Can a Species Keep Pace with a Shifting Climate?
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Abstract
Consider a patch of favorable habitat surrounded by unfavorable habitat and assume that due to a shifting climate, the patch moves with a fixed speed in a onedimensional universe. Let the patch be inhabited by a population of individuals that reproduce, disperse, and die. Will the population persist? How does the answer depend on the length of the patch, the speed of movement of the patch, the net population growth rate under constant conditions, and the mobility of the individuals? We will answer these questions in the context of a simple dynamic profile model that incorporates climate shift, population dynamics, and migration. The model takes the form of a growthdiffusion equation. We first consider a special case and derive an explicit condition by glueing phase portraits. Then we establish a strict qualitative dichotomy for a large class of models by way of rigorous PDE methods, in particular the maximum principle. The results show that mobility can both reduce and enhance the ability to track climate change that a narrow range can severely reduce this ability and that population range and total population size can both increase and decrease under a moving climate. It is also shown that range shift may be easier to detect at the expanding front, simply because it is considerably steeper than the retreating back.
Keywords
Climate change Reaction–diffusion equation Traveling wave Moving favorable patch Comoving population profile Persistence Extinction Principal eigenvalue1 1. Introduction
The area occupied by a species is to a large extent determined by the climatic circumstances with temperature playing a major role. The global warming phenomenon, therefore, has a great impact on survival and location of such species. See Walther et al. (2002) for a review of ecological responses to recent climate change.
We idealize the world by putting the North Pole at +∞ and the equator at −∞. This ignores the finiteness of the Earth, but it offers a good framework for a theoretical analysis. Warming and its effect can be seen as a shift in the profile of local climatic suitability, which the population density profile of a species tries to track. If a species keeps pace, its area expands as much in the north as it loses in the south. However, if it lags behind too much, it will go extinct.
Which of these two scenarios applies? How does the answer depend on the mobility of the species, on the extensiveness of the area, the local population dynamics, and on the speed of climate shift and the way climate actually acts on a species? If the species survives, what happens to the size and form of its population profile?
The recent research of one of us (Nagelkerke, 2004) tackles these issues in the context of a relatively realistic metapopulation model, using simulations as the main tool. The aim of the present paper is to address the same issues for a continuous population using an analytical approach. We study a simplified model, taking the form of a reactiondiffusion equation. Within this framework, our findings confirm the ones that had been observed in simulations. Here, we establish these results for a large class of equations, with rigorous mathematical proofs, thus proving their robustness and shedding light on the mathematical properties behind them.

An explicit formula (23) and in different forms in (24) and (25), for the critical size of the favorable patch for persistence, as a function of the Malthusian parameters, the diffusion constant and the climate speed. The formula pertains to the juxtaposition of two types of homogeneous habitat, the favorable patch being a bounded interval outside of which the environment is unfavorable.

Revelation of a striking asymmetry in the comoving population profile: the north front is much steeper than the south tail and the population maximum occurs near to the northern border of the population profile.

The observation that if the climate does not move too fast, the size of the total population as well as the range of the population may actually grow, relative to the situation in a static climate. But when the climate speed is further increased, an abrupt collapse may follow (see Figs. 7, 8 and 9).
Such questions are a bit reminiscent of the “critical patch length” problem (cf. Okubo and Levin, 2001 and Ludwig et al., 1979), the “traveling wave invasion” problem (cf. Kolmogorov et al., 1937; Fisher, 1937; Aronson and Weinberger, 1978; Berestycki and Hamel, 2009; Thieme and Zhao, 2003; Rass and Radcliffe, 2003) and the “heterogeneous environment” problem (cf. Berestycki et al., 2005a, 2005b; Roques and Stoica, 2007; Shigesada and Kawasaki, 1997; Shigesada et al., 1986; Weinberger, 2002). Yet, the mix of ingredients (in particular, the fact that c is prescribed, and hence amounts to an external forcing) makes it different from each of these and apart from Pease et al. (1989), where a quantitative genetics approach is adopted, we could not find any references. After most of the work described here was finished, however, we came across the preprint version of Potapov and Lewis (2004), which addresses exactly the same question, but with emphasis on the effect of a moving climate on the outcome of competitive interaction between two species. In fact, the special case that we treat in Sections 2 and 3 is also treated by Potapov and Lewis. Yet, we decided to include our analysis of this case in this paper as (1) the method is more geometrical (essentially phase plane analysis), (2) we deliver an analytical solution for the population profile and (3) we present additional results, leading to further biological insights. Other related work can be found in Dahmen et al. (2000), Deasi and Nelson (2005), and Pachepsky et al. (2005). It is known that diffusion enhances invasion speed but is counter productive for population growth on a finite stationary patch. Consequently, for a moving patch, there is a conflict between gain due to colonization of newly favorable habitat and loss due to migration into unfavorable habitat. A key result, formula (23) below, provides a quantitative algorithm for deciding which of these two effects is the stronger one.
