A Appendix

Extinction and spreading of a species under the joint influence of climate change and a weak Allee effect: a two-patch model

Abstract

Many species see their range shifted poleward in response to global warming and need to keep pace in order to survive. To understand the effect of climate change on species ranges and its consequences on population dynamics, we consider a space-time heterogeneous reaction-diffusion equation in dimension 1, whose unknown  stands for a population density. More precisely, the environment consists of two patches moving with a constant climate shift speed : in the invading patch the growth rate is negative and, in the receding patch it is of the classical monostable type. Our framework includes species subject to a weak Allee effect, meaning that there may be a positive correlation between population size and its per capita growth rate. We study the large-time behaviour of solutions in the moving frame and show that whether the population spreads or goes extinct depends not only on the speed but also, in some intermediate speed range, on the initial datum. This is in sharp contrast with the so-called ‘hair-trigger effect’ in the homogeneous monostable equation, and suggests that the size of the population becomes a decisive factor under the joint influence of climate change and a weak Allee effect. Furthermore, our analysis exhibit sharp thresholds between spreading and extinction: in particular, we prove the existence of a threshold shifting speed which depends on the initial population, such that spreading occurs at lower speeds and extinction occurs at faster speeds.

2010 Mathematics Subject Classification: 35B40, 35C07, 35K15, 35K57, 92D25
Keywords:
Climate change, reaction-diffusion equations, travelling waves, long time behaviour, sharp threshold phenomena.

1 Introduction

In this paper we are interested in the following problem

(1.1)

where and will be assumed to be a bounded, nonnegative and compactly supported function, and

(1.2)

Here is monostable in the sense that

(1.3)

This problem is motivated by the study of the effect of climate change on population persistence. In [16, 26], the different authors point out that global warming induces a shift of the climate envelope of some species toward higher altitude or latitude. Therefore, these species need to keep pace with their moving favourable habitat in order to survive. Here we model the evolution of the density  of a population using a reaction-diffusion equation where the reaction term, which accounts for the growth of the species, is heterogeneous with respect to both space and time. That is, we assume that the dispersion of the population is a diffusion process, and in (1.1) the growth term  depends on the shifting variable where the nonnegative parameter  (which we assume to be constant) stands for the speed of the shift of the climate envelope. More precisely, while the growth term is monostable in the favourable environment {, it is negative in the unfavourable zone . We highlight here that by monostable, we only mean as stated in (1.3) that the growth rate of the population is positive when (after renormalization, 1 stands for the carrying capacity in the climate envelope). However, we do not assume that the per capita growth rate is maximal at , that is the population may be subjected to a weak Allee effect, which is a common feature of ecological species arising for instance from cooperative behaviours such as defense against predation [1]. Let us also briefly note that our analysis easily carry out if the growth term in (1.2) is chosen to be any linear function with in the unfavourable patch : indeed our main results would still hold as they stand below, and we choose here just for the sake of simplicity.

Such shifting range models were introduced in several papers to study similar ecological problems. In [22] and [4], the effect of climate change and shifting habitat on invasiveness properties is studied for a system of reaction-diffusion equations. In [5, 6, 7, 15, 25], the authors deal with persistence properties under a shifting climate for a scalar reaction-diffusion equation in dimension 1 and higher, and exhibit a critical threshold for the shifting speed below which the species survives and above which it goes extinct. This problem has also been studied in the framework of integrodifference equations, where time is assumed to be discrete and dispersion is nonlocal [13, 27, 28]. However, all the aforementioned papers hold under KPP type assumptions whose ecological meaning is that there is no Allee effect and which mathematically imply that the behaviour of solutions is dictated by a linearized problem around the invaded unstable state. In the context of (1.1), the KPP assumption typically writes as for all and .

On the other hand a few papers investigate the case when no KPP assumption is made. In [24], the authors analyse numerically the effect of climate change and the geometry of the habitat in dimension 2, again in the framework of reaction-diffusion equations, with or without Allee effect. In [8], Bouhours and Nadin consider (1.1) when the size of the favourable zone is bounded under rather general assumptions on the reaction  in the favourable zone (including the classical monostable and bistable cases). More precisely, they have shown the existence of two speeds such that the population persists for large enough initial data when and goes extinct when . However, for a fixed initial datum, it is not known in general whether there exists a threshold speed delimiting persistence and extinction. We will prove here that, when satisfies the hypotheses (1.2) and (1.3), the answer is positive but also that, unlike in the KPP case [5, 6, 7, 15, 25], the threshold speed depends non trivially on the initial datum.

