 0 such that U2(x) > U1(x) in [0, x0), U1(x0) = U2(x0), and U10(x0) ≥ U20(x0).
From (4.10) we have −D Z x0 0 ¡ U10e(v/D)x¢xU2 = Z x0 0 · g µ I0e−k0xexp(−k1 Z x 0 e(v/D)sU1(s)ds) ¶ − d ¸ U1U2e(v/D)x.
Using integration by parts, we deduce
− DU0 1(x0)e(v/D)x0U2(x0) + D Z x0 0 e(v/D)xU0 1U20dx = Z x0 0 · g µ I0e−k0xexp(−k1 Z x 0 e(v/D)sU 1(s)ds) ¶ − d ¸ U1U2e(v/D)xdx. Similarly, − DU0 2(x0)e(v/D)x0U1(x0) + D Z x0 0 e(v/D)xU0 1U20dx = Z x0 0 · g µ I0e−k0xexp(−k1 Z x 0 e(v/D)sU 2(s)ds) ¶ − d ¸ U1U2e(v/D)xdx. Therefore De(v/D)x0U 1(x0) [U20(x0) − U10(x0)] = Z x0 0 · g µ I0e−k0xexp(−k1 Z x 0 eDvsU1(s)) ¶ − g µ I0e−k0xexp(−k1 Z x 0 eDvsU2(s)) ¶¸ U1U2e v Dx.
The right hand side of the above equality is positive while the left hand side is nonpositive, a contradiction. Thus we complete the proof of Theorem 1. ¤
5. Dependence of d∗(v, L, D) on v: Proofs of Theorems 2 and 3
This section is devoted to proofs of Theorems 2 and 3. Recall that d∗(v, L, D) satisfies
(5.1)
(
− Dϕxx+ vϕx− g(I0e−k0x)ϕ = −d∗(v, L, D)ϕ in (0, L), Dϕx(0) = vϕ(0), Dϕx(L) = vϕ(L), ϕ > 0 in (0, L).
Set ψ = e−(v/D)xϕ. Then, ψ satisfies
(5.2)
(
− Dψxx− vψx− g(I0e−k0x)ψ = −d∗(v, L, D)ψ in (0, L), ψx(0) = ψx(L) = 0, ψ > 0 in (0, L).
Lemma 5.1. ψx < 0 in (0, L).
Proof. Multiplying (5.2) by e(v/D)x, we rewrite the resulting equation as
(5.3) ( − D(e(v/D)xψ x)x− e(v/D)xg(I0e−k0x)ψ = −d∗(v, L, D)ψe(v/D)x in (0, L), ψx(0) = ψx(L) = 0. Integrating (5.3) in (0, L), we have Z L 0 e(v/D)xψ[g(I 0e−k0x) − d∗] dx = 0,
which implies that g(I0e−k0x)−d∗changes sign in (0, L). Since g(I0e−k0x) is strictly decreasing
in (0, L), there exists a unique x0 ∈ (0, L) such that g(I0e−k0x) > d∗ for 0 < x < x0 and
g(I0e−k0x) < d∗ for x0 < x < L. Hence, by (5.3) we see that (e(v/D)xψx)x < 0 for 0 < x < x0
and (e(v/D)xψ
x)x > 0 for x0 < x < L, i.e., e(v/D)xψx is strictly decreasing in (0, x0) and
strictly increasing in (x0, L). Since ψx(0) = ψx(L) = 0, we have ψx < 0 in (0, L). ¤
Lemma 5.2. d∗(v, L, D) is strictly monotone decreasing in v. Proof. Recall that d∗(v, L, D) satisfies
(5.4)
(
Dψxx+ vψx+ g(I0e−k0x)ψ = d∗(v, L, D)ψ in (0, L), ψx(0) = ψx(L) = 0.
We normalize ψ such that R0Lψ2 = 1. It can be shown that d
∗ and ψ are smooth functions of v (see e.g., [1, 2]). For simplicity of the notation, we denote ∂ψ/∂v by ψ0, etc. Differentiating
(5.4) with respect to v, we have (5.5)
(
Dψ0xx+ vψ0x+ ψx+ g(I0e−k0x)ψ0 = d0∗ψ + d∗ψ0 in (0, L), ψx0(0) = ψx0(L) = 0.
Multiplying (5.5) by e(v/D)x, we rewrite the result as
(5.6) D(e(v/D)xψ0
x)x+ e(v/D)xψx+ e(v/D)xg(I0e−k0x)ψ0 = d0∗ψe(v/D)x+ d∗ψ0e(v/D)x in (0, L).
Multiplying (5.6) by ψ and integrating the resulting equation in (0, L), we have
(5.7) − D Z L 0 e(v/D)xψxψx0 + Z L 0 e(v/D)xψψx+ Z L 0 e(v/D)xψ0ψg(I0e−k0x) = d0 ∗ Z L 0 ψ2e(v/D)x+ d ∗ Z L 0 ψψ0e(v/D)x.
Multiplying (5.4) by e(v/D)x, we write the result as
(5.8) D(e(v/D)xψx)x+ e(v/D)xg(I0e−k0x)ψ = d∗e(v/D)xψ.
