Homogenization and influence of fragmentation in a biological invasion model

In this paper, some properties of the minimal speeds of pulsating Fisher-KPP fronts in periodic environments are established. The limit of the speeds at the homogenization limit is proved rigorously. Near this limit, generically, the fronts move faster when the spatial period is enlarged, but the speeds vary only at the second order. The dependence of the speeds on habitat fragmentation is also analyzed in the case of the patch model.


Introduction and main hypotheses
In homogeneous environments, the probably most used population dynamics reactiondiffusion model is the Fisher-KPP model [13,23]. In a one-dimensional space, it corresponds to the following equation (1.1) The unknown u = u(t, x) is the population density at time t and position x, and the positive constant coefficients D, µ and ν respectively correspond to the diffusivity (mobility of the individuals), the intrinsic growth rate and the susceptibility to crowding effects. A natural extension of this model to heterogeneous environments is the Shigesada-Kawasaki-Teramoto model [32],

2)
Definition 1.1 (L-periodicity) Let L be a positive real number. We say that a function h : R → R is L-periodic if ∀ x ∈ R, h(x + L) = h(x).
In this paper, we are concerned with the general equation: The diffusion term a L satisfies a L (x) = a(x/L), where a is a C 2,δ (R) (with δ > 0) 1-periodic function that satisfies On other hand, the reaction term satisfies f L (x, ·) = f (x/L, ·), where f := f (x, s) : R×R + → R is 1-periodic in x, of class C 1,δ in (x, s) and C 2 in s. In this setting, both a L and f L are L-periodic in the variable x. Furthermore, we assume that: The growth rate µ may be positive in some regions (favorable regions) or negative in others (unfavorable regions). The stationary states p(x) of (1.3) satisfy the equation Under general hypotheses including those of this paper, and in any space dimension, it was proved in [4] that a necessary and sufficient condition for the existence of a positive and bounded solution p of (1.6) was the negativity of the principal eigenvalue ρ 1,L of the linear operator with periodicity conditions. In this case, the solution p was also proved to be unique, and therefore L-periodic. Actually, it is easy to see that the map L → ρ 1,L is nonincreasing in L > 0, and even decreasing as soon as a is not constant (see the proof of Lemma 3.1).
In this work, we are concerned with the propagation of pulsating traveling fronts which are particular solutions of the reaction-diffusion equation (1.3). Before going further on, we recall the definition of such solutions: Definition 1.2 (Pulsating traveling fronts) A function u = u(t, x) is called a pulsating traveling front propagating from right to left with an effective speed c = 0, if u is a classical solution of: where the above limits hold locally in t.
This definition has been introduced in [31,32]. It has also been extended in higher dimensions with p L ≡ 1 in [1] and [35], and with p L ≡ 1 in [5].
Under the above assumptions, it follows from [5] that there exists c * L > 0 such that pulsating traveling fronts satisfying (1.9) with a speed of propagation c exist if and only if c ≥ c * L . Moreover, the pulsating fronts (with speeds c ≥ c * L ) are increasing in time t. Further uniqueness and qualitative properties are proved in [14,15]. The value c * L is called the minimal speed of propagation. We refer to [2,3,11,18,25,27,28,34] for further existence results and properties of the minimal speeds of KPP pulsating fronts. For existence, uniqueness, stability and further qualitative results for combustion or bistable nonlinearities in the periodic framework, we refer to [6,7,12,16,17,19,24,26,35,36,37,38].
In the particular case of the Shigesada et al model (1.2), when a(x) ≡ 1, the effects of the spatial distribution of the function µ L on the existence and global stability of a positive stationary state p L of equation (1.2) have been investigated both numerically [30,31] and theoretically [4,8,29]. In particular, as already noticed, enlarging the scale of fragmentation, i.e. increasing L, was proved to decrease the value of ρ 1,L . Biologically, this result means that larger scales have a positive effect on species persistence, for species whose dynamics is modelled by the Shigesada et al model.
The effects of the spatial distribution of the functions a L and µ L on the minimal speed of propagation c * L have not yet been investigated rigorously. This is a difficult problem, since the known variational formula for c * L bears on non-self-adjoint operators, and therefore, the methods used to analyze the dependence of ρ 1,L on fragmentation cannot be used in this situation. However, in the case of model (1.2), when a L ≡ 1, ν L ≡ 1 and µ L (x) = µ(x/L), for a 1-periodic function µ taking only two values, Kinezaki et al [22] numerically observed that c * L was an increasing function of the parameter L. For sinusoidally varying coefficients, the relationships between c * L and L have also been investigated formally by Kinezaki, Kawasaki, Shigesada [21]. The case of a rapidly oscillating coefficient a L (x), corresponding to small L values, and the homogenization limit L → 0, have been discussed in [19] and [38] for combustion and bistable nonlinearities f (u).
The first aim of our work is to analyze rigorously the dependence of the speed of propagation c * L with respect to L, under the general setting of equation (1.3), for small L values. We determine the limit of the minimal speeds c * L as L → 0 + (the homogenization limit), and we also prove that near the homogenization limit, the species tends to propagate faster when the spatial period of the environment is enlarged. Next, in the case of an environment composed of patches of "habitat" and "non-habitat", we consider the dependence of the minimal speed with respect to habitat fragmentation. We prove that fragmentation decreases the minimal speed.

