Introduction

Climate change has resulted in poleward and upslope range shifts in many species across the globe, and it becomes important whether or not species maintain in their current ranges (Parmesan and Yohe [36],

Parmesan [37], Walther et al. [43]). Species may respond to climate change by shifting their distribution or phenology, acclimating or adapting to changes; however the inability to sufficiently adapt will result in extinction (Aitken et al. [1], Cleland et al. [11], Valladares et al. [42]). Species potential to successfully adapt or shift distributions in response to climate change depends on a host of factors, such as the speed and variability of changing conditions, species' dispersal abilities, characteristics of a species climatic niche and species interactions. Several reaction-diffusion models have been developed to explore species persistence using shifting boundary conditions or shifting growth functions (Berestycki et al. [7], Li et al. [24], MacDonald and Lutscher [31], Potapov and Lewis [39]).

The early work by Potapov and Lewis [39] conceptualized the shifting suitable habitat of a species.

The single-species version of their model takes the form

with

Here u(x, t) is the density of a population at location x and time t, f (u, x -ct) describes the population growth at point x at time t, c is the speed at which the habitat shifts, L and r are positive constants, and g satisfies g(0) = g(1) = 0 and g(u) > 0 on (0, 1). g(u) exhibits monostability, and a prototype example is g(u) = u(1 -u). In this model the population grows in the interval [0, L] and declines outside this interval. Berestycki et al. [7] provided the critical length of L and showed that for L above the critical length equation (1.1) has a globally attracting nontrivial forced traveling wave with speed c. The authors extended the results to a general class of equations by studying the eigenvalue problem of a linearized system. MacDonald and Lutscher [31] extended the results in [7] by including individual movement behavior at habitat edges. Piecewise growth functions similar to (1.2) have been also used to study promotion zones and barrier zones for species persistence and spread in heterogeneous environments (Du et al. [15] and Li et al. [27]).

The equation (1.1) has also been investigated for f (u, x) in other forms different from (1.2). Li et al. [24] and Hu et al. [13] considered spreading speeds for (1.1) where f (u, x -ct) = u(s(x -ct) -u) and s(x) is a nondecreasing function for -∞ < x < ∞. Bouhours and Giletti [9] studied the spreading and vanishing dynamics for a general two-dimensional reaction-diffusion equation which includes f (u, x -ct) = u(s(x -ct) -u) as a special case. Berestycki and Fang [8] established the existence and stability of traveling waves for a one-dimensional reaction-diffusion equation with a general nonlinear growth function f (x -ct, u). The results in [8] allow both s(∞) and s(-∞) to have same sign in the case of f (x-ct, u) = u(s(x-ct)-u). For more results regarding traveling waves for reaction-diffusion equations with a shifting habitat, the reader is referred to Berestycki et al. [7], Berestycki and Rossi [5,6],

Hamel [19], Hamel and Roques [20], and Fang et al. [16]. Mathematical models have been developed

in other forms that are used to describe species development in shifting habitats; see, for example, Zhou and Kot [45], Li et al. [25,27], and Li et al. [29], where integro-difference equations and integrodifferential equations are involved.

The aforementioned papers assume no Allee effect in species growth. An Allee effect arises when the per-capita birth rate increases at lower population densities, and a strong Allee effect is an Allee effect with a critical population density [2,35]. There are cases where Allee effects occur when species distributions shift in response to climate change (Livshultz et al. [30], Samuel and Chandler [40], Shanks et al. [41], Wood et al. [44]). It is of great interest to explore the population dynamics of species with a strong Allee effect in a shifting habitat. In this paper we study (1.1) and (1.2) where g has a strong

Allee effect, i.e., bistability. The reaction-diffusion equation

with g(u) exhibiting bistability has been well studied (see Fife [17] and references cited therein). It is well-known that there exists a real number c * which is the unique speed of traveling waves connecting zero to the carrying capacity, and the sign of c * is the same as that of the integral of g(u) from zero to the carrying capacity. c * can be calculated using variation techniques (Benguria and Depassier [4])

when it is positive.