Multiplying (5.8) by ψ0 and integrating it in (0, L), we have
(5.9) −D Z L 0 e(v/D)xψ xψ0x+ Z L 0 e(v/D)xψ0ψg(I 0e−k0x) = d∗ Z L 0 ψψ0e(v/D)x.
It follows from (5.7) and (5.9) that
(5.10) d0 ∗ = RL 0Re(v/D)xψψxdx L 0 e(v/D)xψ2 .
This together with Lemma 5.1 and the positivity of ψ imply that d0
∗ < 0. ¤
To study the asymptotic behavior of d∗ for sufficiently large v (either positive or negative),
we first recall the following result (Theorem 1, [4]): Lemma 5.3. Let λ(v) denote the principal eigenvalue of (5.11)
½ − ∆ψ − v∇m · ∇ψ + c(x)ψ = λψ in Ω,
∇ψ · n|∂Ω= 0,
where Ω is a domain in RN with smooth boundary ∂Ω and n is the outward unit normal vector on ∂Ω. Suppose that m ∈ C2( ¯Ω) and c ∈ C( ¯Ω), and all critical points of m are
non-degenerate. Then
lim
v→∞λ(v) = minM c, where M is the set of local maximum of m(x).
Lemma 5.4. We have lim
v→∞d∗(v, L, D) = g(I0e
−k0L), lim
v→−∞d∗(v, L, D) = g(I0).
Proof. Applying Lemma 5.3 with Ω = (0, L) and m(x) = x, we see that M = {L} and
lim
v→∞(−d∗(v, L, D)) = minM (−g(I0e
−k0x)) = −g(I
0e−k0L).
Similarly, applying Lemma 5.3 with Ω = (0, L) and m(x) = −x, we see that M = {0} and
lim
v→−∞(−d∗(v, L, D)) = minM (−g(I0e
−k0x)) = −g(I
0),
which completes the proof. ¤
Lemma 5.5. Suppose that 0 < d < g(I0e−k0L). Then for any v ∈ (−∞, ∞),
Z L 0 P (x; v) dx > 1 k1 lnI0e−k0L g−1(d) > 0. Proof. Integrate the equation of P (x; v) in (0, L), we have
Z L
0
P (x; v) [g(I(x)) − d] dx = 0.
Hence, g(I(x)) − d changes sign in (0, L). Since I(x) is strictly decreasing, g(I(x)) − d must be negative at x = L. That is,
g(I0e−k0Le−k1 RL 0 P (x;v) dx) < d, which is equivalent to Z L 0 P (x; v) dx > 1 k1 lnI0e −k0L g−1(d) > 0,
where the last inequality follows from 0 < d < g(I0e−k0L). ¤
Proofs of Theorems 2, 3. Theorem 2 follows from Lemmas 5.2 and 5.4. Theorem 3 follows from Theorems 1, 2 and Lemma 5.5. ¤
6. Dependence of d∗(v, L, D) on L: Proofs of Theorems 4 and 5
In this section we investigate the dependence of d∗ on L. First, we establish the
mono-tonicity of d∗ in L.
Lemma 6.1. d∗(v, L, D) is strictly monotone decreasing in L.
Proof. Given any 0 < L1 < L2, we show that d∗(v, L1, D) > d∗(v, L2, D). For simplicity, we
write d∗(v, Li, D) as di, and denote corresponding eigenfunctions ψ(x; v, Li, D) as ψi, i = 1, 2.
Rewrite the equations of ψi as
(6.1)
(
D(e(v/D)xψ
i,x)x+ g(I0e−k0x)e(v/D)xψi = diψie(v/D)x in (0, Li), ψi,x(0) = ψi,x(Li) = 0.
Multiplying the equation of ψ1 by ψ2, the equation of ψ2 by ψ1, and subtracting, we have
(d1− d2)ψ1ψ2e(v/D)x= D £ (e(v/D)xψ 1,x)xψ2− (e(v/D)xψ2,x)xψ1 ¤ .
Integrating the above equation in (0, L1) and applying boundary conditions of ψ1, ψ2 at
x = 0, we have
(d1− d2)
Z L1
0
ψ1ψ2e(v/D)xdx = −De(v/D)L1ψ2,x(L1)ψ1(L1).
Since ψi > 0 for i = 1, 2 and ψ2,x(L1) < 0 (Lemma 5.1), we see that d1 > d2. ¤
The next two results concern the limiting behaviors of d∗ for small and large L.
Lemma 6.2. limL→0+d∗(v, L, D) = g(I0).
Proof. Set x = Ly and w(y) = ψ(x). Then w satisfies
(6.2)
(
Dwyy+ vLwy+ L2g(I0e−k0Ly)w = d∗(v, L, D)L2w in (0, 1), wy(0) = wy(1) = 0.
We normalize w such that max[0,1]w = 1. It is easy to show that as L → 0+, passing
to a subsequence if necessary, w → w0 in C2[0, 1], where w0 satisfies w0,yy = 0 in (0, 1),
w0,y(0) = w0,y(1) = 0, and max[0,1]w0 = 1. Hence, w0 ≡ 1; i.e., w → 1 in C2[0, 1].
