Persistence versus extinction for two competing species under a climate change

Abstract. This paper considers effects of a climate-induced range shift on outcomes of two competitive species, which is modeled by a reaction-diffusion system with the increasing growth rates of species along a shifting habitat gradient. Analytical conditions are established for the coexistence or competitive exclusion of two-competitors under the climate change, which present the control strategies to maintain the persistence of species.


Introduction
In real ecological systems, issues of persistence of competing species and invasive spread of one species is fundamental.Ecologically, an understanding of spreading speeds of species can provide insights into invasion process [4,22], for example, how quickly an introduced species can move into a novel landscape or how rapidly an extirpated species can recover to its previous range.Whether introduced purposely, accidentally or by natural means, the invasion of new species can have serious economic and ecological consequences.
There is an increasing acknowledgement that we are experiencing a period of rapid climate change.Meanwhile, an increasing appreciation that ecosystem responses to climate change are complex and widespread has promoted a focus on understanding and predicting biological impact [7,9,15,18,19,26].Whether populations and species will persist at the local and global scale, respectively, depends on their abilities to endure future climate shifts.Thus, determining the rate of climate change that populations can cope with, therefore, is urgently needed.
Mathematical modeling is a powerful approach to understand the ecological effects of climate change.Early on, the issue of persistence in a changing landscape investigated by a number of papers [1,3,16,23,29] has primarily focused on the established species that exist at equilibrium distributions in a bounded domain prior to the onset of climate change.Few studies have considered the spread or invasion of an introduced species and how this might be impacted by shifts in habitat suitability.Recently, a number of studies have used analytical approaches to address the potential for the population persistence in asymmetric flow [8,[12][13][14]25] or in the situation of changing climatic conditions, where the medium is static, but appropriate habitat is shifting [2,5,21,32].
To explore the issue of species spread in the context of climate change, Li et al. [11] studied the spread of a single species over a region with varying habitat suitability, that is, shifting in time taking the form of Here c > 0 and r(ξ) is continuous and nondecreasing and bounded with r(−∞) < 0 and r(∞) > 0. Thus, r(x − ct) divides the spatial domain into two parts: the region with the good-quality habitat suitable for growth (i.e., r(x − ct) > 0), and the region with the poor-quality habitat unsuitable for growth (i.e., r(x − ct) < 0).The edge of the habitat suitable for species is shifting at a speed c. Results in [11] have shown that the persistence and spreading dynamics of (1) depend on c and c * (∞), that is, determined by the maximal linearized growth rate and the diffusion coefficient.More specifically, if c > c * (∞), the species will become extinct in the habitat; if c < c * (∞), then the species will persist and spread along the shifting habitat gradient at an asymptotic spreading speed c * (∞).These results can be interpreted as whether a species is able to persist depend on its ability to outrun an encroaching boundary.
The purpose of the present paper is to extend the work in [11] to a two-competing species model.We will analyze the influence of climate change on the competition outcomes and characterize the spreading speeds of biological populations in response to both biological competitions and the shifting habitat under the climate change.We derive sufficient conditions for the coexistence of two species and show that there are switches in competitive dominance induced by different speeds of the shifting habitat edge.More precisely, if two species are initially restricted to and distributed over a band of suitable habitat, persistence then requires that the two species keep pace with the movement of its suitable habitat band.Indeed, we find that the interspecific competition slows the invasion speed of invasive species.Moreover, if the shifting of climate change moves slower than the minimum invasion rate of two species, coexistence of two species happens when the competition is weak; replacement happens when the shifting rate of climate change is medium; if the shifting rate of climate change is lager than the maximum invasion rate of two species, both species will go extinct.https://www.mii.vu.lt/NAThe rest of this paper is organized as follows.In Section 2, we present our mathematical model and analyze its spatial dynamics.In Section 3, we provide a number of simulations to illustrate the results from Section 2. Section 4 gives a brief discussions.