Making use of the linearization at w = 0 we thus derive rather explicit answers to the key questions.

Either no positive traveling wave exists and zero is the global attractor,

Or such a wave does exist and it attracts all orbits starting from nonnegative (≢ 0) initial data.
The biological insights derived from our analysis are explained in detail in Sections 2 and 3 while taking for granted that the results of Section 4 demonstrate the correctness and the robustness of the conclusions. More ecological consequences are discussed in Section 5. Readers who are looking for theorems and proofs will find the rigorous theory for a general class of equations presented in Section 4.
2 2. Glueing phase portraits
Already at this stage we can conclude that K only sets the scale for u, but that it is irrelevant for the answers to our questions.
This system has equilibria (w, v) = (0, 0) and (w, v) = (1, 0). (Incidentally, orbits connecting these two equilibria yield the classical KPPFisher traveling waves Fisher, 1937; Kolmogorov et al., 1937. These exist if and only if c ≥ 2√r. The lowest possible wave speed, 2√r, is the invasion speed Aronson and Weinberger, 1978; Berestycki and Hamel, 2009.)
If on the other hand, Open image in new window , then (0, 0) is a stable node. Since μ_{−} < σ_{±}, orbits of (10) that approach the origin from the positive half plane w>0 do so “above” the line υ = μ_{−} w. It is known (see Hadeler and Rothe, 1975; Aronson and Weinberger, 1978; Volpert et al., 1994, or Diekmann and Temme, 1982) that the unstable manifold of (1, 0) that lies in the region w ≤ 1 does, in fact, approach the origin in this manner, and that it lies entirely above the line v = μ_{−} w (in fact above v = σ_{−} w).
In view of the results of Schaaf on twopoint boundary value problems (Schaaf, 1990), it is to be expected that the length of the ζinterval of an orbit piece connecting the two lines increases with increasing distance (along either line) from the origin (with limit +∞ if we approach the connection via pieces of the stable—and unstable manifold of (1, 0)).
Indeed, let us now prove this monotonicity property.
Proof: By shifts of the solutions w ^{1} and w ^{2}, (taking w ^{ i }(x + a _{ i })) there is no loss in generality in assuming that a ^{1} = a ^{2} = 0. Then we want to show that b ^{2} > b ^{1}.
Indeed, g(w ^{1} > g(w ^{2}) in (0,α). Now, if α < min⨑ub;b ^{1}, b ^{2}⫂ub; is such that w ^{1} <w ^{2} in (0,α) while w ^{1}(α) = w ^{2}(α), then (15) yields w ^{2} _{ x }(α) α w ^{1} _{ x } ^{1} and w ^{2} do not cross each other in (0, min⨑ub;b ^{1}, b ^{2}}⫂).
So, the shortest feasible L is obtained in the limit where the points on the line υ = μ_{±} w approach the origin. In that limit, we can replace the nonlinear term −rw(1 − w) in the second equation of (11) by its linearization −rw.
From this it easily follows that in order to see persistence of the species, the diffusion constant should be neither too small nor too large, depending on c. This is illustrated in Fig. 5 which shows L(D) achieving a minimum at a positive value of D, depending on c, for any c > 0. Figure 6 displays the curve Open image in new window for various values of r with the asymptotic value as Open image in new window given by formula (26) above. It further shows that for given L, r has to have a minimum size for persistence.
The (counter intuitive) conclusion is that an increase in c may lead to an increase of the total population size whenever the unsuitable area outside the favorable core area is not too harsh. This is due to a lag effect in the left tail: the decay of the population in the region that was favorable until recently may be slow while meanwhile, the rise of the population in the right region that just became favorable is relatively fast. This possibility of increases in both range and population size was not shown in the related work of Potapov and Lewis (2004).
In conclusion of this section, we formulate an insight deriving from (23): for small c, an increase of D entails an increase of the minimal interval length, since diffusion creates a net loss over the boundary of the favorable region. For larger c, however, the influence of D on the minimal interval length may be opposite, since increased mobility helps to track the moving climate.