Multiplying (6.2) by e(v/D)Ly, we can rewrite (6.2) as
(6.3) ( D¡e(v/D)Lyw y ¢ y+ L 2e(v/D)Lyg(I 0e−k0Ly)w = d∗(v, L, D)L2e(v/D)Lyw in (0, 1), wy(0) = wy(1) = 0.
Integrating (6.3) in (0, 1) and dividing the result by L2, we have
(6.4) Z 1 0 e(v/D)Lyg(I 0e−k0Ly)w dy = d∗ Z 1 0 e(v/D)Lyw dy.
By letting L → 0 in (6.4) and applying w → 1, we see that d∗ → g(I0) as L → 0+. ¤
Lemma 6.3. limL→∞d∗(v, L, D) = d∞, where d∞ := d∞(v, D) ≥ 0, and is monotone de-creasing function of v ∈ R1. Moreover, there exists some v
0 > 0 such that d∞(v, D) > 0 for v < v0.
Proof. Since d∗ is monotone decreasing in L and d∗ > 0, we see that limL→∞d∗(v, L, D) = d∞(v, D) for some d∞ = d∞(v, D) ≥ 0. Since d∗ is monotone decreasing in v, we see that d∞ is also monotone decreasing in v. It remains to show that d∞ > 0 for v ∈ (−∞, v0) for
some v0 > 0. Recall that
−d∗ = inf ϕ∈H1((0,L)) RL 0 e(v/D)x £ Dϕ2 x− g(I0e−k0x)ϕ2 ¤ dx RL 0 e(v/D)xϕ2 ≤ inf ϕ∈H1((0,L)) RL 0 e(v/D)x(Dϕ2x− aI0γe−k0γxϕ2) dx RL 0 e(v/D)xϕ2 ,
where the last inequality follows from assumption g(I) ≥ aIγ for I ∈ [0, I
0]. Choose the test
function ϕ(x) = e−(v/D)x. By direct calculation, −d∗ ≤ v2 D − aI0γ(v/D) k0γ + v/D 1 − e−(v/D+k0)L 1 − e−(v/D)L .
By letting L → ∞ in the above inequality, we have
−d∞ ≤ v 2 D − aI0γ(v/D) k0γ + v/D < 0,
where the last inequality holds provided that v(k0γ + v/D) < aI0γ. Clearly, if
v0 := min{aI0γ/(2k0γ),
p
aI0γD/2},
then d∞(v, D) > 0 for v < v0. ¤
Proofs of Theorems 4, and 5. Theorem 4 follows from Lemmas 6.1, 6.2 and 6.3; Theorem 5 follows from Theorems 1 and 4. ¤
7. Dependence of d∗(v, L, D) on D: Proofs of Theorems 6, 7 and 8
In this section we investigate the dependence of d∗ on D. The proof of the following result
is similar to that of Lemma 5.2:
Lemma 7.1. For any v ≤ 0 and L > 0, d∗(v, L, D) is strictly monotone decreasing in D. Proof. For simplicity of notation, we denote ∂ψ/∂D by ψ0, etc, where ψ satisfies (5.4).
Differentiate (5.4) with respect to D, we have (7.1) ( Dψ0 xx+ ψxx+ vψx0 + g(I0e−k0x)ψ0 = d0∗ψ + d∗ψ0 in (0, L), ψ0 x(0) = ψx0(L) = 0.
Multiplying (7.1) by e(v/D)xψ and integrating the resulting equation in (0, L), we have
(7.2) − D Z L 0 e(v/D)xψ xψx0 + Z L 0 e(v/D)xψψ xx+ Z L 0 e(v/D)xψ0ψg(I 0e−k0x) = d0 ∗ Z L 0 ψ2e(v/D)x+ d ∗ Z L 0 ψψ0e(v/D)x.
Similarly, multiplying (5.4) by e(v/D)xψ0 and integrating it in (0, L), we have
(7.3) −D Z L 0 e(v/D)xψ xψ0x+ Z L 0 e(v/D)xψ0ψg(I 0e−k0x) = d∗ Z L 0 ψψ0e(v/D)x.
It follows from (7.2) and (7.3) that (7.4) d0 ∗ = RL 0R e(v/D)xψψxx L 0 e(v/D)xψ2 . By Lemma 5.1, ψx < 0 in (0, L). Hence, Z L 0 e(v/D)xψψxx = − Z L 0 ψx ¡ e(v/D)xψ¢x = − Z L 0 e(v/D)x[ψ2x+ (v/D)ψψx] < 0,
where the last inequality holds for v ≤ 0. Hence, d0
∗ < 0 for any v ≤ 0 and D, L > 0. ¤
Lemma 7.2. Given any v ∈ (−∞, ∞) and L > 0,
(7.5) lim D→∞d∗(v, L, D) = 1 L Z L 0 g(I0e−k0x) dx.
Proof. Recall that ψ satisfies (5.4). We normalize that ψ such that max[0,L]ψ = 1. By
standard elliptic regularity and Sobolev embedding theorem, ψ is uniformly bounded in
C2[0, L] for all D ≥ 1. Therefore, passing to some sequence if necessary, we may assume that
ψ → Ψ in C1, where Ψ satisfies Ψ
xx = 0 in [0, L], Ψx(0) = Ψx(L) = 0, and max[0,L]Ψ = 1.