In this paper, we study whether or not a species governed by (1.1) with (1.2) and a strong Allee effect can keep pace with a shifting habitat. We find that the wave speed c * and the habit shift speed c both play important roles in determining species persistence. We particularly establish the existence forced positive traveling waves. A positive traveling wave is a nonnegative traveling wave which is not uniformly zero valued. We show that if c * > c > 0 there exists a positive number L * (c) such that for L > L * (c) there are two positive traveling waves and for L < L * (c) there is no positive traveling wave, and if c > c * for any L > 0 there is no positive traveling wave. We provide numerical simulations to further examine the behavior of the system. Our numerical results demonstrate that the larger traveling wave attracts solutions with proper initial data so that persistence takes place, and in case of no traveling wave, solutions approach zero and extinction occurs.

This paper is organized as follows. The analytical results regarding the existence of traveling waves are presented in Section 2. The numerical simulations are given in Section 3. Some concluding remarks are provided in Section 4.

Main results

We begin with the following hypotheses to address the presence of a strong Allee effect in g:

< 0, and there is a number a such that 0 < a < 1, g(a) = 0, g(u) < 0 on (0, a), and g(u) > 0 on (a, 1).

ii. g (0) > -r.

Hypotheses 2.1 (i) indicates that g(u) exhibits a strong Allee effect with a the Allee threshold and 1 the (scaled) carrying capacity. A protype example is g(u) = u(u -a)(1 -u) with 0 < a < 1.

Hypotheses 2.1 (ii) assumes that the population decay rate at low densities in the moving patch is less than that outside the patch. Biologically this means the environment outside the growth zone is harsher than that inside the patch. We shall consider c > 0 as the case of c < 0 can be treated in a similar way.

The function f (u, x) is discontinuous at x = 0, L. Following Berestycki et al. [7], we seek traveling wave solutions for (1.1), which are globally of class C 1 (indeed, to guarantee that diffusion conserves mass, the flux u x should be continuous [7]) and piecewise of class C 2 and satisfy the equation at each point with x = 0, L. It is known that the problem of (1.1) with appropriate initial data u(x, 0) has a unique, globally defined, solution u = u(x, t), which, as a function of x has such smoothness (Berestycki et al. [7] and Du et al. [15]). Berestycki et al. [7] determined the critical patch size and established the existence of a positive traveling wave for g(u) = ru(1 -u k ) with r > 0 and k > 0 by glueing phase portraits. There is long history of using phase portrait analysis to study traveling waves; see for example Fife [17] and references cited therein. For some recent work on phase portrait analysis for systems with a strong Allee effect and stationary habitat, the reader may refer to Pouchol et al. [38] and Li et al. [27,28].

We have the following proposition (see Fife [17]). For g(u) = u(u -a)(1 -u), the unique traveling wave speed is c * = 1 -2a √ 2 (see Hadeler and Rothe [18] and Nagumo et al. [34]). In general for 1 0 g(u)du > 0 (i.e., c * > 0), c * can be obtained using the following variational formula given by Benguria and Depassier [4]:

with z = x -ct. This is equivalent to the planar system

Phase plane analysis for the existence of a traveling wave for (2.2) can be found in [17]. We further analyze (2.2) in order to study traveling waves for (1.1) with (1.2). System (2.2) has three equilibria (0, 0), (a, 0) and (1, 0). For c ≥ 0, both (0, 0) and (1, 0) are saddles. For convenience, we use S c 0 and U c 0 to denote the stable and unstable manifolds of (0, 0) corresponding to c for 0 < w < 1, respectively, and use S c 1 and U c 1 to denote the stable and unstable manifolds of (1, 0) corresponding to c for 0 < w < 1, respectively.

The Jacobian matrix of (2.2) is

.

At (0, 0) the eigenvalues are

. The corresponding eigenvectors are given by 1 λ ± 0 .

At (1, 0) the eigenvalues are

, and the corresponding eigenvectors are given by 1 λ ±

1

. Clearly, λ + 0 > 0, λ + 1 > 0, λ - 0 < 0, and λ - 1 < 0. All the eigenvalues λ ± 0 and λ ± 1 decrease in c.

