SPATIAL DYNAMICS OF A LOTKA-VOLTERRA MODEL WITH A SHIFTING HABITAT

In this paper, we study a Lotka-Volterra competition-diffusion model that describes the growth, spread and competition of two species in a shifting habitat. Our results show that (I) if the competition between the two species are either mutually strong or mutually weak against each other, the spatial dynamics mainly depend on environment worsening speed c and the spreading speed of each species in the absence of the other in the best possible environment; (II) if one species is a strong competitor and the other is a weak competitor, then the interplay of the species’ competing strengths and the spreading speeds also has an effect on the spatial dynamics. Particularly, we find that a strong but slower competitor can co-persist with a weak but faster competitor, provided that the environment worsening speed is not too fast.

1. Introduction.It is well known that spatial heterogeneity and diffusion play an important role when considering the interaction of biological species that can diffuse in the real world (see, e.g., [5,6,7,16,26,34,35]).Such a role can be well demonstrated even by diffusive Lotka-Volterra system for two biological species or two strains of the same species.For example, when studying the evolution of dispersals, Hastings [15] and Dockery et al [9] considered the following R-D system where u 1 (t, x) and u 2 (t, x) represent the populations of two competing species with respective diffusion rates d 1 and d 2 .Here, because the goal is to see whether slower or faster diffusion will have a selection advantage, the authors assumed that all biological characteristics of the two species are the same except for the diffusion rates (d 1 = d 2 ).A scenario for such a case is that one species (or strain) is mutated from the other with different diffusion rate.Hence, in (1), the two species share the same competition strength for each against the other (normalized to 1) and the same growth rate r(x) which reflects the intrinsic production and the habitat environment (e.g., richness of resources and quality of the living habitat etc.)It turns out that if Ω is a bounded set and ( 1) is associated with the zero flux boundary condition, then the species with slower diffusion rate will win the competition, e.g., if d 1 < d 2 , then all positive solutions of (1) converge to (u * 1 (x), 0) where u * 1 (x) is the unique positive solution of the boundary value problem When allowing different competition strengths between the two species, the Lotka-Volterra system (1) is modified to where the constant a 1 > 0 (a 2 > 0) represents the competition strength of species 2 (species 1) against species 1 (species 2).Now for such a slightly more general model (than (1)), each species faces choices in dispersal rate and local competition strength, and the interplay of the two sets of parameters {d 1 , d 2 } and {a 1 , a 2 } has revealed some very interesting and surprising results on the asymptotic behaviours of the solutions, including the possibility of a globally asymptotically stable positive (co-persistent) steady state.For details, see, e.g., [9,15,17,18,19,23,24,25,31,32] and the references therein.
On the other hand, the climate change in recent years has been a major concern of the scientific community, including ecologists and applied mathematicians, see, e.g., [2,3,4,14,20,27,29,30,33,37,38,39,40,41,42,45,50].A simple climate change pattern is the shifting of environment quality with a constant speed.This would translate to the shifting of the habitat quality which would be reflected by the shifting of the growth rate for a species.With such a consideration, Li et al [27] considered the following R-D equation for a single species living in the 1-dimensional whole space R: SPATIAL DYNAMICS OF A LOTKA-VOLTERRA MODEL 5635 Here, the way the growth rate function r depends on the time and location is through a moving pattern with a constance speed c, and r(•) is assumed to satisfy (A) r(x) is continuous, nondecreasing, bounded and piecewise continuously differentiable for all x ∈ R with 0 < r(∞) < ∞ and −∞ < r(−∞) < 0.