Therefore, Ψ ≡ 1; i.e., ψ → 1 in C1[0, L]. Integrating (5.4) in [0, L], we have
D[ψx(L) − ψx(0)] + v[ψ(L) − ψ(0)] + Z L 0 g(I0e−k0x)ψ dx = d∗ Z L 0 ψ.
Since ψx(0) = ψx(L) = 0 and ψ → 1 as D → ∞, by letting D → ∞ in the above equation,
we obtain (7.5). ¤
Lemma 7.3. Suppose that v ≤ 0. Then lim
D→0+d∗(v, L, D) = g(I0). Proof. Recall that
(7.6) −d∗ = inf ψ∈H1(0,L) RL 0 e(v/D)x £ Dψ2 x− g(I0e−k0x)ψ2 ¤ dx RL 0 e(v/D)xψ2dx . For ² ∈ (0, L/4), set ψ(x) = 1, 0 ≤ x ≤ ², 2 −x ², ² ≤ x ≤ 2², 0, 2² ≤ x ≤ L. Hence, −d∗ ≤ DR²2²e(v/D)xψ2 x R2² 0 e(v/D)xψ2 − R2² 0 e(v/D)xR g(I0e−k0x)ψ2 2² 0 e(v/D)xψ2 ≤ D ²2 e2v²/D− ev²/D ev²/D− 1 − g(I0e −2k0²).
By letting D → 0+, as v ≤ 0, we have lim infD→0+d∗ ≥ g(I0e−2k0²). By letting ² → 0, we
obtain lim infD→0+d∗ ≥ g(I0). As d∗ < g(I0), we see that limD→0+d∗ = g(I0). ¤
Lemma 7.4. Suppose that v > 0. Then lim
D→0+d∗(v, L, D) = g(I0e −k0L).
Proof. Recall that d∗(v, L, D) satisfies
(7.7)
(
Dϕxx− vϕx+ g(I0e−k0x)ϕ = d∗ϕ in (0, L),
Dϕx(0) = vϕ(0), Dϕx(L) = vϕ(L), ϕ > 0 in (0, L).
Set w(x) = e−(v/D)ηxϕ, where η is some constant which will be chosen differently for
different purposes. Then, w satisfies
(7.8) Dwxx+ v(2η − 1)wx+ w · v2 Dη(η − 1) + g(I0e −k0x) − d ∗ ¸ = 0 in 0 < x < L, Dwx = v(1 − η)w at x = 0, L.
Set η = 1 − C1D/v2, where C1 is some positive constant to be chosen later. Then w
satisfies (7.9) Dwxx+ v(1 − 2C1D v2 )wx+ w[−C1(1 − C1D v2 ) + g(I0e −k0x) − d ∗] = 0, 0 < x < L, wx = (C1/v)w at x = 0, L.
Let x∗ ∈ [0, L] such that w(x∗) = max
0≤x≤Lw(x). Since wx(0) > 0, x∗ 6= 0. If x∗ ∈ (0, L), wxx(x∗) ≤ 0 and wx(x∗) = 0. By (7.9) we have
−C1(1 − C1D/v2) + g(I0e−k0x ∗
) − d∗ ≥ 0,
which is impossible if we choose C1 = 2g(I0) and D < v2/(4g(I0)). Therefore, x∗ = L; i.e.,
w(x) ≤ w(L) for every x ∈ [0, L]. Hence, ϕ(x) ϕ(L) ≤ e
−v
D(1−C1Dv2 )(L−x).
Next, we choose η = 1 + C2D/v2, where C2 > 0 is to be chosen later. By (7.8), w satisfies
(7.10) Dwxx+ v(1 +2C2D v2 )wx+ w[C2(1 + C2D v2 ) + g(I0e −k0x) − d ∗] = 0, 0 < x < L, wx= −(C2/v)w at x = 0, L.
Let x∗ ∈ [0, L] such that w(x∗) = min0≤x≤Lw(x). Since wx(0) < 0, x∗ 6= 0. If x∗ ∈ (0, L), wxx(x∗) ≥ 0 and wx(x∗) = 0. By (7.10) we have
C2(1 + C2D/v2) + g(I0e−k0x∗) − d∗ ≤ 0,
which implies that d∗ > C2. Choose C2 = g(I0). As d∗ < g(I0), we must have x∗ = L; i.e., w(x) ≥ w(L) for every x ∈ [0, L]. Therefore,
ϕ(x) ϕ(L) ≥ e
−v
D(1+C2Dv2 )(L−x).
Integrating (7.7) in (0, L) and dividing the result by ϕ(L), we have (7.11) Z L 0 ϕ(x) ϕ(L) £ g(I0e−k0x) − d∗ ¤ dx = 0.
Set y = (L − x)/D. Then ϕ satisfies
(7.12) e−v(1+C2Dv2 )y ≤ ϕ(L − Dy) ϕ(L) ≤ e −v(1−C1Dv2 )y. We can rewrite (7.11) as (7.13) Z L/D 0 ϕ(L − Dy) ϕ(L) £ g(I0e−k0(L−Dy)) − d∗ ¤ dy = 0.