Note that here a stable or unstable manifold of an equilibrium is tangent to the line passing through the equilibrium with the slope determined by the corresponding eigenvector. These lead to the following lemma.

Lemma 2.1. Assume that Hypotheses 2.1 hold and c ≥ 0. We have the following statements for (2.2):

i. Near (0, 0), S c 0 lies below the w-axis and U c 0 lies above the w-axis. Furthermore for c 2 > c 1 ≥ 0,

ii. Near (1, 0), S c 1 lies above the w-axis and U c 1 lies below the w-axis. Furthermore for

Lemma 2.2. Assume that Hypotheses 2.1 hold. We have the following statements for (2.2):

Proof. The proof of the statement (i) is similar to that of Lemma 4.14 in Fife [17] (also see Kanel' [21]).

For the sake of completeness, we provide the proof here. When v = 0, v can be viewed as a function of w, and

we have

Since dD(w) dw > 0 and v 1 (w 0 ) -v 2 (w 0 ) > 0, the function D(w) increases for w > w 0 and thus is positive whenever w > w 0 , v 1 < 0 and v 2 < 0. It follows that v 1 (w) > v 2 (w) whenever w > w 0 , v 1 (w) < 0 and v 2 (w) < 0. This proves the first part of statement (i). The proof of the second part of statement (i) is similar and omitted.

We

we choose w 0 to be a positive number close to but less than 1 so that v 1 (w) < v 2 (w) for w 0 ≤ w < 1.

(2.4) still holds with these v 1 and v 2 . Since D(w) increases for w < w 0 , v 1 (w) < v 2 (w) whenever w 0 > w ≥ 0, v 1 (w) > 0, and v 2 (w) > 0. This proves the first part of statement (ii). The proof of the second part of statement (ii) is similar and omitted.

be the two solutions of (2.3) corresponding to c i , respectively. Let w 0 be a number such that 1 > w 0 > 0.

The proof of this lemma is similar to that of Lemma 2.2 and is omitted.

When c = 0, the system (2.2) becomes

This system is integrable, and 1 2 v 2 = -g(w)dw.

Assume 1 0 g(u)du > 0. S 0 0 coincides with U 0 0 between 0 < w < 1 forming a homoclinic orbit given by S 0 0 and U 0 0 :

S 0 1 and U 0 1 between 0 < w < 1 are given by For c > 0, (2.2) is not integrable. In the w -v plane, the isocline for dw dz = 0 is v = 0, and the isocline for dv dz = 0 is v = -g(w) c . See Fig. 2 for a graphical demonstration of the direction field. (a) (b) Figure 2: (a) The graph of y = g(u). (b) Direction field: the isocline for dw dz = 0 is v = 0, and the isocline for dv dz = 0 is v = -g(w) c (dashed curve). Lemma 2.4. Assume that Hypotheses 2.1 hold and c * > c > 0. We have the following statements: i. S c 1 lies above S 0 1 , and below a line v = -m(w -1) for some m > 0.

ii. U c 1 lies above U 0 1 and below T * , and U c 1 and S c 0 do not intersect.

iii. S c 0 lies outside the loop S 0 0 and U 0 0 , below S c 1 , and above U c 1 .

iv. U c 0 lies inside the loop S 0 0 and U 0 0 and approaches (a, 0).

Proof. The first part of statement (i) follows from Lemma 2.2 (ii). S c 1 is tangent to the line passing through (1, 0) with the slope

< 0. For a small δ > 0, there exists a point (w 1 , v 1 ) on S c 1 such that for w 1 < w < 1, S c 1 is below the line passing through (1, 0) with the slope λ - 0 -δ.

Let (w 1 , ṽ1 ) be the point on S 0 1 . Since S c 1 lies above S 0 1 , we may choose w 1 sufficiently close to 1 such that S c 1 is above the line v = ṽ1 whenever w 1 > w > 0. Consequently along S c 1 , g(w) v ≤ 1 ṽ1 whenever w 1 > w > 0. In view of (2.3), we find that along S c

whenever 1 > w > 0. This leads to the second part of statement (i).

The first part of statement (ii) follows from Lemma 2.2 (ii). Since c * > c > 0, there is no traveling wave and thus U c 1 and S c 0 do not intercept. This proves the statement (ii).