Before stating our main results, let us start by giving some basic properties of problem (1.1). Using sub and super solution method and comparison principle we know that, for bounded and nonnegative initial conditions, problem (1.1) has a unique global and bounded solution. Because of the discontinuity of at , by solution we will always mean a function which is with respect to  for all , and which satisfies the equation in a classical sense on both and . The initial condition is understood as follows:

In this paper we are interested in the asymptotic behaviour of the solution of (1.1) as time goes to infinity, and in particular whether the solution goes extinct or spreads, in the sense of Definition 1.1 below. To do so we will study the previous problem in the moving frame, i.e. letting . If we define to be the solution in the moving frame, then is solution of the parabolic equation

(1.4)

such that , for all .

Notation

In the following we will denote by the space variable in the moving frame and the variable in the non moving frame. With some slight abuse of notations, we will write when we consider the solution in the non moving frame and (instead of ) for the solution in the moving frame.

Let us also introduce the classical homogeneous monostable equation

(1.5)

and denote by the minimal speed for existence of travelling wave solutions of (1.5), namely particular solutions of the type where . It is well-known that and that, under the KPP hypothesis for all , then . We refer to the seminal paper of Aronson and Weinberger [3] for details.

1.1 Main results

As we are interested in the long time behaviour of the solution of problem (1.4), a first step is to classify the stationary solutions of (1.4). As we will see more precisely in Theorem 2.1 in Section 2, there are three types of nonnegative stationary solutions of (1.4):

  • the trivial solution 0;

  • ‘ground states’, namely positive stationary solutions of (1.4) such that , among which there is a ‘critical ground state’ which is characterized by the fact that it has the largest value at as well as the fastest decay as ;

  • a unique ‘invasion state’, namely a positive stationary solution of (1.4) such that and .

Let us highlight here that these are stationary solutions in the moving frame with the same speed  as the shifting favourable zone, which may thus be seen as travelling waves in the original non moving frame. In particular, while we refer to as the ‘invasion state’, the invasion is of course restricted to the favourable zone. In a similar fashion, by ‘ground state’ we mean that the population migrates but does not spread away from the point .

While the invasion state always exists, there may exist either none or an infinity of ground states depending on the value of . In the latter case, ground states all lie below the invasion state, and each of them will be characterized by its value at . We will denote by the ground state such that . The supremum of the admissible such that exists will be denoted by and, as mentioned above, we call the critical ground state. Proposition 2.3 describes the asymptotics of the ground states as depending on , which will justify our statement that the critical ground state has the fastest decay at infinity. The discussion above will be made more rigorous in Section 2.

From this, we can be more precise about what we mean by extinction or spreading of the solution of (1.4):

Definition 1.1.

Let be a nonnegative, bounded and compactly supported initial datum, and be the solution of (1.4) with initial condition . We say that

  • extinction occurs if converges uniformly with respect to to 0 as time  goes to infinity;

  • grounding occurs if converges uniformly with respect to to the critical ground state as time  goes to infinity;

  • spreading occurs if converges locally uniformly to the invasion state as time goes to infinity.

Note that the fact that the convergence is or is not uniform is a consequence of the choice of compactly supported initial data (clearly the solution may never converge uniformly to the invasion state). Furthermore, we narrow grounding to the large-time convergence to the critical ground state (we again refer to Theorem 2.1 for a more precise definition). The reason is that our arguments will largely rely on the fact, which is contained in our main statements below and which will be proved in Section 4, that the solution may never converge to a non critical ground state when the initial condition is compactly supported.

Our first theorem investigates the large-time behaviour of the solution of (1.4) in various speed ranges:

Theorem 1.2.

Let be the solution of (1.4) with a nonnegative, non trivial, bounded and compactly supported initial datum .

  1. If , then spreading occurs.

  2. If (provided such exists), then both spreading and extintion may occur depending on the choice of . To be more precise, there exist initial data such that spreading occurs for , and extinction occurs for .