By (7.12), we can apply Lebesgue dominant convergent theorem and pass to the limit in (7.13) to obtain lim D→0+d∗ = limD→0+ RL/D 0 ϕ(L−Dy) ϕ(L) g(I0e−k0(L−Dy)) dy limD→0+ RL/D 0 ϕ(L−Dy) ϕ(L) dy = R∞ 0 e−vyR g(I0e−k0L) dy ∞ 0 e−vydy = g(I0e−k0L).
This completes the proof. ¤
Lemma 7.5. For any L > 0, there exists some v1 > 0 such that if v < v1, then
(7.14) d∗(v, L, D) > 1 L Z L 0 g(I0e−k0x) dx
for sufficiently large D.
Proof. Let ψ1 be the unique solution of
(7.15) ψ1,xx = 1 L Z L 0 g(I0e−k0x) dx − g(I0e−k0x), 0 < x < L, ψ1,x(0) = ψ1,x(L) = 0, Z L 0 ψ1(x) dx = 0.
In particular, multiplying the first equation of (7.15) by ψ1 and integrating the result in
(0, L), we have (7.16) Z L 0 g(I0e−k0x)ψ1(x) dx = Z L 0 ψ21,xdx > 0,
where the last strict inequality follows from the fact that g(I0e−k0x) is non-constant.
Set ψ = 1 + ψ1/D in (7.6), we have (7.17) d∗ ≥ RL 0 e(v/D)x £ −Dψ2 x+ g(I0e−k0x)ψ2 ¤ dx RL 0 e(v/D)xψ2dx . By direct calculations, Z L 0 e(v/D)x£−Dψ2 x+ g(I0e−k0x)ψ2 ¤ dx = Z L 0 g + 1 D · v Z L 0 xg(I0e−k0x) dx − Z 1 0 ψ2 1,x+ 2 Z L 0 g(I0e−k0x)ψ1 ¸ + O(1/D2) = Z L 0 g + 1 D · v Z L 0 xg(I0e−k0x) dx + Z 1 0 ψ2 1,x ¸ + O(1/D2),
where the last equality follows from (7.16); Similarly, Z L 0 e(v/D)xψ2dx = L + v 2DL 2+ O(1/D2). Hence, (7.18) d∗− 1 L Z L 0 g(I0e−k0x) ≥ 1 DL ·Z L 0 ψ2 1,x− v µ L 2 Z L 0 g(I0e−k0x) − Z L 0 xg(I0e−k0x) ¶¸ + O(1/D2).
We claim that (7.19) Λ := L 2 Z L 0 g(I0e−k0x) − Z L 0 xg(I0e−k0x) > 0.
To establish this assertion, note that
(7.20) Λ = Z L 0 g(I0e−k0x)( L 2 − x) = Z L 0 [g(I0e−k0x) − g(I0e−k0L/2)](L 2 − x), where the last equality follows from
Z L 0 g(I0e−k0L/2)( L 2 − x) = g(I0e −k0L/2) Z L 0 (L 2 − x) = 0.
Since functions g(I0e−k0x) − g(I0e−k0L/2) and L/2 − x are strictly monotone decreasing, and
both vanish at x = L/2, we see that [g(I0e−k0x) − g(I0e−k0L/2)](L2 − x) > 0 for any x 6= L/2.
This together with (7.20) imply that Λ > 0, i.e., (7.19) holds. Set v1 := RL 0 ψ21,x L 2 RL 0 g(I0e−k0x) − RL 0 xg(I0e−k0x) .
By (7.19), v1 > 0. Hence, by (7.18) and the definition of v1 we see that, for any v < v1,
(7.14) holds for sufficiently large D. ¤
Proofs of Theorems 6 and 7. Theorem 6 follows from Lemmas 7.1, 7.2, 7.3, 7.4 and 7.5; In particular, (3.5) follows from Lemmas 7.2, 7.4, 7.5 and the fact that g(I0e−k0L) <
1
L
RL
0 g(I0e−k0x). Theorem 7 follows from Theorems 1 and 6. ¤
Proof of Theorem 8. Parts (a) and (c) follow from Theorem 1 and the definitions of d and d. Hence, it suffices to show part (b). Given L > 0 and 0 < v < v1. Set f (D) =
d − d∗(v, L, D). By Lemma 7.4 we have
lim
D→0+f (D) = d − g(I0e
−k0L) > 0,
where the last inequality follows from assumption on d. Choose ˜D such that d∗(v, L, ˜D) =
sup0<D<∞d∗(v, L, D). By our assumption d < sup0<D<∞d∗(v, L, D), f ( ˜D) < 0. Let Dmin ∈
(0, ˜D) be such that f (Dmin) = 0, f (D) ≥ 0 for D ∈ (0, Dmin) and there exists some δ > 0
such that f (D) < 0 for D ∈ (Dmin, Dmin + δ). Choose D = Dmin+ δ. By the definition of f , we have d ≥ d∗(v, L, D) for 0 < D ≤ Dmin and d < d∗ for d ∈ (Dmin, D). By Theorem 1,
(3.1) has no positive steady state for 0 < D ≤ Dmin and a unique positive steady state for d ∈ (Dmin, D). Similarly, we can show that there exist Dmaxand D such that D ≤ D < Dmax
and (3.1) has no positive steady state for D ≥ Dmin and a unique positive steady state for d ∈ (D, Dmax). ¤
8. Asymptotic behaviors of steady states P (x; v) for large |v|
This section is devoted to proofs of Theorems 9 and 10. Let P (x; v) denote the unique positive steady state of (3.8), i.e.,
(8.1)
½P
xx− vPx+ (g(I) − d) P = 0, 0 < x < 1, Px(0) − vP (0) = Px(1) − vP (1) = 0,
Lemma 8.1. If v ≤ 0, then Px < 0 in (0, 1).