The non-decreasing property of r(x) assumes that the environment gets worse as time goes, and the negativity of r(−∞) accounts for a scenario that the environment is shifting to a very severe level.It is shown in [27] that if the environment's worsen speed c > c * (∞) := 2 dr(∞), then the species will go to extinction in the habitat; while when c < c * (∞), then the species satisfying some condition on its initial distribution will persist and spread along the gradient of the shifting habitat at an asymptotic spreading speed being precisely c * .The feature of "shifting with constant speed" represented by the moving frame, allows one to explore the traveling wave solutions of the form u(t, x) = U (x − ct) = U (ξ) governed by a second order non-autonomous ODE with the moving coordinate ξ = x − ct as the independent variables.In this regards, for the case c > c * (∞), by proving the existence of extinction wave solutions, Hu and Zou [21] confirmed that the extinction indeed occurs for (4) through the form of waves with the constant speed c.If the −∞ < r(−∞) < 0 is replaced by r(−∞) > 0 in (A), the biological scenario is changed from "severely worsening environment" to "mildly worsening environment".The recent work [3] has investigated traveling waves for a general KPP type reaction diffusion equation with mildly worsening environment that includes (4) as a special case.The much earlier work [2] also explored the traveling waves of a general R-D equation that has (4) as special case but with the growth function r(ξ) having support only on an finite interval.When considering a pathogen's population under the shifting host population, Fang et al [10] also obtained a scalar equation of the form (4) satisfying (A).In addition to spreading speed of the pathogen in comparison the spread speed of host, they also discussed the existence of traveling wave front solutions, leading to a better understanding of spatial-temporal patterns under such a shifting environment.
When two competing species face a shifting environment represented by r(x − ct) as in Li et al (4), the model system (3) is naturally revised to Now there arises a question: when a species not only faces an environment worsening represented by the shifting pattern given by r(x − ct) with r satisfying the assumption (A), but also encounters a competition from another species, how should the species choose its diffusion strategy?On the one hand, in light of the results in Li et al [27], faster diffusion (so that c * (∞) = 2 d r(∞) > c) would help the species to escape the environment worsening and thereby, help the species to persist.On the other hand, in view of the results in Hastings [15] and Dockery et al [9], slower diffusion should be favoured.It seems that there should be some strategy for a species which can balance the pressure from environment shifting and competition so that the species can persist.It is also possible that if the two species diffuse improperly, they will both go to extinction, whereas if they diffuse properly, they can both persist in some manner, surviving the environmental shifting and the competition.Exploring conditions for each of the above possibilities, particularly 5636 YUEDING YUAN, YANG WANG AND XINGFU ZOU conditions for co-persistence, constitutes the goal of of this paper.We will achieve this goal by studying the spatial dynamics of model ( 5), with particular interest in conditions for the two species to co-persist.Note that model ( 5) is heterogeneous in space and time.The heterogeneity described by r(x − ct) makes the existing theory and results for the case of constant r (see, e.g., [8,28] and the reference therein) not applicable to (5).In order to prove our main result, for the case of weak competition meaning that 0 < a j < 1 for j = 1, 2, we will use the fluctuation method, developed in Li et al. [27], and for the case of strong competition meaning that a j ≥ 1 for j = 1, 2, we will develop an alternative approach which enables us to obtain the spatial dynamics of model ( 5) in such a case.
Under assumption (A) and in the case of either weak competition or strong competition, the limit system of (5) has a positive co-persistence (positive) constant steady state (u + 1 , u + 2 ) where We show that the spatial dynamics mainly depend on c, c , then two competing species will both go extinct in the habitat; , then species 1 will become extinct in the habitat and species 2 will persist and spread; (iii) if 0 < c < c * 1 (∞) and a j ≥ 1 for j = 1, 2, then species 1 will become extinct in the habitat and species 2 will persist and spread; (iv) if 0 < c < ĉ * (∞) and 0 < a j < 1 for j = 1, 2, then two competing species will coexist.
The rest of this paper is organized as follows.In Section 2, we present the main mathematical results regarding the spatial dynamics of (5).In Section 3 we give some numerical simulation results that help illustrate the results from Section 2 and motivate some conjectures.Section 4 contains some discussion of the results and comparison with some recent work.To make the reading smoother, we leave the proof of Lemma 2.5 to the Appendix.
2. Mathematical results.In this section, we present mathematical results on the spatial dynamics of model (5) for the casen Ω = R.We first introduce some notations.Let R and R + be the sets of all reals and nonnegative reals, respectively.For any u For any constant k, we denote by k the vector (k, k).Define and Then we may rewrite (5) as the following more convenient form with given non-negative initial function: Throughout the paper, we assume that the function r satisfies the assumption (A) and one can easily verify the following Lipschitz condition: Clearly, if 0 ≤ φ(x) ≤ r(∞), then u ≡ 0 and u ≡ r(∞) are coupled upper and lower solutions of (6) (see [36] for definition), where 0 = (0, 0), r(∞) = (r(∞), r(∞)).The theory on the existence and uniqueness of solutions for reaction-diffusion systems has been well established (e.g.,Theorem 2.1 in [36]), by which, it is known that the initial value problem (6) For convenience of notations, we follow [27] to introduce some related functions.For r(x) > 0, define where The infimums occur at µ * i (x) = r(x)/d i , i = 1, 2. By [27], c * i (∞) is nothing but the asymptotic spread speed for species i in the absence of species j (j = i).Next, we focus on three case: The following result shows that if the two species initially live only on a bounded domain and their respective spreading speed is less than the habitat's worsening speed c, then both species will go to extinction, regardless of the competition strength.