According to Lemma 2.2 (i), for v < 0, S c 0 lies below S 0 0 . On the other hand, since c * > c > 0, S c 0 does not intercept U c 1 to form a heteroclinic orbit, and thus S c 0 lies above U c 1 . Consequently S c 0 intercepts the w-axis at a number w 1 between B and 1. Since the flow crosses the w-axis from above and since w = v > 0 whenever v > 0, S c 0 lies on the left-hand side of line w = w 1 whenever v > 0. If

U 0 0 is above S c 0 whenever w is between w 2 and w 1 . This leads to that U 0 0 is above the w-axis whenever w is between w 2 and w 1 , so that U 0 0 cannot intercept the w-axis at B. This contradiction shows that S c 0 does not intersect with U 0 0 whenever w > 0 and v > 0. We therefore conclude that S c 0 is above U 0 0 whenever v > 0, and thus S c 0 stays outside the loop determined by U 0 0 . Finally the uniqueness of solutions implies S c 0 does not intercept with S c 1 nor U c 1 , so that S c 0 lies below S c 1 , and S c 0 lies above U c 1 .

This completes the proof of the statement (iii).

By Lemma 2.2 (i), U c 0 stays below S 0 0 and U 0 0 whenever v > 0. If U c 0 intercepts U 0 0 and S 0 0 below w-axis, we use (w 0 , v 0 ) to denote the first point at which U c 0 intercepts U 0 0 and S 0 0 from above such that v 0 < 0 and U c 0 lies above U 0 0 and S 0 0 whenever v < 0 and w > w 0 . On the other hand, by Lemma 2.3 (ii), U c 0 lies below S 0 0 whenever w > w 0 and v < 0. This leads to a contradiction. It follows that U c 0 stays inside the loop S 0 0 and U 0 0 for v > 0 and v ≤ 0. Since g (a) ≥ 0, (a, 0) is stable for c > 0. Furthermore for system (2.2),

By Dulac's criterion, there is no limit cycle. It follows from the Poincaré-Bendixson Theorem that U c 0 approaches (a, 0). The proof is complete.

According to this lemma, S c 1 lies above S c 0 , and below a line v = -m(w -1) with m > 0. In view of the Poincaré-Bendixson Theorem, S c 1 must intersect the v-axis at a number v 1 > 0. Since S c 0 lies outside the loop S 0 0 and U 0 0 , below S c 1 , and above U c 1 , S c 0 must intersect the v-axis at a number v 0 > 0.

See a graphical demonstration of the statements in Lemma 2.4 in Fig. 3 (a). Consider a trajectory T c of (2.2) for c * > c > 0 that starts at a number between v 0 and v 1 on the v-axis; see Fig. 3 (b) for a graphical description.

To study traveling waves for (1.1) and (1.2), we glue the phase portraits of (2.2) inside the patch to those outside the patch. Using phase plane analysis, we determine a critical patch size for the existence of traveling waves. Using z = x -ct and the substitution u(x, t) = ū(z, t), (1.1) becomes ūt = ūzz + cū z + f (ū, z).

(2.6)

A traveling wave of (1.1) is a steady solution of (2.6) satisfying with

(

The corresponding planar system is

and

with w(0

Throughout this paper we consider bounded traveling waves with values between 0 and 1. Bounded solutions with w > 0 for linear system (2.10) are given by

where

, and k 1 and k 2 are positive constants. Here w(-∞) = w(∞) = 0. It follows that v = m + w for z < 0 and v = m -w for z > L.

In the w -v plane, a bounded positive traveling wave described by (2.9) and (2.10) to a path involving v = m + w, T c (a trajectory of (2.2) or (2.9)), and v = m -w. We use (w c 0 , v c 0 ) and (w c

to denote the interception points of S c 0 with line v = m + w, and S c 1 with line v = m + w, respectively.

We use (p c 1 , q c 1 ) and (p c 2 , q c 2 ) to denote the interception points of T c and line v = m + w, and T c and line v = m -w, respectively. Note that w c 0 < p c 1 < w c 1 . See Fig. 4 (a) for a graphical description. The graph of the corresponding traveling wave is provided in Fig. 4 (b).