Proof. Integrating the equation of P (x; v) in (0, 1), we have
Z 1
0
P [g(I(x)) − d] dx = 0.
Since I(x) is strictly deceasing in (0, 1), there exists some x0 ∈ (0, 1) such that g(I(x)) > d
in (0, x0) and g(I(x)) < d in (x0, 1). By the equation of P , Pxx− vPx < 0 in (0, x0) and
Pxx − vPx > 0 in (x0, 1). Hence, Px − vP in strictly monotone decreasing in (0, x0) and
strictly increasing in (x0, 1). As Px= vP at x = 0, 1, Px− vP < 0 in (0, 1). Since v ≤ 0 and
P > 0, Px < 0 in (0, 1). ¤
Set
w(x) = e−vηxP (x; v),
where η is some constant which will be chosen differently for different purposes. Clearly,
Px = evηx(vηw + wx) and Pxx = evηx(v2η2w + 2vηwx+ wxx). Then, w satisfies (8.2) ( wxx+ v(2η − 1)wx+ w £ v2η(η − 1) + g(I(x)) − d¤ = 0 in 0 < x < 1, wx = v(1 − η)w at x = 0, 1.
Lemma 8.2. If v > 2pg(I0) − d, then Px > 0 for 0 ≤ x ≤ 1. Proof. Set η = 1/2. Then, w satisfies
(8.3) wxx+ w · −v2 4 + g(I(x)) − d ¸ = 0 in 0 < x < 1, wx = v 2w at x = 0, 1. If v > 2pg(I0) − d, then v2 4 − g(I(x)) + d > 0
in (0, 1), i.e., wxx > 0 in (0, 1). Since wx(0) > 0, we have wx> 0 in [0, 1]. This implies that Px = e(v/2)x[(v/2)w + wx] > 0
in [0, 1]. ¤
Lemma 8.3. There exist positive constants Ci (i = 1, 2), both independent of v, such that
(a) if v ≥ C1, e−C2v (1−x) ≤ P (x; v) P (1; v)e−v(1−x) ≤ e C2 v (1−x) for every x ∈ [0, 1]; (b) if v ≤ −C1, then eC2v x ≤ P (x; v) P (0; v)evx ≤ e −C2v x for every x ∈ [0, 1].
Proof. We first set η = 1 − C3/v2, where C3 is some positive constant to be chosen later. Then w satisfies (8.4) ( wxx+ v(1 − 2C3/v2)wx+ w[−C3(1 − C3/v2) + g(I(x)) − d] = 0 in 0 < x < 1, wx = (C3/v)w at x = 0, 1.
Let x∗ ∈ [0, 1] such that w(x∗) = max
0≤x≤1w(x). If x∗ ∈ (0, 1), wxx(x∗) ≤ 0 and wx(x∗) =
0. By (8.4) we have
−C3(1 − C3/v2) + g(I(x∗)) − d ≥ 0,
which is impossible if we choose C3 = 2g(I0) and v > 2
p
g(I0). Hence, for such choices of
C3 and v, x∗ = 0 or x∗ = 1. We consider two cases:
Case 1. v > 0. For this case, since wx(0) > 0, x∗ 6= 0. Therefore, x∗ = 1; i.e., w(x) ≤ w(1)
for every x ∈ [0, 1]. Therefore,
P (x; v) ≤ P (1; v)e−v(1−C3/v2)(1−x),
which can be written as
P (x; v)
P (1; v)e−v(1−x) ≤ e
C3 v(1−x).
Case 2. v < 0. Since wx(1) < 0, x∗ 6= 1. Therefore, x∗ = 0; i.e., w(x) ≤ w(0) for every x ∈ [0, 1], which can be written as
P (x; v) P (0; v)evx ≤ e
−C3vx.
For the other side of inequalities, set η = 1 + C4/v2, where C4 > 0 is to be chosen later.
By (8.2), w satisfies (8.5)
(
wxx+ v(1 + 2C4/v2)wx+ w[C4(1 + C4/v2) + g(I(x)) − d] = 0 in 0 < x < 1,
wx = −(C4/v)w at x = 0, 1.
Let x∗ ∈ [0, 1] such that w(x∗) = min0≤x≤1w(x). If x∗ ∈ (0, 1), wxx(x∗) ≥ 0 and wx(x∗) =
0. By (8.5) we have
C4(1 + C4/v2) + g(I(x∗)) − d ≤ 0,
which implies that d > C4. Hence, if C4 = d, we must have x∗ = 0 or x∗ = 1. Next we
consider two cases:
Case 1. v > 0. Since wx(0) < 0, x∗ 6= 0. That is, x∗ = 1; i.e., w(x) ≥ w(1) for every x ∈ [0, 1]. Therefore,
P (x; v) ≥ P (1; v)e−v(1+C4/v2)(1−x),
which can be written as
P (x; v)
P (1; v)e−v(1−x) ≥ e
−C4v (1−x).