Main results
In this section, we describe the main results of this paper. Unless otherwise mentioned, we make the assumptions of Section 1. The first theorem gives the limit of c * L as L goes to 0. Theorem 2.1 Let c * L be the minimal speed of propagation of pulsating traveling fronts solving (1.9). Then, denote the arithmetic mean of µ and the harmonic mean of a over the interval [0, 1].
Formula (2.1) was derived formally in [33] for sinusoidally varying coefficients. Theorem 2.1 then provides a generalization of the formula in [33] and a rigorous analysis of the homogenization limit for general diffusion and growth rate profiles.
Remark 2.2 The previous theorem gives the limit of c * L as L → 0 when the space dimension is 1. Theorem 3.3 of El Smaily [11] answered this issue in any dimensions N , but under an additional assumption of free divergence of the diffusion field (in the one-dimensional case considered here, this assumption reduces to da/dx = 0 in R). Lastly, we refer to [6,7,16] for other homogenization limits with combustion-type nonlinearities.
Our second result describes the behavior of the function L → c * L , for small L values. Theorem 2.3 Let c * L be the minimal speed of propagation of pulsating traveling fronts solving (1.9). Then, the map L → c * L is of class C ∞ in an interval (0, L 0 ) for some L 0 > 0. Furthermore, Lastly, γ > 0 if and only if the function is not identically equal to 2.

Corollary 2.4
Under the notations of Theorem 2.3, it follows that if a is constant and µ is not constant, or if µ is constant and a is not constant, then γ > 0 and the speeds c * L are increasing with respect to L when L is close to 0.

Remark 2.5
The question of the monotonicity of the map L → c * L had also been studied under different assumptions in [11] (see Theorem 5.3). The author answered this question for a reaction-advection-diffusion equation over a periodic domain Ω ⊆ R N , under an additional assumption on the diffusion coefficient (like in Remark 2.2, this assumption would mean again in our present setting that the diffusion coefficient a(x) is constant over R). Our result gives the behavior of the minimal speeds of propagation near the homogenization limit for general diffusion and growth rate coefficients. The condition γ > 0 is generically fulfilled, which means that, roughly speaking, the more oscillating the medium is, the slower the species moves. But the speeds vary only at the second order with respect to the period L. Based on numerical observations which have been carried out in [21] for special types of diffusion and growth rate coefficients, we conjecture that the monotonicity of c * L holds for all L > 0.
Lastly, we give a first theoretical evidence that habitat fragmentation, without changing the scale L, can decrease the minimal speed c * . We here fix a period L 0 > 0.
We assume that a ≡ 1, and that µ L 0 := µ z takes only the two values 0 and m > 0, and depends on a parameter z. More precisely: With this setting, the region where µ z is positive, which can be interpreted as "habitat" in the Shigesada et al model, is of Lebesgue measure l in each period cell [0, L 0 ]. For z = 0, this region is simply an interval. However, whenever z is positive, this region is fragmented into two parts of same length l/2 (see Figure 1). Our next result means that this fragmentation into two parts reduces the speed c * . Remark 2.7 Note that, whenever z > (L 0 − l)/2, the two habitat components in the period cell [l/2 + z, L 0 + l/2 + z] are at a distance smaller than (L 0 − l)/2 from each other. In fact, Theorem 2.6 proves that, when z varies in (0, L 0 − l), c * z is all the larger as the minimal distance separating two habitat components is small, that is as the maximal distance between two consecutive habitat components is large.