3 3. The linearization at zero
In this section, we investigate formally the stability of the extinct state. We find that the principal eigenvalue switches sign exactly at the codimension one manifold in parameter space that separates the domain of existence of a nontrivial solution from the domain of nonexistence. In the next section, we shall see that the principal eigenvalue for a general equation characterizes the existence and nonexistence of nontrivial solutions and determines as well the large time dynamics of this model. Note that within the domain of nonexistence this eigenvalue further yields information about the rate of decay to zero, i.e., the rate at which the population declines on its way to extinction.
In the next section, we will see that the sign of the principal (or dominant) eigenvalue of this operator, when properly defined, yields the long term dynamics in Eq. (1). Its sign gives a criterion for either extinction or persistence.
Therefore, methods to determine the sign of the dominant eigenvalue are of great interest, as are methods to give more quantitative estimates in case it is negative (at which time scale will the extinction happen?).
4 4. Analysis of a general class of equations
So far, we have considered a particular type of heterogeneity, that which is obtained by juxtaposing two homogeneous media—the favorable and unfavorable ones—with an abrupt transition at the two end points of the favorable interval. Are the results which we have derived previously robust? And is the comoving nontrivial solution stable if it exists? Here we give very strong affirmative answers to both these questions in a rather general setting. The motivation for considering general types of nonlinearities is twofold. First, the assumptions made in Section 2 are rather contrived from a modeling point of view and one would like to consider more complex transitions e.g. gradual transitions between recognizable but not necessarily strictly homogeneous zones. Second, a general mathematical theory sheds much more light on the underlying mechanisms, since the proofs reveal the role that various assumptions play in yielding the conclusions. Here, for instance, the linearization at the trivial steady state, in particular the sign of the associated principal eigenvalue, will be seen to fully account for the ability to keep pace with a shifting climate.
In this section, we consider Eq. (1). As was already mentioned, there is no loss in generality in assuming that D = 1, which we do henceforth.
 (a)
Negative density dependence: u ↦ g(u, x) is decreasing for all x ∈ ℝ and strictly decreasing for x ∈ I _{0}, where I _{0} is a nonempty open interval.
 (b)
Allow for multiple discontinuities, e.g., several patches: there is a finite set of points F = {a _{1}, … , a _{ p }} in ℝ such that g is continuous on ℝ_{+} × (ℝ\F) and both Open image in new window g(u, x) and Open image in new window g(u, x) exist, uniformly for u in compact subsets of ℝ_{+}.
 (c)
Existence of a linearization: There existsδ > 0 such that u ↦ g(u, x) is C ^{1} on [0,δ] for all x ∈ ℝ, g _{ u } is continuous on [0,δ] × (ℝ∖F) and both Open image in new window gu(u, x) and Open image in new window gu(u, x) exist, uniformly for u ∈ [0,δ].
 (d)
Unfavorable outer regions: g(0, x) → −1 as x → ±∞.
 (e)
Saturation: There existsM > 0 such that g(u, x) ≤ 0 for all x ∈ ℝ whenever u ≥M.
The properties formulated in (a)–(e) above are the standing hypotheses on the function g throughout this section. The last one means that everywhere the population declines when it exceeds some level M, i.e., negative density dependence guarantees that the population stays bounded. The limits at ±∞ in (d) are taken to be the same in order to simplify the formulation, but the statements and proofs can readily be adapted to the case of different limits. Note that the values of the limit can be changed by a scaling of the time variable t. Accordingly, the value −1 is representative for general negative values. Similarly it is no restriction that we take D = 1, as this can always be achieved by a scaling of the spatial variable x (after the scaling of time). Since g may have discontinuities with respect to x, we consider generalized solutions. These are functions u which, as a function of x, are globally of class C ^{1} and piecewise of class C ^{2} and satisfy the equation at each point with x ≠ a _{ i }, i = 1, … , p.
For studies of solutions of (1) on bounded domains, without an imposed translation speed (i.e., c = 0), we refer to Murray and Sperb (1983), Cantrell and Cosner (1991, 1998, 2003), CanoCasanova and LópezGómez (2003) and the references given there. Recently, the effect of a heterogeneous but spatially periodic environment has been studied by Berestycki et al. (2005a, 2005b). Lastly, periodic stochastic environments are considered by Roques and Stoica (2007).
The problem we study here involves a lack of compactness (the problem is set on the whole real line) as well as the difficulty deriving from the fact that c is imposed.