Mathematical results
We consider the following competition model: with the initial conditions u(0, x) = u 0 (x), v(0, x) = v 0 (x).Here t ∈ R + , −∞ < x < +∞; u := u(t, x), v := v(t, x) are the densities of two competing species respectively; d 1 , d 2 > 0 are their respective diffusion coefficients; a 1 , a 2 > 0 are interspecific competition coefficients; c > 0 is a speed at which the edge of the habitat suitable for species growth is shifting.The per capita growth rates r 1 (x − ct) and r 2 (x − ct) satisfy the following standing hypothesis: Hypothesis 1. r i (x), i = 1, 2, are continuous, nondecreasing, and piecewise continu- where Thus, the reaction terms in (2) are Lipschitz continuous in u, v.By adding dominant linear terms ρu, ρv to both sides of (2a) and (2b) respectively, model ( 2) can be written as ) are a pair of lower and upper solutions of (2).It follows from [17,20,24,27,31] that the initial problem of (2) with (u(0, x), v(0, x)) = (u 0 (x), v 0 (x)), where u 0 (x), v 0 (x) are continuous and 0 We will consider only these solutions in what follows.
It is easily seen that a solution (u(t, x), v(t, x)) of ( 2) takes the form of where To motivate our main results, we consider the homogeneous nonspatial case of system (2): There are four constant equilibria: the unpopulated state (0, 0); the first-species monoculture state (r 1 (∞), 0); the second-species monoculture state (0, r 2 (∞)); and the coexistence state (u * , v * ), where We summarize the well-known facts for the existence and stability of the equilibria in the following: Furthermore, the coexistence state If v is the resident species and u is the invasive species, then (6) implies that the resident species is excluded by the invasive species because positive solutions of (3) tend to (r 1 (∞), 0) as t approaches ∞.From (4), it is also easy to see that We now study how the outcomes of competing species u and v are affected by the climate changes.For r i (x) − a i r j (∞) > 0, i = j, i, j = 1, 2, we define It is easily seen that where The infimum occurs at µ * i (x) = (r i (x) − a i r j (∞))/d i .We also define It is easily seen that φ i (x; µ) > ψ i (µ) for 0 < µ < µ * i (x) and φ i (x; The main results of this paper are stated in the following theorem.
Theorem 1. Suppose that Hypothesis 1 is satisfied.Assume Then the following statements are valid: r 2 (∞) on a closed interval, and u(0, x) = v(0, x) ≡ 0 for all sufficiently large x, we have , u(0, x), v(0, x) are zero for all sufficiently large x, v(0, x) > 0 for other values of x, then for every ε > 0, there exists T > 0 such that the solution (u(t, x), v(t, x)) of (2) satisfies , and u(0, x) = v(0, x) ≡ 0 for all sufficiently large x, then for every ε > 0, there exists T > 0 such that for t > T , the solution (u(t, x), v(t, x)) of (2) satisfy u(t, x) < ε and v(t, x) < ε for all x.
Before the proof of Theorem 1, we present a few preliminaries.The first one is an auxiliary function proposed by Weinberger [28].For γ > 0 and µ > 0, we define Clearly, w(µ; x) is a continuous function in x, and its second-order derivative in x exists and is continuous when x = 0, π/γ.The maximum of w(µ; x) occurs at σ(µ) = (1/γ) tan −1 (γ/µ), which is a strictly decreasing function of µ.
We are now able to present the proof of Theorem 1.
Note that for t 0 t t 0 + h.It follows from ( 13) and ( 14) that for t 0 t t 0 + h, we have where By similar arguments to those in [11], we see that ( 15) is valid for all t t 0 .