Case 2. v < 0. Since wx(1) > 0, x∗ 6= 1. That is, x∗ = 0; i.e., w(x) ≥ w(0) for every x ∈ [0, 1], which can be written as
P (x; v) P (0; v)evx ≥ e
C4 vx.
This completes the proof. ¤
Lemma 8.4. For any y ≥ 0, lim v→∞ v P (1; v) Z 1−y/v 0 P (s; v) ds = e−y
and lim v→−∞ v P (0; v) Z −y/v 0 P (s; v) ds = e−y− 1.
Proof. First of all, we establish the first limit. By part (a) of Lemma 8.3, P (s; v) P (1; v) ≤ e C2/ve−v(1−s). Hence, Z 1−y/v 0 P (s; v) P (1; v)ds ≤ e C2/v Z 1−y/v 0 e−v(1−s)ds = eC2/ve −y− e−v v ,
which can be written as
v P (1; v)
Z 1−y/v
0
P (s; v) ds ≤ eC2/v[e−y − e−v].
Similarly, by part (a) of Lemma 8.3,
P (s; v) P (1; v) ≥ e −C2/ve−v(1−s). Hence, v P (1; v) Z 1−y/v 0 P (s; v) ds ≥ e−C2/v[e−y− e−v].
This proves the first limit.
For the proof of the second limit, by part (b) of Lemma 8.3, for v ≤ −C1,
eC2/vevs ≤ P (s; v) P (0; v) ≤ e −C2/vevs. Hence, eC2/ve −y− 1 v ≤ Z −y/v 0 P (s; v) P (0; v)ds ≤ e −C2/ve −y− 1 v ,
which can be written as
eC2/v[1 − e−y] ≤ −v
P (0; v)
Z −y/v
0
P (s; v) ds ≤ e−C2/v[1 − e−y].
This completes the proof. ¤
Lemma 8.5. Suppose that d ∈ (0, g(I0e−k0)). Then,
lim
v→∞
P (1; v)
v = κ
∗, where κ∗ > 0 is uniquely determined by
Z 1
0
g(I0e−k0−k1κ ∗z
) dz = d.
Proof. Dividing (3.1) by P (1; v), integrating in (0, 1) and applying the boundary condition
in (3.1), we have Z
1 0
P (x; v)
P (1; v)[g(I(x)) − d] dx = 0.
Set x = 1 − y/v. We can rewrite the above equation as (8.6) Z v 0 P (1 − y/v; v) P (1; v) h g( ˜I(y)) − d i dy = 0,
where
˜
I(y) = I0e−k0(1−y/v)−k1 R1−y/v
0 P (s;v)ds.
We claim that P (1; v)/v is uniformly bounded for all v. To establish this assertion, we argue by contradiction: If not, passing to a sequence if necessary we may assume that
P (1; v)/v → ∞ as v → ∞. Then by Lemma 8.4, Z 1−y/v 0 P (s; v)ds = P (1; v) v · v P (1; v) Z 1−y/v 0 P (s; v) ds → ∞
pointwisely in y as v → ∞. Hence, ˜I(y) → 0 pointwisely as v → ∞. As e−C2/ve−y ≤ P (1 − y/v; v)
P (1; v) ≤ e
C2/ve−y
for every y ∈ (0, v), we see that
P (1 − y/v; v) P (1; v) → e −y pointwisely in y as v → ∞. Moreover, P (1 − y/v; v) P (1; v) ¯ ¯ ¯g( ˜I(y)) − d ¯ ¯ ¯ ≤ eC2/ve−y[g(I 0) + d]
for every y ∈ (0, v). Hence, we can apply the Lebesgue Dominant Convergent Theorem and let v → ∞ in (8.6) to conclude that
Z ∞
0
e−y(g(0) − d) = 0, which is a contradiction as g(0) = 0 and d > 0.
Hence, P (1; v)/v is bounded uniformly for large v. Passing to a sequence if necessary, we may assume that P (1; v)/v → κ as v → ∞ for some constant κ ≥ 0. For this case,
Z 1−y/v 0 P (s; v)ds = P (1; v) v · v P (1; v) Z 1−y/v 0 P (s; v) ds → κe−y. Hence, ˜ I(y) → I0e−k0−k1κe −y
pointwisely in y as v → ∞. Following the same argument as before, we can apply the Lebesgue Dominant Convergent Theorem and let v → ∞ in (8.6) to conclude that
(8.7) Z ∞ 0 e−y[g(I 0e−k0−k1κe −y ) − d] dy = 0.
We claim that κ > 0: if κ = 0, then from (8.7) we obtain g(I0e−k0) = d, which contradicts
our assumption d < g(I0e−k0). By the new variable z = e−y, (8.7) can be rewritten as
F (κ) = d, where
F (κ) :≡
Z 1
0
g(I0e−k0−k1κz) dz.