  3. If , then extinction occurs.

Recall from [3] that the minimal wave speed of (1.5) is also the spreading speed of solutions of the Cauchy problem with compactly supported initial data. Therefore, the third statement of Theorem 1.2 simply means that, when the climate shifts faster than the species spreads in a favourable environment, then the species cannot keep pace with its climate envelope and goes extinct as time goes to infinity. On the other hand, when is less than (statement of Theorem 1.2) which is the speed associated with the linearized problem around , then any small population is able to follow its habitat and thrive. In particular, when , we retrieve a threshold speed between persistence and extinction, which as in the KPP framework of [5, 6, 7, 15, 25] does not depend on the initial datum (let us note here that, while the KPP assumption implies that , the converse does not hold [12]).

Nonetheless, a striking feature of Theorem 1.2 is the fact that when , whether the solution persists or not depends on the initial datum, see statement above. This is in sharp contrast with the so-called ‘hair-trigger effect’ for the classical homogeneous monostable equation (1.5), whose solution spreads as soon as the initial datum is non trivial and nonnegative. This result also highlights qualitative differences with the KPP framework of [5, 6, 7, 15, 25], where the persistence of the population depends only on the value of . The biological implication is that under the combination of a weak Allee effect and a shifting climate, the size of the initial population becomes crucial for the survival of the species.

This new behaviour can be understood from the appearance, in the range of speeds , of intermediate stationary solutions between 0 and . It turns out that, although the equation (1.4) is of the monostable type in the half line , it shares some features with the usual bistable case as the trivial state 0 becomes stable with respect to some small enough compactly supported pertubations. This leads to the following dichotomy, or sharp threshold between extinction and spreading, in the same spirit as the results of [10, 20, 21, 29] in the spatially homogeneous framework:

Theorem 1.3.

Assume that and choose some . Then for any strictly ordered and continuous (in the -topology) family of initial data satisfying the same assumptions than in Theorem 1.2, there exists some such that spreading occurs for , extinction for , and grounding for whenever .

Theorem 1.3 further highlights that spreading and extinction are the two reasonable outcomes. Indeed, the only other possibility is grounding, and as can be seen when looking at an ordered family of initial data , this may only occur for a critical choice of the parameter .

Note that this threshold phenomenon strongly relies on our choice of compactly supported initial data. Indeed, one may for instance check that for any initial datum which does not decay to 0 as , spreading necessarily occurs. A similar threshold phenomenon was also exhibited in [20] in the homogeneous but degenerate monostable framework, namely equation (1.5) where satisfies (1.3) except that .

The above two theorems describe what happens in the different range speeds depending on the initial data. Let us now adopt a different approach where the initial datum is fixed and the speed  varies. Our last theorem writes as follows:

Theorem 1.4.

For all nonnegative, non trivial, bounded and compactly supported function , there exists such that the following three statements hold true.

  1. For all , spreading occurs.

  2. For all , extinction occurs.

  3. If :

    1. if , grounding occurs and ;

    2. if , there may be either grounding or extinction;

    3. if , extinction occurs.

Theorem 1.4 shows that there still exists, in the general monostable framework, a threshold forced speed below which spreading occurs and above which the solution goes extinct. As mentioned above, the existence of such a threshold for persistence was already known in the KPP framework. However, Theorem 1.4 together with Theorem 1.2 clearly imply that depends in a non trivial way on the initial datum as soon as . From an ecological point of view, this means that the persistence of the population is determined by the value of the climate shift speed with respect to this threshold. When the per capita growth rate of the population is optimal at zero density (no Allee effect), this threshold speed is independent of the initial datum, whereas in the presence of a weak Allee effect, this threshold depends on the size of the initial datum. More precisely, it follows from Theorem 1.3 that is nondecreasing with respect to the initial condition , so that a large population will be less sensitive to climate change in the sense that it can keep pace with a faster shifting habitat than a smaller population.


Remark 1.5.

The regularity of plays an important role in our main results whenever . Indeed assume for instance that in a neighbourhood of . From the phase plane analysis of the ODE

and proceeding as in Section 2, one may check that there does not exist any nonnegative and bounded stationary solution other than 0 and the invasion state . While we do not study such a case here, this leads us to formally expect that spreading then occurs for all initial data when .