The next result indicates that if one of the species spreads with a speed faster than the habitat's worsening speed, then that species is able to persist in the spreading sense, as long as that species' initial presence is significant in a very mild sense as stated in the following theorem.
If c < c 1 (∞), then obviously (ii) and (iii) also holds for species 1, as stated in the following theorem.
(i) For any ε > 0, (ii) If φ(x) ≡ 0 for all sufficiently large x, then for any ε > 0, Theorem 2.3 shows that if the spread speeds for the two species are both larger than the habitat's worsening speed (i.e., c < c * 1 (∞) due to the assumption d 1 < d 2 ), then an observer moving toward the right direction with a speed less than the habitat's worsening speed c, or moving with a speed larger than c but with the two species initially living only in a bounded domain, will not be able see individuals of the two species as t → ∞.In such a case, it is very natural and interesting to ask if the conclusion (iv) in Theorem 2.2 remains true for species 1 as well when c < c * 1 (∞), meaning that species 1 can also persist in the moving mode as stated in Theorem 2.2-(iv) for species 2. This problem turns out to be very challenging due to the presence of competition between the two species.In the sequel, we study two cases: the case of week competition and the case involving strong competition.In the case of week competition, we shall show that under a stronger condition (ĉ * (∞) > c), the answer to the question is affirmative, with the persistence levels for each species modified to reflect the effect of competition (see Theorem 2.7 ). But, in the case involving strong competition, the answer to the question is negative, with species 1 becoming extinct in the habitat and species 2 persisting and spreading (see Theorem 2.11).To proceed toward the goal, the rest of this section is organized as follows: we consider the case of week competition in Subsection 2.1.In Subsection 2.2, we will give some results on the case of strong competition.
For any finite T > 0, we consider (11) in the finite time interval (0, T ): We need a notion of weak upper/lower solutions adopted from [27].
Definition 2.4.We call a vector function v ≡ (v 1 , v 2 ) a continuous weak upper (lower) solution of (12 and in the distributional sense, i.e., for any for This definition is a slight variation/modification of Definition 1.1 in [46].Continuous weak upper and lower solutions were used in [1,27,46] in studying reactiondiffusion systems. Lemma 2.5.Assume that (12) has a weak upper solution The proof is similar to that of Lemma 1.2 in [46] with some minor modifications, and is given in the appendix for readers' convenience. Let . For fixed γ > 0, consider the following function of x on the interval [0, π/γ] parameterized by µ > 0: Such functions were used in [46,47] in studying reaction-diffusion systems, in addition to [27].Obviously, ϕ 1 (µ; x) and ϕ 2 (µ; x) are continuous in x and their second order derivatives in x exist and are continuous at x = 0, π/γ.The maximum of The following Lemma is a generalization of Lemma 2.3 in [27], and will play an important role in establishing the persistence of (11)., let be the number such that ĉ * ( ) = ĉ * (∞) − a * , where and sufficiently small β > 0 and γ > 0, there holds with ϕ 1 and ϕ 2 given by ( 14).
To prove the second inequality of (15), we transfer (11) to the following quasimonotone system by the translation where , and Clearly, the initial value problem (22) Then Repeating the proof of the first half of ( 15) with v replaced by v * , v by v * and θ by φ * , we have which leads to the second half of (15), completing the proof.
The following theorem shows that if ĉ * (∞) > c, two competing species can both persist in space and spread to the right at the asymptotic spreading speed is larger than ĉ * (∞).