4.1 4.1. A priori estimates for the “far out” asymptotic behavior of traveling waves
This observation motivates us to formulate that w(x) behaves like e^{μ+x } for x→ − ∞ and like e^{μ−x } for x→ +∞in a weaker sense that we now make precise.
Next, we want to derive lower bounds. We first formulate an auxiliary result that will also be used later.
To conclude this subsection, we formulate an estimate for the derivative of w.
4.2 4.2. The eigenvalue problem
As another preparatory step, we shall make precise how, in the present case, the principal (or dominant) eigenvalue of the linearized problem at w ≡ 0 is defined. Since we consider solutions defined on the whole real line, some special care is needed.
Note that the ϕ that we consider here are allowed to grow beyond any bound for x → ∞. By restricting the “test” functions ϕ to those that are bounded on ℝ, we obtain a different generalized eigenvalue that may be smaller. For instance, if g(0, x)=−1 for all x, then Open image in new window while the additional requirement that the functions be bounded yields a generalized eigenvalue equal to 1; so when c ≠ = 0, these are not equal. We refer to Berestycki et al. (2007) for a general study of these themes.
We conclude this subsection by describing, in a crude manner, the “far out” asymptotic behavior of ϕ^{∞} when λ_{∞} is either negative or zero. We only state the properties that we shall use in the next subsection.
So the inequality (57) follows from Proposition 4.2, provided the argument of the squareroot is positive. This requires thatδ > − 1 − λ_{∞} and since we already required thatδ >0, we should restrict toδ > max {0, −1 − λ_{∞}}.
Proposition 4.7. Let ϕ ^{∞} be positive and satisfy (56) on ℝ\F with λ^{∞} = 0. Then ϕ ^{∞} has the properties formulated in terms of w as (49), (50), and in Proposition 4.4.
4.3 4.3. The solvability condition
Theorem 4.8. Equation (48) has a bounded positive solution if and only if λ^{∞} > 0.
Assumption (e) guarantees that the constant function taking the value M is a supersolution. Clearly, υ(x) < M for small ɛ. We conclude that a solution exists.
Indeed, this follows by combining (57) with (48) and (58) with (49), if we chooseδ in Proposition 4.6 such that not onlyδ > max{0, −1 −λ_{∞}} but alsoδ < −λ_{∞}. The point is that in this case Open image in new window so that by choosing next ɛ in Proposition 4.1 sufficiently small, the quotient ϕ^{∞}(x)/w(x) has a positive exponent for large positive x and a negative exponent for large negative x.
4.4 4.4. Uniqueness of traveling waves
Theorem 4.9. Equation (48) has at most one bounded positive solution.
Proof: The argument follows some ideas in Berestycki (1981). The new difficulty is that here we have to deal with an unbounded domain.
Corollary 4.10. Equation (48) has exactly one bounded positive solution if λ_{∞} > 0 and no such solution if λ_{∞} ≤ 0.
4.5 4.5. Large time behavior
 (i)
If λ_{∞} ≤ 0, then u(t, x) → 0 for t → ∞, uniformly for x ∈ ℝ. That is, any population is bound to go extinct, no matter what the initial distribution is.
 (ii)
If λ_{∞} > 0 and u _{0} is nontrivial, then u(t, x) − w(x − ct) → 0 for t → ∞, uniformly for x ∈ ℝ. Here, w is the unique bounded positive solution of (48). So, any population is bound to persist by traveling along with the shifting climate.
It remains to prove (ii). If u _{0} is nontrivial, u(δ, x) is strictly positive forδ > 0, and hence so is υ(δ, x). So, for any given R > 0 and ɛ sufficiently small we have υ(δ, x) ≥ ɛϕ^{ R }(x) for −R ≤ x ≤ R. Now assume that λ_{∞} > 0. Recall from the proof of Theorem 4.8 that for R large enough, we obtain a subsolution if we extend ɛϕ^{ R }(x) by zero outside the interval [−R, R]. Accordingly, z(t, ·) cannot converge to zero for t → ∞ when λ_{∞} > 0 and, therefore, the limit must be the unique bounded positive solution w of (48), (compare Corollary 4.10). Likewise, the subsolution converges to w and since υ is sandwiched in between; it too must converge to w.
So, the assumption that z does not converge uniformly to w for t → ∞ leads to a contradiction and we conclude that the convergence is, in fact, uniform. The proof that the subsolution converges uniformly to w follows exactly the same pattern. Hence, the true solution υ, which lies in between, must converge uniformly to w and the proof of (ii) is completed.