We now discuss how L is related to a traveling wave. In the first quadrant, by (2.9), w = v > 0, so that w(x) increases in x and thus v can be viewed as a function of w. We use v = v + (p c 1 ; w) to describe the upper part of T c above the w-axis. Similarly v can be viewed as a function of w on T c in the fourth quadrant. We use v = v -(p c 1 ; w) to denote the lower part of T c below the w-axis. We use w c * to denote the intersection of T c and the w-axis. Observe w c * > B > a so that g(w c * ) > 0.

The first equation of (2.9) shows that L is given by

(2.12)

It should be noted that this is an improper integral due to v + (p c 1 , w) and v -(p c 1 , w) being zero at w = w c * .

Lemma 2.5. H(p c 1 ) is continuous in p c 1 for p c 1 ∈ (w c 0 , w c 1 ).

Proof. Since g(w) ∈ C 1 [0, 1], the solution (w(z), v(z)) along T c continuously depends on the initial values w(0) = p c 1 and v(0) = m + p c 1 and thus on p c 1 (see Theorem 1.3.1 in [14]). Consequently, p c 2 = w(L)

w) > 0, the first equation of (2.9)

shows dw dz > 0, so that w(z) is strictly increasing and thus x is a continuous function of w. We conclude

In view of (2.9), near (w c * , 0), v -g(w c * ) < 0, so that w is a function of v on T c . Furthermore,

It follows that for any small > 0, there exists γ > 0 such that for w

This particularly is true when v is replaced by v ± (p c 1 ; w). Therefore H(p c 1 ) given by (2.12) is well-defined.

Consider a trajectory T c of (2.9) with pc 1 as the w-coordinate of the intersection point with v = m + w, pc 2 as the w-coordinate of the intersection point with v = m -w, and wc * as the w-intercept. Assume

shows that T c is above T c for v > 0, and consequently w c * > wc * .

For any small ε > 0, continuity and (2.13) imply that there exists δ 1 > 0 such that for p c 1 -pc

.14) Use ṽ+ (p c 1 ; w) (ṽ -(p c 1 ; w)) to denote the part of Tc above (below) the w-axis. Choose w 0 with p c 1 < w 0 < wc * and w 0 sufficiently close to wc * , such that wc * w 0 1 ṽ+ (p c 1 ; w) dw < ε 3 , (2.15) and ṽ+ (p c 1 ; w) ≥ ṽ+ (p c 1 ; w 0 ) for pc 1 ≤ w ≤ w 0 . For δ 2 = ṽ+ (p c 1

Combining (2.14)-(2.17), we obtain that for p c 1 -pc 1 < δ with δ = min 1≤i≤3 {δ i },

Similarly, for the given ε, there exists δ > 0 such that for p c 1 -pc 1 < δ,

We have shown that H(p c 1 ) is a continuous function of p c 1 . The proof is complete.

Lemma 2.6. Assume that Hypotheses 2.1 hold and c * > c > 0. Then H(p c 1 ) given by (2.12) satisfies

Proof. The tangent line to

< 0. For any small number δ > 0, there exists a number w 1 < 1 and w 1 close to 1 such that S c 1 lies below v = (λ - 1 -δ)(w-1) for w 1 ≤ w < 1. Consider a trajectory T c of (2.9) close to S c 1 near (1, 0) with w c * , the w-intercept satisfying w 1 < w c * < 1. Since T c lies below S c 1 , T c has a point with w 1 as the w-coordinate. Since T c lies below v = (λ - 1 -δ)(w -1) for w 1 ≤ w < 1, (2.12) shows

dw.

This shows H(p c 1 )→∞ as w c * approaches 1 from the left. Due to continuous dependence of a solution on its initial value, w c * →1 from the left if p c 1 →w c 1 from the left. We therefore have H(p c 1 )→∞ as p c 1 →w c 1 -.