Organisation of the paper

In Section 2 we study the stationary problem in the moving frame, i.e. the stationary solutions of (1.4). In Theorem 2.1 we classify all the different stationary solutions, and then in Section 2.2 we describe the decay of ground states to 0 at infinity. In Section 3 we examine the different speed ranges and prove Theorem 1.2. In particular we show that the hair-trigger effect does not hold in the intermediate speed range , see Theorem 3.4. In Section 4, we prove the convergence of the solution of (1.4) to a stationary solution as time goes to infinity. Furthermore, combining the uniform in time exponential estimates of Section 4.1 and the decay properties of ground states from Section 2.2, we show that the limiting stationary solution may only be 0, the invasion state or the critical ground state. In the last Section 5, we deal with the sharp threshold phenomena and prove both Theorems 1.3 and 1.4. Lastly, we include in an Appendix A some results on the so-called ‘zero number argument’ from [2, 10], which we use in Section 4 and in particular for the large-time convergence to a stationary solution.

2 Stationary solutions in the moving frame

In this section, we are interested in the stationary solutions of the equation (1.4), which may also be seen as travelling wave solutions of the original problem in the sense that they are also entire solutions of (1.1) moving with constant speed and profile. We will prove the following theorem which classifies all the stationary solutions of (1.4), making more rigorous the discussion in the beginning of Section 1.1.

Theorem 2.1.

All the positive and bounded stationary solutions of (1.4) can be classified as follows.

  1. For any , there exists a maximal positive solution , which we call invasion state and satisfies

  2. If , there exists and a family of positive stationary solutions , that will be called ground states, which satisfy:

    (2.1)

    and is nondecreasing. Moreover:

    1. if , then and there exists a positive stationary solution satisfying the same properties (2.1), which we call the critical ground state;

    2. on the other hand, if , then and locally uniformly as .

There exists no other positive and bounded stationary solution of (1.4) than the ones defined above.

This theorem already highlights three different situations, depending on the forcing/shifting speed . First, if , then there exists a unique positive and bounded stationary solution, which intuitively means that the equation retains its monostable feature. However, as soon as , some intermediate stationary solutions emerge in a neighbourhood of the trivial steady state 0, which thus becomes stable with respect to small enough perturbations. This leads to the loss of the hair-trigger effect, and the more complex dynamics stated in our Theorems 1.21.3 and 1.4. Furthermore, when , these intermediate stationary solutions even form some sort of foliation from 0 to the maximal stationary solution , which completely prevents the propagation, at least for compactly supported initial data. One can look at Figure 1 for an illustration of the different stationary solutions in the phase plane depending on the value of .

2.1 Proof of Theorem 2.1

Note that any stationary solution of (1.4) satisfies the second-order ODE

(2.2)

This equation is homogeneous on each half interval and of the domain, thus we will construct stationary solutions by ‘glueing’ phase portraits as in Berestycki et al [5].

Figure 1: Phase portrait of the discontinuous ODE (2.2) for different values of . All the trajectories start on the line before entering the phase plane of the homogeneous equation (2.4). The trajectories in dashed lines are the ones that cross the -axis, the one in bold dashed-dotted line is the critical ground state and the one in bold dashed line is the maximal solution . These simulations were conducted using Matlab and the function .
Proof of Theorem 2.1.

For , a stationary solution of (1.4) satisfies

As we are only interested in positive and bounded stationary solutions, it immediately follows that

(2.3)

where is defined as

Let us now ‘glue’ this with a solution of the homogeneous monostable equation

(2.4)

on . The identity (2.3) means that we are now looking at trajectories of (2.4) starting from some point on the half line , as illustrated in Figure 1. We argue in the phase plane of (2.4) and construct in the three steps below all the stationary solutions of (1.4). We refer to Aronson and Weinberger [3] for more complete and detailed arguments on the phase plane analysis of (2.4).

Step 1: existence of a maximal solution . We start by proving the first point in Theorem 2.1, and take any . It is easy to check that the steady state is a saddle point. In particular, it admits a one dimensional stable manifold and we can consider the unique trajectory which converges to from the upper phase plane . This trajectory clearly crosses the half line and, up to some shift, we can assume without loss of generality that the associated solution satisfies and converges monotonically to as . Glueing it with the exponential (2.3), we obtain a positive and bounded stationary solution as in Theorem 2.1.