Remark 1. Theorems 2.1, 2.2 and 2.7 imply that in the case of weak competition, two competition species will coexist if 0 < c < ĉ * (∞); species 1 will become extinct in the habitat and species 2 will persist and spread if c * 1 (∞) < c < c * 2 (∞), and the two competing species will both go extinct in the habitat if c > c * 2 (∞).Remark 2. Theorem 2.7 implies that if the environment worsening speed c is less than ĉ * (∞) (a parameter reflecting the two species' individual spreading capability and their competition strengths), then the two species can co-persist by spreading to the right at certain speeds ĉ1 and ĉ2 respectively, where, by Theorems 2.3 and 2.7, ĉi Here the condition c < ĉ * (∞) is only a sufficient condition for co-persistence; some numerical simulations presented in Section 3 (see Figure 4) shows that it is not a necessary condition for co-persistence.Remark 3. We point out that in the case of strong competition (i.e.,the case of a 1 ≥ 1 and a 2 ≥ 1), the functions v(µ; t, x) and v(µ; t, x) constructed in Lemma 2.6 are no longer continuous weak upper and lower solutions of (11), and hence, the above fluctuation method does not seem to apply (at least directly) in the case of strong competition.It remains an interesting and challenging problem to explore the spatial dynamics of model ( 5) subject to the strong competition.In the next subsection, we explore an alternative approach to deal with this case.

The case involving strong competition.
In this subsection, we consider the case of strong competition, i.e., the case of a 1 ≥ 1 and a 2 ≥ 1 in (6) with the scenario 0 < c < c * 1 (∞) (< c * 2 (∞)).In this case, ĉ * (∞) is no longer defined.Note that the model ( 6) is heterogeneous in space and time.Due to heterogeneity described by r(x − ct), the existing theory and results on spreading speeds of diffusive strong competition system with constant growth rate (see e.g., [8] and the references therein) are not (at least directly) applicable to (6).We also point out the fluctuation method used in Subsection 2.1 does not seem to apply in the case of strong competition.In this subsection, we develop a new approach that enables us to obtain the spatial dynamics of the model ( 6) subject to the strong competition.More specifically, we firstly design a subtle iteration scheme which will generate a sequence {u 2 } ∞ k=0 functions satisfying some required properties.Then, by carefully analyzing this sequence, and with the help of Egorov's Theorem, we show that this sequence converge to a limit function u * , uniformly on every bounded subset of R + × R, implying that u * is the solution to (6) which also satisfies the required properties.
Remark 4. From the proof of Lemma 2.8, we easily see that in the case of strong competition, the conclusion of Lemma 2.8 remains valid even if c ≥ c * 1 (∞) (assuming c < c * 2 ).Also, we have seen that we have only used the condition a 1 ≥ 1, meaning that a 2 ≥ 1 is actually not required.
Similarly, we consider the system 2 (t, x, φ)) be the solution of the system (90).Then by u (0) 1 ≥ 0 and the comparison principle, we obtain that u Moreover, by making use of Lemma 2.8, we can establish the following result.
Lemma 2.9.For every ε with 0 Proof.By Lemma 2.8, for any δ with 0 < δ < r(∞)/a 2 , there exists T > 0 such that u is continuous, nondecreasing and bounded, and piecewise continuously differentiable 2 (T, x, φ) is also positive on some closed interval.Thus, it follows from (90), ( 93), (94), the comparison principle and Theorem 2.2-(iii) in [27] that for every ε with 0 where c * δ = 2 d 2 r δ (∞).Because δ is arbitrary, we have actually shown that for any ε with 0 This together with the fact that u (1) Let δ 1 = ε/5 h 1 .In the case of t ≤ δ 1 , we have In the case of t > δ 1 , we obtain that In order to estimate the integral on the last line of (105), for given z ∈ Λ we define Obviously, Λ z ⊂ Λ 1 where is bounded.Thus, by Egorov's Theorem (see, e.g., Theorem 3.2.8 in [48]), for the above ε > 0 there exists a measurable subset uniformly for all z ∈ Λ ε , where m(Λ 1 − Λ ε ) is the measure of the set Λ 1 − Λ ε .Thus, for the above ε > 0 there exist K ε > 0 and P ε > 0 such that (107) It follows from ( 104) and (107) that By ( 105) and (108), we obtain that where Thus, the limit lim k→∞ u uniformly holds for all z ∈ Λ.Similarly, we obtain that the limit lim k→∞ u uniformly holds for all z ∈ Λ.Because T and M are arbitrary, we have actually shown that the function sequence { u uniformly on every bounded subset of R + ×R.This together with the iteration scheme (95) (e.g., writing (95) as some integral form and applying the Lebesque's dominated convergence theorem) implies that (u * 1 , u * 2 ) is the solution to (6).