For the chosen > 0, we select L > 0 large enough such that Note that for any s > 0, https://www.mii.vu.lt/NALet t 1 > t 0 be a sufficiently large number.For t > t 1 , the solution (u(t, x), v(t, x)) satisfies the integral equations ) for all t 0 and all x, it follows from ( 17) and ( 15) that for t > t 1 , and we have Nonlinear Anal.Model.Control, 22(3):285-302 This, together with ( 16), implies that for x satisfying (19), It follows from (18) that for x satisfying (19), and Here we use the simple fact that It follows from ( 20), ( 21) and ( 22), ( 23) that for t t 1 and x satisfying (19), where Equations ( 17) and ( 18) and induction arguments show that for t t 1 and x satisfying we have where ũ(n) (t) and ṽ(n Direct calculations and induction show that where Furthermore, b n (t) and e n (t) in ( 26) are the sums of polynomials, and products of polynomials and exponential functions in the form of e −j(t−t1) with j a positive integer.Observe that lim t→∞ ũ(n) (t) = f n and lim t→∞ ṽ(n) (t) = g n .Since f n is increasing and g n is decreasing, it is easy to see that the limits of f n and g n as n → ∞ exist.Taking the limits to both sides of the first equation and third equation of ( 27) and setting lim n→∞ f n = f * , lim n→∞ g n = g * , we get If ( 5) holds, since is sufficiently small, from (28) we get as n → ∞.If ( 6) holds, we have r 2 (∞) − a 2 r 1 (∞) < 0. Note that the limit of g n can not be negative due to the positivity of the solution (u(t, x), v(t, x)) for t > 0 and that > 0 is sufficiently small.It follows from ( 28) that as n → ∞.These and (26) indicate that there exist a positive integer N and t 2 > t 1 such that for t > t 2 and x satisfying (24) with n replaced by N , either 6) holds.We choose t 1 sufficiently large such that for t t 1 , For any given ε with 0 < ε < (c * 1 (∞) − c)/2, we choose sufficiently small such that < ε/2.Since ψ 1 (µ 1 ) = c + and ψ Figure 1 displays the numerical solutions when c = 1.4 < c * 1 (∞), which shows that two species with the weak competition coexist in an asymptotic region t(c + ε) x t(c * 1 (∞) − ε) by expanding its spatial range to the right at the asymptotic speed c * 1 (∞).Figure 2 demonstrates the numerical solutions when c = 1.4 < c * 1 (∞) with a 1 = 0.21, a 2 = 1.5 satisfying (6), which shows that in a region of asymptotic size (c * 1 (∞) − c)t, the superior competitor u displaces the well established species v.
In Fig. 3, we consider the case where c = 1.9 ∈ (2 is the asymptotic invasion speed of u in (1) without competition (in the absence of individuals of species v).Thus, u becomes extinct and only species v persists in the suitable habitat through expanding its spatial range to the right at the asymptotic speed 2 d 2 r 2 (∞).

Discussion
One crucial measure of a species' invasiveness is the rate at which it spreads into a competitor' environment, so that the spreading speed which reflecting the invasion speed of the invader becomes a hot topic in the past decades [6,10,30].In this paper, we focus on the effect of climate change on two species interacting through Lotka-Volterra competition with different dispersal and competitive abilities with varying habitat suitability, that is, shifting in time.
By applying the methods developed in Li et al. [11], we have determined the critical invasion speed for each species, and have found that two competing species become extinct if the habitat boundary moves at speeds greater than the fastest speed of expansion of species population.On the other hand, two competing species coexist and spread if the habitat boundary moves at speeds lower than the slowest speed of expansion of species population when the competition is weak.If the rate of shifting habitat edge is medium, then more adapted species survives.As we have pointed out earlier, the persistence conditions in our theorem requires weak interactions between two competitors (i.e., a 1 a 2 < 1), and we shall consider the strong competition in future.
there exists t 3 > t 2 such that Then profiles of advantageous reproduction areas of two competing species move to the right at a speed c, and the two species are initially distributed on the interval [20, 20 + π].Let us fix d 1 https://www.mii.vu.lt/NA