Step 2: non existence of positive bounded solutions with . Now let be any stationary solution that satisfy (2.3) with , and consider the trajectory in the phase plane of (2.4) starting from . Clearly it lies above the trajectory of and, therefore, it crosses the vertical line above . Then, as for all , it is straightforward that the trajectory goes to infinity. In other words, any stationary solution of (2.2) satisfying is unbounded. So there does not exist any bounded and nonnegative solution of the stationary problem (2.2) that lies above .

Step 3: existence of non trivial intermediate solutions (ground states) if and only if . Next let be any stationary solution satisfying (2.3) with . Let us first note that the trajectory of in the phase plane of (2.4), which starts from , crosses the horizontal axis and enters the part of the phase plane. As for all , clearly it cannot cross back the horizontal axis. Thus the trajectory may only leave the set by crossing the vertical axis below the origin and, if it does not, then it converges to the equilibrium .

We now divide the proof into three parts depending on the value of . If , it is rather straightforward that the linearized operator around the steady state only admits two complex eigenvalues. Thus, any stationary solution  which is nonnegative for and satisfies changes sign in . Putting this together with the step 2 above, we are now able to conclude that there exists no positive and bounded stationary solution other than .

Next, we assume that . Let us recall that is the smallest such that there exists a trajectory in the phase plane, connecting the two steady states and , whose associated solution of (2.4) is exactly the travelling wave with speed  [3]. In particular, if , then clearly the trajectory starting from with must lie above that of , hence does not cross the vertical axis and converges to 0 as . For each we obtain a (unique) positive and bounded stationary solution, as announced in Theorem 2.1.

It now only remains to consider the case . In this case, the steady state is a node, namely trajectories locally converge to along the lines of slope . Note that, in the critical case , this crucially relies on the regularity of . Furthermore, it can be shown that there exists a trajectory converging to from the part of the phase plane, which is extremal in the sense that any trajectory lying below it must cross the vertical axis before converging to . We temporarily denote by the solution of (2.4) associated with this extremal trajectory ( is defined up to any shift). Following backward this extremal trajectory, it is clear that it must leave the part of the phase plane, either through the horizontal axis between and , or through the vertical line . The latter contradicts the fact that, since , the trajectory originating from the unstable manifold of may not converge to without crossing the vertical axis. Moving further back on the trajectory, it becomes clear that it also intersects the half line below the point . Denoting by the horizontal coordinate of this intersection, it is now straightforward that, for any , the stationary solution of (2.2) satisfying remains positive and converges to 0 as , while for it changes sign. Glueing all the trajectories which do not cross the vertical axis with the exponentials (2.3), we obtain all the stationary solutions described in Theorem 2.1, and by construction there exists no other positive and bounded stationary solution of (1.4).

We are now in a position to conclude the proof of Theorem 2.1. We first show that is nondecreasing, and consider two different speeds . Choose then any . From a phase plane analysis and using the fact that is decreasing, one can prove that the trajectory of (2.4) with and starting at lies above (respectively below) the trajectory of (2.4) with and starting at in the part of the phase plane (respectively the part of the phase plane). It is then straighforward that the trajectory of (2.4) with starting at converges to without crossing the vertical axis. It follows that and, as could be chosen arbitrarily close to , we conclude that .

It only remains to check that the intermediate stationary solutions constructed in the step 3 above satisfy the inequality on the whole line. For , it immediately follows from the fact that all stationary solutions are exponential, see (2.3). Moreover, it is straightforward from our construction in the phase plane (see also Figure 1) that there does not exist a such that and , which immediately implies that for . In particular, if and , then converges locally uniformly to a stationary solution such that , hence by the strong maximum principle. Note, however, that this convergence could also be deducted directly from the phase plane. This ends the proof of Theorem 2.1. ∎

Remark 2.2.

We mention here that, in [3], the minimal wave speed is obtained as the smallest such that the extremal trajectory defined above crosses the vertical line below . In particular, even though we treated above the cases and separately, one may see that both arguments actually follow from the same idea.