By (101), for any δ > 0, there exists a positive constant T δ such that For the above δ > 0, there exists K δ,Λ0 > 0 such that Thus, for the above δ > 0 there holds Because τ and L are arbitrary, we have actually shown that Applying the similar arguments to (98), ( 99) and (100), we can confirm (ii), (iii) and (iv) of the theorem.The proof is completed.Remark 6.From Remarks 4 -5 and the proof of Theorem 2.11, we know that the conclusions of Theorem 2.11 remains true even if c * 1 (∞) ≤ c < c * 2 (∞).Moreover, for the two competition coefficients, we have only used the condition a 1 ≥ 1 in obtaining our results in Theorem 2.11, meaning that the condition a 2 ≥ 1 is actually not required.In other words, the key conditions in our results in this section are (A) conditions on speeds: c * 2 > c and c * 2 > c * 1 ; and (B) the condition on the competition strength: a 1 ≥ 1.An explanation for this observation is that the condition c * 2 > c means that the larger spreading speed of species 2 enables it to survive the worsening environment, while the condition a 1 ≥ 1 shows that species 2 have strong competition against species 1, and it is this strong competition from surviving species 2 that drives the the species 1 to extinction, regardless of whether c * 1 < c or c * 1 > c, whether a 2 < 1 or a 2 ≥ 1.It is interesting to ask what happens if c * 2 > c (also (c * 2 > c * 1 ) but a 1 < 1 ≤ a 2 , a scenario that species 2 can persist by spreading to the right in the absence of species 1 and its spread speed is larger than that of species 1, yet species 2 has competition disadvantage.In such a scenario, which species can persist by spreading and by what speed?Under this scenario, there are two cases: c * 1 > c and c * 1 < c.The latter has been covered by Theorems 2.1 and 2.2; for the former, we are not able to answer these questions theoretically at the present, but numerical simulations in Section 3 (see Figure 6) shows that co-persistence is possible by spreading to the right.
3. Some numeric simulations.In this section, we present some numerical simulations for model (6) for two purposes: (I) numerically confirm the theoretical results obtained in Section 2; (II) numerically explore some parameter ranges that have not been covered in the results of Section 2, by which we hope to gain some intuition and suggestions for further theoretical investigation of (6).To this end, we choose the growth function , a scenario that the environment is worsening very fast.Not surprisingly, the two species will eventually go to extinction, as claimed in Theorem 2.1 and illustrated in Figure 1.Next, we consider a case that worsening speed c is a little bit slower: ), a scenario that the spreading capability of species 1 without competition is not enough to allow this species to survive the environment worsening speed, but the spreading capability of species 2 without competition enables it to survive the environment worsening speed.The numerical results, presented in Figure 2, indicate that species 1 eventually becomes extinct in the habitat and the species 2 persists by spreading to the right with spread speed c * 2 (∞) = 2.2, confirming the result in Theorem 2.2.
We further consider an even smaller value of c, c = 1.65.Then, c < ĉ * (∞), a scenario of Theorem 2.7.The numeric simulations (see Figure 3) confirm that the two competing species co-persist in a spreading pattern, and their respective asymptotical spreading speeds seem to be c * 1 (∞) = 2 and c * 2 (∞) = 2.2, respectively.In Remark 2, we mentioned that c < ĉ * (∞) is a sufficient condition but may not be necessary condition for the two species to co-persist by spreading to the right.We now demonstrate this by considering c = 1.8, which satisfies c > ĉ * (∞) but c < c * (∞).As is seen in the simulations presented in Figure 4, the two competing species can still co-persist by spreading to the right, and their asymptotic spread speeds still seem to be c * 1 (∞) = 2 and c * 2 (∞) = 2.2, respectively.Next, we consider the case involving strong competition.For convenience, we still use d 1 = 1 and d 2 = 1.21 leading to c * 1 (∞) = 2 and c * 2 (∞) = 2.2.Now we also take c = 1.8 as for Figure 4, but we replace the weak competition coefficients a 1 = 0.19 and a 2 = 0.36 by strong ones a 1 = 3 > 1 and a 2 = 2 > 1.The numerical simulations in Figure 5 show that, species 1 will go to extinction while species 2 persists -copersistence is no longer the outcome.This confirms the conclusion of Theorem 2.11, and is in strong contrast to the results for weak competition illustrated in Figure 4 where co-persistence is observed.