The tangent line to S c 0 at (0, 0

< 0. For any small positive number δ with δ < |λ - 0 |, there exists a point (w 0 , v 0 ) on S c 0 close to (0, 0) with 0 < w 0 < a and v 0 < 0 such that S c 0 lies below the line v = (λ - 0 + δ)w for 0 < w ≤ w 0 , and g(w) > (g (0) -δ)w for 0 < w ≤ w 0 . Consider a trajectory T c of (2.9) close to S c 0 near (0, 0) with p c 2 < w 0 and q c 2 > v 0 . Since T c is below S c 0 for v < 0, T c has a point (w 0 , v 1 ) with v < 0, v 1 < v 0 . In view of (2.3), along T c for p c 2 ≤ w ≤ w 0 ,

We therefore use (2.12), a variable change, and (2.3) to obtain

This shows that H(p c 1 )→∞ as q c 2 →0 + . Due to continuous dependence of a solution on its initial value,

where H(p c 1 ) is given by (2.12). Lemma 2.5 and Lemma 2.6 show that L * (c) is well-defined. Clearly, L * (c) ≥ 0. Ecologically speaking, L * (c) is the critical patch size for a population to persist. We will further explore L * (c) in the rest of this section.

We shall show that L * (c) is positive. To this end, we introduce an integral operator. Integral operators have been proven useful in studying traveling waves for reaction-diffusion systems [23]. Consider Q c defined by

where

The next lemma shows that w(x -ct) is a bounded traveling wave of (

1.1) if and only if w is a fixed point of Q c . Lemma 2.7. Assume that Hypotheses 2.1 hold. Then w satisfies (2.7) and (2.8) with w(

If w satisfies (2.7) and (2.8) with w(-∞) = w(∞) = 0, a slightly revised version of the first part of the proof Theorem 3.2 in [23] shows that w(z) is a fixed point of Q c . If w(z) is a fixed point of Q c with w(-∞) = w(∞) = 0, a proof similar to that of Lemma 3.1 in [27] and the second part of Theorem 3.2 in [23] show that w satisfies (2.7) and (2.8). We omit the details here.

Lemma 2.8. Assume that Hypotheses 2.1 hold and c * > c ≥ 0. Then there exists a positive number L 0 (c) such that for L < L 0 (c), there is no positive traveling wave connecting 0 and 0 for (1.1).

Proof. We consider

Let wn = sup -∞<z<∞ w n (z). If follows from (2.19) that

For z ≤ 0,

e -λ 1 z z -∞ e λ 1 y ρ-r ρ wn dy + e λ 2 z 0 z e -λ 2 y ρ-r ρ wn dy + e λ 2 z L 0 e -λ 2 y g( wn)+ρ wn ρ dy

wn .

Here we have used the simple fact g( wn)+ρ wn ρ ≤ 2 wn . For z ≤ 0, e λ 2 z ≤ 1, 0 < 1 -e λ 2 z + e λ 2 (z-L) < 1.

We therefore have for z ≤ 0,

e -λ 1 z 0 -∞ e λ 1 y ρ-r ρ wn dy + e -λ 1 z z 0 e λ 1 y g( wn)+ρ wn ρ dy + e λ 2 z L z e -λ 2 y g( wn)+ρ wn ρ dy

(2.21)

e -λ 1 z 0 -∞ e λ 1 y ρ-r ρ wn dy + e -λ 1 z L 0 e λ 1 y g( wn)+ρ wn ρ dy + e -λ 1 z z L e λ 1 y ρ-r ρ wn dy + e λ 2 z ∞ z e -λ 2 y ρ-r ρ wn dy 22) show that there exist positive numbers σ < 1 and L 0 (c) such that for L ≤ L 0 (c) and any z, w n+1 (z) ≤ σ wn leading to wn+1 ≤ σ wn . Therefore for L ≤ L 0 (c) , wn → 0 as n → ∞. We conclude that for L ≤ L 0 (c), the solution w n (z) of (2.19) approaches 0 as n → ∞. This and Lemma 2.7 imply that for L ≤ L 0 (c), system (1.1) has no positive traveling wave. The proof is complete. Proof. Assume that w(x -ct) is a traveling wave connecting 0 and 0 for (1.1) and (1.2) in the case of 1 0 g(u)du > 0 and c > c * > 0. Since 0 ≤ w(x) ≤ 1 and 0 ≤ f (w(x),x)+ρw(x) ρ ≤ 1 for all x, and since T 1 T 2 L=8 -5 0 5 10 0.0 0.2 0.4 0.6 0.8 x-ct w (a) T min L * =6.79 -5 0 5 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x-ct w (b) In Fig. 9 we numerically examine the stability of the T 1 equilibrium. In (a) we initiate with u(x, 0) = 1.05T 1 (x). We see that the solution asymptotically approaches the T 2 equilibrium. In (b)