2.2 Exponential decay of ground states

Theorem 2.1 states that, when , there exist infinitely many ground states between 0 and . In order to understand the dynamics of the time evolution problem, one may want further insight on these intermediate steady states and, for instance, may wonder whether ground states are ordered or not. It follows from Theorem 2.1 and Proposition 2.3 below that they cannot since the critical ground state (when it exists) satisfies both and for all large enough, for any .

Furthermore, it turns out that the way ground states decay as plays an essential role in determining whether they may appear in the large-time behaviour of solutions under our choice of compactly supported initial data.

Proposition 2.3.

Assume that and define

  1. If , then for any , the ground state satisfies

    while (provided that ), the ground state satisfies

    where in both cases and are positive constants.

  2. For and any , there exists , and such that

    more precisely, if , while if .

Figure 2: Phase portrait of the steady states of problem (1.4) in the moving frame when . The figure on the right is a zoom of the left figure. One can notice, as stated in Proposition 2.3, that for all , the stationary solutions (defined in Theorem 2.1) converge to 0 asymptotically to the line , whereas converges to asymptotically to the line .
Figure 3: Phase portrait of the steady states of problem (1.4) in the moving frame when . The figure on the right is a zoom of the left figure. Here the stationary solutions all converge to 0 asymptotically to the line .
Proof of Proposition 2.3.

We briefly sketch the proof, which follows from the construction of ground states above, and from standard phase plane analysis of the homogeneous monostable equation (2.4). We also refer to the Figures 2 and 3 for an illustration of the argument.

When , the linearized problem around the steady state admits two eigenvectors . Noting that , it is well-known (see for instance [9]), that all trajectories converging to do so along the line , except for a single trajectory which goes through along the line . In particular, this latter trajectory lies below all the others which converge to from the part of the phase plane without crossing the vertical axis. It is also what we refered to earlier as the extremal trajectory which, by construction (see the proof of Theorem 2.1), coincides with the trajectory of the critical ground state when . Note that, if , the extremal trajectory lies below the trajectory of the travelling wave with speed (with which it actualy coincides if ). Therefore, it does not cross the horizontal axis between and and thus does not coincide with any positive and bounded stationary solution of (2.2).

It follows from the discussion above that, if , then converges to as with the exponential rate , while any other ground state converges to 0 with the exponential rate . The more accurate asymptotics stated in Proposition 2.3 are a classical consequence of the -regularity of  [9].

The critical case follows from a similar argument. While the linearized problem around  admits a unique eigenvector and, therefore, all trajectories converge to along the same line , it is still known that trajectories behave in a similar fashion as in the linear problem thanks to the -regularity of . More precisely, there exists an extremal trajectory which decays to 0 with the asymptotics with , while all the other trajectories decay to 0 with the slightly slower asymptotics where and . The end of the proof of Proposition 2.3 is again a simple consequence of the construction of as the unique stationary solution of (1.4) associated with the extremal trajectory. ∎

3 Large-time behaviour: spreading and extinction

In this section we prove most of Theorem 1.2 about the asymptotic behaviour of the solution of problem (1.4), and in particular we show how the hair-trigger effect disappears for intermediate speeds. The results presented in this section mostly rely on direct comparison methods. However, we will leave partly open the critical case , which will be dealt with later on (see Theorem 3.3 and Proposition 4.7).

Before starting the proof, we state some lemma which will be used extensively in the following sections:

Lemma 3.1.

Let be any nonnegative, bounded and compactly supported initial datum, and be the associated solution of (1.4). Then

Proof.

Note that any constant is a supersolution of (1.4). Thus, the solution of (1.4) with initial datum is decreasing in time and, by standard parabolic estimates, it converges locally uniformly to a stationary solution. As is the largest positive and bounded stationary solution, it easily follows that the solution actually converges to . Choosing large enough so that for all and by the parabolic comparison principle, we can already conclude that

(3.1)

Let us now improve the above estimate. Recall that

is such that the exponential satisfies (1.4) for all and any . One can then choose such that for all , and such that for all thanks to the fact that for all . By the parabolic maximum principle, we can infer that

(3.2)

for all and .