In Remark 6, we raised the question of what happens if c * 2 > c * 1 > c and a 1 < 1 ≤ a 2 , which we are unable to answer theoretically.Now we present some numerical results.We still use seem to suggest that c < c * (∞) is the necessary and sufficient condition for the two species to be able to co-persist by spreading to the right.Actually the theoretical results obtained in Section 2 and the simulation results presented in Section 3 seem to suggest that (i) c * i (∞) > c is the necessary and sufficient condition for the species i to be able to persist by spreading to the right ( assuming reasonable initial distribution); and (ii) if c * i (∞) > c then the species i will persist by spreading to the right with the asymptotic spread speed c * i (∞) (see, Remark 1).If the above conjectures are true, then the weak competition does not affect the outcome and it is the individual intrinsic spreading capability compared to the environment worsening speed that determines the long time spatial dynamics of the species' population.We point out that when studying traveling wave fronts of autonomous Lotka-Velterra type diffusive cooperative/competitive systems that connect the extinction equilibrium and the co-existence equilibrium (assuming existence), the minimal wave speed is also independent of the competition strengths (see, e.g., [22,43,44]), and this offers another motivation to the above conjectures.Noticing that c * i (∞) = 2 d i r(∞) is increasing in d i , the above discussion indicates that in such a situation (in whole space R with the environment worsening at a constant speed), evolution would favour a faster dispersal rate, in strong contrast to the situation of bounded domain with heterogeneous static environment in which slower dispersal is favoured.This is reasonable, because when there is no limit in space, weak competition is less relevant and spatial invasion/spread plays the dominant role for a species to survive.In the case involving strong competition, situation would be different.Our results indicate that the interplay of the species' competing strengths and the spreading speeds also has an effect on the spatial dynamics.Theorem 2.11 shows that if the faster species is also a strong competitor (a i > 1) and if its spreading speed is faster than the environment worsening speed, then slower species will go to extinction, regardless of whether it is a strong or weak competitor and whether it spreads faster or slower than the environment worsening speed.A ecological explanation for this given in Remark 6.An particular interesting scenario is that one species is a strong competitor while the other is a weak competitor (i.e., ((a 1 − 1)(a 2 − 1) < 0).Particularly, we find that a strong but slower competitor can co-persist with a weak but faster competitor, provided that the environment worsening speed is not too fast, as demonstrated in Figure 6.Such an outcome of co-persistence ought to be a result of balancing the capabilities of spreading and competition.Similar phenomenon is also observed in [13] where a system of form (5) (or (3) with constant r with a 1 < 1 < a 2 and d 1 < d 2 is considered.Particular attention of [13] is to the effect of initial functions.
Our results are obtained for both cases of weak and strong competition.The methods developed in [27] (also in subsection 2.1 of this paper) for weak competition and in subsection 2.2 of this paper for the case involving strong competitio do not seem to apply (at least directly) to the case of c < c * 1 (∞) and 0 < a 1 < 1 ≤ a 2 .We have to leave the theoretical analysis of this case for a future work, using the numeric result in Figure 6 as an intuition, but developing some new method.