Remark 2.8
Here, the function µ z does not satisfy the general regularity assumptions of Section 1. However, c * z can still be interpreted as the minimal speed of propagation of weak solutions of (1.9), whose existence can be obtained by approaching µ z with regular functions.
The main tool of this paper is a variational formulation for c * L involving elliptic eigenvalue problems which depend strongly on the coefficients a and f. Such a formulation was given in any space dimension in [3] in the case where the bounded stationary state p of the equation (1.3) is constant, and in [5] in the case of a general nonconstant bounded stationary state p(x).

The homogenization limit: proof of Theorem 2.1
This proof is divided into three main steps.
Step 1: a rough upper bound for c * L . For each L > 0, the minimal speed c * L is positive and, from [5] (see also [3] in the case when p ≡ 1), it is given by the variational formula where λ * L > 0 and, for each λ ∈ R and L > 0, k(λ, L) denotes the principal eigenvalue of the problem with L-periodicity conditions. In (3.2), ψ λ,L denotes a principal eigenfunction, which is of class C 2,δ (R), positive, unique up to multiplication by a positive constant, and L-periodic.
Furthermore, it follows from Section 3 of [5] that the map λ → k(λ, L) is convex and that ∂k ∂λ (0, L) = 0 for each L > 0. Therefore, for each L > 0, the map λ → k(λ, L) is nondecreasing under the notations of Section 1. Multiplying (3.2) by ψ λ,L and integrating by parts over [0, L], we get, due to the Lperiodicity of a L and ψ λ,L : for all λ > 0 and for all L > 0. Consequently, Step 2: the sharp upper bound for c * L . For any λ > 0 and L > 0, consider the functions Since ψ λ,L is unique up to multiplication, we will assume in this step 2 that The above choice ensures that We are now going to prove that the families (ψ λ,L ) λ,L and (ϕ λ,L ) λ,L remain bounded in H 1 (0, 1) for L small enough and as soon as λ stays bounded. For each L > 0, we call where [1/L] stands for the integer part of 1/L. Multiplying (3.2) by ψ λ,L and integrating by parts over [0, M L L], we get that
It follows from the above computations that the sequences (ψ n ) and (v n ) are bounded in H 1 (0, 1). Hence, up to extraction of a subsequence, ψ n → ψ and v n → w as n → +∞, strongly in L 2 (0, 1) and weakly in H 1 (0, 1). By Sobolev injections, the sequence (ψ n ) is bounded in C 0,1/2 ([0, 1]). But since each function ψ n is L n -periodic (with L n → 0 + ), it follows from Arzela-Ascoli theorem that ψ has to be constant over [0, 1]. Moreover, the boundedness of the sequence (k(λ n , L n )) n∈N implies that, up to extraction of another subsequence, We denote this limit by k(λ), we will see later that indeed it depends only on λ. It follows now, from (3.11) after replacing (λ, L) by (λ n , L n ) and passing to the limit as n → +∞, that Consequently, Actually, since the functions ψ n are L n -periodic (with L n → 0 + ) and converge to the constant ψ strongly in L 2 (0, 1), they converge to ψ in L 2 loc (R). But where M = sup n∈N λ n . Hence, ψ = 0 and By uniqueness of the limit, one deduces that the whole sequence (k(λ n , L n )) n∈N converges