Since F (0) = g(I0e−k0) > d, limκ→∞F (κ) = 0, and F is strictly decreasing in (0, ∞) we see
that there exists a unique κ∗ such that F (κ∗) = d. Since κ∗ is independent of the choice of
Lemma 8.6. Suppose that d ∈ (0, g(I0)). Then,
lim
v→−∞
P (0; v) v = κ∗, where κ∗ < 0 is uniquely determined by
Z 1
0
g(I0ek1κ∗(1−z)) dz = d.
Proof. Dividing (3.1) by P (0; v), integrating in (0, 1) and applying the boundary condition
in (3.1), we have
Z 1
0
P (x; v)
P (0; v)[g(I(x)) − d] dx = 0.
Set x = −y/v. We can rewrite the above equation as (8.8) Z −v 0 P (−y/v; v) P (0; v) h g( ˆI(y)) − di dy = 0, where ˆ I(y) = I0ek0y/v−k1 R−y/v 0 P (s;v)ds.
We claim that P (0; v)/v is uniformly bounded for all large negative v. If not, we may assume that P (0; v)/v → ∞ as v → −∞. Then by part (b) of Lemma 8.4,
Z −y/v 0 P (s; v)ds = P (0; v) v · v P (0; v) Z −y/v 0 P (s; v) ds → ∞
pointwisely in y as v → −∞. Hence, ˜I(y) → 0 pointwisely as v → −∞. As eC2/ve−y ≤ P (−y/v; v)
P (0; v) ≤ e
−C2/ve−y
for every y ∈ (0, −v), we see that P (−y/v; v)/P (0; v) → e−y pointwisely in y as v → −∞.
Moreover, P (−y/v; v) P (0; v) ¯ ¯ ¯g( ˆI(y)) − d ¯ ¯ ¯ ≤ e−C2/ve−y[g(I 0) + d]
for every y ∈ (0, −v). By the Lebesgue Dominant Convergent Theorem and let v → −∞ in (8.8) we have that R0∞e−y(g(0) − d) = 0, which is a contradiction as g(0) = 0 and d > 0. Hence, P (0; v)/v is bounded uniformly for large negative v. Passing to a sequence if
necessary, we may assume that P (0; v)/v → κ∗ as v → −∞ for some constant κ∗ ≤ 0. For
this case, Z −y/v 0 P (s; v)ds = P (0; v) v · v P (0; v) Z −y/v 0 P (s; v) ds → κ∗[e−y− 1].
Hence, ˆI(y) → I0ek1κ∗[1−e −y]
pointwisely in y as v → −∞. Following the same argument as before, we can let v → −∞ in (8.8) to conclude that
(8.9) Z ∞ 0 e−y[g(I 0ek1κ∗[1−e −y] ) − d] dy = 0.
We claim that κ∗ < 0: if κ∗ = 0, from (8.9) we obtain g(I0) = d, which contradicts our
assumption d < g(I0). By the new variable z = e−y, (8.9) can be rewritten as G(κ∗) = d,
where
G(κ) :≡
Z 1
0
Since G(0) = g(I0) > d, limκ→−∞G(κ) = 0, and G is strictly increasing in (−∞, 0) we see
that there exists a unique κ∗ < 0 such that G(κ∗) = d. Since κ∗ is independent of the choice
of sequence, we see that P (0; v)/v → κ∗ as v → −∞. ¤
Lemma 8.7. There exist positive constants C5, C6, both independent of v, such that
(a) if v ≥ C5, ¯ ¯ ¯ ¯P (x; v)P (1; v) − e−v(1−x) ¯ ¯ ¯ ¯ ≤ Cv26 for every x ∈ [0, 1]. (b) if v ≤ −C5, ¯ ¯ ¯ ¯P (x; v)P (0; v) − evx ¯ ¯ ¯ ¯ ≤ Cv26 for every x ∈ [0, 1].
Proof. We first establish part (a). By part (a) of Lemma 8.3 we have g1(x; v) ≤
P (x; v) P (1; v) − e
−v(1−x)≤ g
2(x; v),
where gi(x; v) (i = 1, 2) are given by
g1(x; v) = (e−C2(1−x)/v− 1)e−v(1−x)
and
g2(x; v) = (eC2(1−x)/v− 1)e−v(1−x).
It is easy to check that
∂g1(x; v)
∂x = ve
−v(1−x)[e−C2(1−x)/v(1 + C
2/v2) − 1].
For large v, the only critical point (denoted by x1) of g1 in [0, 1] is determined by
eC2(1−x1)/v = 1 + C
2/v2,
which implies that x1 = 1 − (1/v)(1 + o(1)) for large v. Hence,
g1(x1; v) ≥ −
C2
v2e
−v(1−x1)≥ −C7
v2
for some positive constant C7 independent of v. As g1 attains the global minimum at x = x1
in [0, 1], we see that P (x; v) P (1; v) − e −v(1−x)≥ −C7 v2. For g2 we have ∂g2(x; v) ∂x = (v − C2/v)e −v(1−x) · eC2(1−x)/v− 1 1 − C2/v2 ¸ .
For large v, the only critical point (denoted by x2) of g2 in [0, 1] is determined by
eC2(1−x2)/v = 1
1 − C2/v2
,
which implies that x2 = 1 − (1/v)(1 + o(1)) for large v. Hence,
g2(x2; v) = C2/v 2
1 − C2/v2
e−v(1−x2) ≤ C8