we initiate with u(x, 0) = 0.95T 1 (x). We see that the solution asymptotically approaches zero. This strongly suggests that T 1 acts as a separatrix, where solutions initiated below it go extinct, and solutions initiated above it grow to the T 2 equilibrium. Similar results were seen with other parameter choices.

In Fig. 10 we show an example of extinction occurring when L < L * and when c > c * . In Fig.

10-(a) the habitat size used is L = 6 which is less than L * which is 6.79 for the parameters used (c = 0.5c * , a = 0.2, r = 1). The initial condition used is u(x, 0) = 1. We see by t = 38 the maximum density has fallen below the Allee threshold of 0.2. In Fig. 10-(b), c = 1.2c * so the habitat shift speed is greater then species spread speed and by the theory developed we would expect even for large values of L extinction will eventually occur. We use the parameters a = 0.2, r = 1 and L = 10. We see that by t = 55 the maximum density has already fallen below the Allee threshold. we see that the solution converges to the T 2 equilibrium from above. The initial condition used was u(x, 0) = 1. In (b) we see the solution converges to the T 2 equilibrium from below. The initial condition used was u(x, 0) = 0.6e -0.1(x-4) 2 . In (b) we see that a solutions initiated 5% below T 1 asymptotically approach extinction.

steady states through a fold bifurcation with the bigger steady state stable and smaller one unstable; see Figure 17.8 in Kot [22]. We conjecture that for c * > c, L * (c) is the fold bifurcation value, there is exactly one positive traveling wave if L = L * (c), and there are exactly two positive traveling waves if L > L * (c) with the bigger (smaller) traveling wave stable (unstable) in (1.1) with (1.2) under Hypotheses 2.1. This conjecture is supported by our numerical simulations which show solutions initiated near but below the lower equilibrium go extinct while those initiated slightly above the lower equilibrium converge to the upper equilibrium (see Fig. 8 and Fig. 9).

It was shown in Berestycki et al. [7] that for (1.1) with no Allee effect, the unique travelig wave (traveling pulse) is globally attracting. As shown in the present paper, the presence of a strong Allee effect leads to the multiplicity of traveling waves (Theorem 2.1(iii)). Our simulations show that the critical patch size L * (c) increases as c increases (see Fig. 7). This implies that a species persisting in a stationary habitat may eventually die out when the habitat shifts.

The methodology developed in this paper might work for reaction-diffusion models in a shifting patch with other growth functions and boundary conditions. Inside the patch the growth function g(u) could exhibit a weak Allee effect, with an example given by g(u) = u 2 (1 -u). MacDonald and Lutscher [31] considered a model in a form similar to (1.1) and (1.2) with no Allee effect, different matching boundary conditions and more general movement behavior. The matching conditions in [31] are determined by the probability with which an individual at a boundary point decides to move into or out of the suitable habitat. One may consider a model with a strong Allee effect and the matching conditions given in [31]. The phase plane analysis presented in this paper might be extended to study the existence of positive traveling waves for these models. This paper considered the case of a bounded shifting habitat. It would be of interest to investigate persistence and spread in a reaction-diffusion model with an unbounded shifting habitat and strong Allee effect. The most cited and obvious cause of the Allee effect is the difficulty of finding mates at low population sizes in sexually reproducing species (Boukal and Berec [10], Courchamp et al. [12]). There are works on traveling wave solutions, spreading phenomena, and critical patch sizes for two-sex populations (Ashih and Wilson [3], Maciel et al [32],

Miller et al. [33]). It is worth of studying two-sex species models with shifting habitats by extending the framework developed in this paper.