Finally, we note that the proof that z(t, ·) converges uniformly to zero when λ_{∞} ≤ 0 is based on precisely the same arguments as used above.
5 5. Concluding remarks
Mathematical studies of simplified models can yield ecological insights, and at the same time, shed light on basic mechanisms. In that spirit, we have analyzed the effect that a shifting climate may have on the persistence of a species. A patch of favorable habitat, surrounded by unfavorable habitat, is able to sustain a population provided the gain by reproduction can balance the losses due to mortality inside the patch and dispersal away from the patch. If the patch itself moves in space, an additional loss term is created, since individuals may be left behind. Dispersing individuals, on the other hand, may be fortunate enough to land where conditions are changing for the better. As a result, the critical size that a patch should have in order to sustain a population, does not only depend on reproduction, mortality and dispersal rates, but also on the speed with which the patch moves through space. In Section 2, we have derived an explicit expression in formulas (23), (24), and (25) for the dependence which produces valuable insights. In Section 4 we have rigorously established several mathematical properties for a large class of models.
Persistence in a moving patch is facilitated when the rate of climate change is low, the rate of population growth within the patch is high, and the climate outside the patch not too hostile (Fig. 5). Migration, however, is a doubleedged sword. Both too much and too little dispersal can lead to extinction and the optimal dispersal rate increases with patch speed (Fig. 5). The results imply that a small latitudinal range diminishes the maximal rate of climate change a species should be able to track. This means that the conventional approach (see Skellam, 1951) of using the invasion (Fisher) speed as an estimate of this maximal rate can lead to a severe overestimation when ranges are small or D is large.
A moving climate can have dramatic effects on the size and form of the population profile. When the favorable region moves to the north, the population becomes more concentrated toward the north end of the population profile. Interestingly, if the habitat outside the favorable patch is not too hostile, the south tail becomes considerably thicker and longer as a result of the movement, since it takes a while before the marooned local population disappears. As a consequence, movement may result in increases in both the total population size and the population range (Figs. 7, 8 and 9).
In unpublished simulations of a metapopulation model, Nagelkerke (2004) obtained results similar to those reported here on our continuous population model. This demonstrates the structural robustness of our sometimes counterintuitive findings. For example, he modeled jump dispersal of propagules. This leads us to believe that our results are not restricted to movement by simple diffusion. Distance dispersal is relevant for many organisms.
Here, we have concentrated on the long time dynamics. Nagelkerke (2004) also studied the transient dynamics shortly after the climate starts to move. He found that generally the northern border initially moves faster than the southern border, both for surviving populations and for those that were doomed to go extinct. In the case of ultimate extinction, the southern border catches up after a while and then moves even faster than the climate, until it collides with the northern border. Note that another reason for not being too confident about an increasing range is the threshold behavior shown in Fig. 8. A small additional increase in climate speed can cause total collapse. The initial asymmetry between the velocities of both borders is in agreement with the outcome of an extensive analysis of butterfly data by Parmesan et al. (1999) that found more evidence for moving northern borders than for southern borders, suggesting that this is a transient phenomenon (see also Collingham et al., 1996). In addition, it could be easier to observe the move of the steep north front than that of the far less steep south back.
Our analysis was relatively simple, since we considered a onedimensional spatial domain. Twodimensional models give rise to new subtleties. Some mathematical issues involved in higherdimensional versions of this problem will be discussed in Berestycki and Rossi (2008). Of particular interest is to understand the effect of the geometry on the ability to persist despite a climate change. For instance, a bottleneck may occur when the extension of the patch in the lateral direction has a local minimumgiving rise to a narrow strait. Actually, one could mimic this effect in the onedimensional setting by allowing the diffusion coefficient to depend on the spatial variable x; there would then be both an x and an xct dependence, making the problem inhomogeneous even modulo time translation. We plan to analyze such problems in further works.
Notes
Acknowledgement
Part of the research presented in this paper was carried out while Henri Berestycki was visiting the Department of Mathematics at the University of Chicago, which he thanks for its hospitality. Odo Diekmann thanks Frithjof Lutscher for bringing the work of Potapov and Lewis to his attention and A.B. Potapov for stimulating discussions during a Spatial Ecology meeting in Miami organized by Cantrell, Cosner, and DeAngelis. Lastly, the authors are indebted to Lionel Roques (INRA, Avignon, France), for providing the computations shown in Figs. 5, 6, 7 and 8.
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