On the other hand, let us now denote by the solution of the ordinary differential equation with Clearly it converges to 1 as and, since it is a supersolution of (1.4), we get that

(3.3)

We are now in a position to conlude the proof. Choose any . Then let be large enough such that

This implies, together with (3.2) and (3.3), that

Putting this together with (3.1), we conclude that

Recalling that can be chosen arbitrary small, and noting that for all , the lemma is proved. ∎

3.1 In the case

We want to prove that the solution converges in the moving frame to the maximal positive stationary solution of (1.4), namely statement of Theorem 1.2. In particular, this means that the population manages to survive, and even to expand through the favorable zone. In other words, when the shifting speed of the favorable area is slow enough, the hair-trigger effect is still valid. Statement of Theorem 1.2 follows from the next theorem.

Theorem 3.2.

Assume that and is a non trivial, nonnegative and bounded initial datum.

Then the solution of (1.4) with initial datum converges locally uniformly to as goes to infinity.

Proof.

Thanks to Lemma 3.1, we only need to prove that

where the limit is understood to be locally uniform with respect to .

To do so, we exhibit a non trivial but arbitrarily small subsolution of (1.4). We argue in the phase plane of (2.4):

Since , the eigenvalues of the linearized problem around are in . Therefore, for any , the trajectory in the phase plane going through the point crosses the vertical axis twice. Cutting this trajectory and extending it by , it is then straightforward to construct a nonnegative compactly supported subsolution of (1.4), which tends to 0 as and, up to some shift, whose support is included in . Note also that the size of the compact support stays bounded as , as it is given by the non trivial imaginary part of the complex eigenvalues of .

Letting be the solution of (1.4) with the initial datum we know that for all . Moreover, using again Lemma 3.1 or the fact that any constant is a supersolution of (1.4), the function is bounded uniformly in time. Thus, by standard parabolic estimates, it converges locally uniformly to a nonnegative, non trivial and bounded stationary solution. By uniqueness (see Theorem 2.1), it is clear that the limit of must be .

Now consider a non trivial, nonnegative and bounded initial datum. Then, by the strong maximum principle, the associated solution is positive for any positive time. In particular and one can choose small enough so that, for all ,

Here we used the fact that the size of the support of remains bounded as . Using the parabolic comparison principle together with Lemma 3.1, we conclude that

where the limits are understood to be (at least) locally uniform with respect to . ∎

3.2 In the case

In this section we prove statement of Theorem 1.2 . Note that in the particular case , here we only prove that spreading does not occur.

Theorem 3.3.

Let be a non trivial, nonnegative, bounded and compactly supported initial datum.

  • If , then the solution of (1.4), with initial datum , converges uniformly to as .

  • If , then the solution does not spread. Furthermore, there exists no time sequence such that locally uniformly with respect to .

To conclude that extinction occurs in the critical case is slightly more intricate. This will be performed in the next section by a combination of Theorem 3.3 and some exponential bounds: we refer more precisely to Proposition 4.7 and the subsequent discussion.

Proof.

In both cases, the proof relies on the fact that solutions of the homogeneous monostable reaction-diffusion equation

(3.4)

are supersolutions of (1.1).

Classical results from Aronson-Weinberger [3] imply that the solution  of (3.4) with a (compactly supported) initial datum spreads with speed , in the sense that

In particular, it would follow by the comparison principle that

(3.5)

where in this equation is the solution of (1.1) in the non moving frame. In fact, the result of Aronson and Weinberger was restricted to the special case . Here we will prove (3.5) by a similar argument, which has also been known to extend Aronson and Weinberger’s result to the general case under our assumption that for all .

First recall from Lemma 3.1 that

(3.6)

as . Moreover, letting

it is straightforward that, for any and , the function

is a supersolution of (1.4) which also satisfies . We can now conclude from the above and the comparison principle that, for all and ,

Next, we define small enough so that

and

and choose a nonincreasing smooth function which is identically equal to 1 when and identically equal to 0 when .

Then, we introduce the travelling wave solution with minimal speed of the homogeneous problem (3.4), shifted so that . We recall from [3] that is a decreasing function and that for all . In particular,

and

Now we define

(3.7)

where is a large enough constant so that

(3.8)

Clearly is a supersolution of (3.4) for all and . On the other hand, for any and :

If moreover , it immediately follows from the fact that and for all that

while if