We point out that after we have written up our results into the first draft of this paper, Dr. Wendi Wang drew our attention (in an conference) to the publication of the most recent paper [49] by him and his co-authors (we thank Dr. Wang for this).In [49], the authors considered a system that is a little bit more general than (5) in the sense that the resource related growth functions for the two species can be different, that is, Here we would like to comment on some differences.Firstly, [49] is mainly motivated by [27], aiming to extend the work [27] to competitive system of Lotka-Volterra type.However, our work is motivated by those results on evolution of dispersals such as [9,15,17,18,19,23,24,25,31,32] as well as [26], intending to compare the impact of different dispersal rates on the temporal-spatial dynamics in the Lotka-Volterra system when encountering a special shifting habitat.As such, we assume the same grow function for the two competing species (as in the most of above mentioned works) and focus on comparing the distinct dispersal rates and competition strengths.This setting enables us to obtain some more detailed results.For example, in the weak competition case ([49] solely deals with weak competition case), while the only theorem of [49] (Theorem 1) contains a result that is the corresponding version of our Theorem 2.7, it does not contain a results corresponding to Theorem 2.3.Consequently, [49] only established a lower bound ĉ * (∞) for the spreading speed of the species i when c ∈ (0, ĉ * i (∞)), however, because of Theorem 2.3, our results here give not only the lower bound but also an upper bound for the spreading speed ĉi of the species i when 0 < c < ĉ * i (∞) (see Remark 1).Moreover, for the case of r 1 (x) = r 2 (x) the corresponding co-persistence portion of the results in [49] (i.e., Theorem 1-a), under the assumption d 1 < d 2 , requires that the initial distributions for the two species satisfy 0 < φ 1 (x) < u + 1 , u + 2 < φ 2 (x) ≤ r 2 (∞) on a closed interval, while our Theorem 2.7 only requires 0 < φ i (x) ≤ r(∞) on a closed interval.
Appendix.In this appendix, we give the detailed proof of Lemma 2.5.
Proof of Lemma 2.5.Firstly, we consider the bounded case, i.e., the case in which R is replaced by a bounded interval [−L, L], where L > 0. Therefore, the boundary value problem (12) can be written as  Obviously, φ(0, x) = θ(x) for all x ∈ R and v ≤ φ ≤ v.
A vector function v ≡ (v 1 , v 2 ) is said to be a continuous weak upper (lower) solution of (118) if v is continuous on Q, v| Γ ≥ (≤)φ and , i = 1, 2. Let v ≡ ( v 1 , v 2 ) and v ≡ ( v 1 , v 2 ) be the continuous weak upper and lower solutions of (118).Clearly, they are also the continuous weak upper and lower solutions of (12) as L → ∞.Let where i = 1, 2. G i and v i,Φi can be obtained by the Perron method (see [12]).Define v i ), where .By the strong maximum principle for weakly subparabolic functions, we have v ≤ v (1) ≤ v on Q (see [11]).Define v (j) i = v i,Φi + T i (v (j−1) i ), i = 1, 2. Then similarly as above, we have on Q, where v (j) ≡ (v 2 ), j = 1, 2, and v ≡ (v 1 , v 2 ).Then by the Lebesgue Dominated Convergence Theorem, lim j→∞ T i (v and by the same reasoning regarding v (1) in the previous paragraph, we can prove that v = (v 1,Φ1 + T 1 (v 1 ), v 2,Φ2 + T 2 (v 2 )) is a continuous weak solution of (118).Then a bootstrap argument implies that v is a classical solution of (118).Obviously, v ≤ v ≤ v. Thus, we have completed the proof of Lemma 2.5 in case of bounded interval [−L, L].
By the conclusion for bounded interval proved above, (122) has a classical solution v (j) ≡ (v 2 ) with v ≤ v (j) ≤ v for each j ≥ 1.Then applying L P interior estimates and Schauder interior estimates to v (j) , we have where M (Q ) is independent of j, 0 < α < 1 and i = 1, 2. From this and a diagonalization argument, there is a subsequence of {v (j) } j=∞ j=1 (still denote it by {v (j) } j=∞ j=1 ) such that v (j) i → v i in C 1,2 loc ((0, T ) × R), where i = 1, 2. Obviously, v satisfies the differential equation in (12) and v ≤ v ≤ v on (0, T ) × R. By the same arguments as in the proof of [46] (Lemma 1.2), we have v i ∈ C([0, T ] × R) and v i (0, x) = θ i (x) for all x ∈ R, where i = 1, 2. The proof of Lemma 2.5 is completed.

Figure 5 .
Figure 5. Numerical simulations on (6) with (113)-(116) with a 1 = 0.19 and a 2 = 0.36 replaced by a 1 = 3 and a 2 = 2 respectively.The environment worsening rate is very small (c=1.8) in the sense that c < c * 1 (∞), species 1 becomes extinct in the habitat and species 2 persist by spreading to the right with the asymptotic speed c * 2 (∞) = 2.2.