to this quantity k(λ) as n → +∞, which proves the claim (3.12). Now, take any sequence L n → 0 + such that c * Ln → lim sup L→0 + c * L as n → +∞. For each λ > 0 and for each n ∈ N, one has Since this holds for all λ > 0, one concludes that (3.14) Step 3: the sharp lower bound for c * L . The aim of this step is to prove that which would complete the proof of Theorem 2.1. For each L > 0, the minimal speed c * L is given by (3.1) and the map (0, +∞) λ → k(λ, L)/λ attains its minimum at λ * L > 0.We will prove that, for L small enough, the family (λ * L ) is bounded from above and from below by λ > 0 and λ > 0 respectively. Namely, one has Lemma 3.1 There exist L 0 and 0 < λ ≤ λ < +∞ such that The proof is postponed at the end of this section. Take now any sequence (L n ) n such that 0 < L n ≤ L 0 for all n, and L n → 0 + as n → +∞. From Lemma 3.1, there exists λ * > 0 such that, up to extraction of a subsequence, λ * Ln → λ * as n → +∞. One also has (3.12) and (3.13).
and the proof of Theorem 2.1 is complete.
Proof of Lemma 3.1. Observe first that, for λ = 0 and for any L > 0, k(0, L) is the principal eigenvalue of the problem and we denote φ L = ψ 0,L a principal eigenfunction, which is L-periodic, positive and unique up to multiplication. In other words, k(0, L) = −ρ 1,L under the notations of Section 1.
Dividing the above elliptic equation by φ L and integrating by parts over [0, L], one gets On the other hand, as already recalled, ∂k ∂λ (0, L) = 0 and the map λ → k(λ, L) is convex for all L > 0. Therefore, Assume here that there exists a sequence (L n ) n∈N of positive numbers such that L n → 0 + and λ * Ln → 0 + as n → +∞. One then gets This is contradiction with (3.14). Thus, for L > 0 small enough, the family (λ * L ) L is bounded from below by a positive constant λ > 0 (actually, these arguments show that the whole family (λ * L ) L>0 is bounded from below by a positive constant). It remains now to prove that (λ * L ) L is bounded from above when L is small enough. We assume, to the contrary, that there exists a sequence L n → 0 + as n → +∞ such that λ * Ln → +∞ as n → +∞. Call for all n ∈ N and x ∈ R. Rewriting (3.10) for λ = λ * Ln and for L = L n , one consequently gets ∀ n ∈ N, (a Ln ϕ n ) + µ Ln ϕ n = k n ϕ n in R.  But, for each n ∈ N, M Ln ∈ N while a Ln and ψ n are L n -periodic. Hence, a Ln (θ n + M Ln L n ) = a Ln (θ n ), ψ n (θ n + M Ln L n ) = ψ n (θ n ), and ψ n (θ n + M Ln L n ) = ψ n (θ n ) = 0. Then, 0 is given by (1.4)), (3.17) whenever n is large enough so that 2 ≤ e 2λ * Ln M Ln Ln (remember that λ * Ln → +∞ as n → +∞, by assumption). Meanwhile, for all n ∈ N,  since ψ n is L n -periodic. One has We refer now to equation (3.2). Taking λ = λ * Ln , dividing this equation (3.2) by the L nperiodic function ψ n and then integrating by parts over the interval [0, L n ], we get Owing to (1.4), it follows that ∀ n ∈ N, Putting the above result into B(n), we obtain, for all n ∈ N, where β = 2α 2 √ a M µ M /α 1 × C and C is a positive constant such that Lastly, let us rewrite equation ( Together with (3.17), (3.18), (3.19) and (3.20), one concludes that there exists n 0 ∈ N such that for n ≥ n 0 , ≤ β × ψ 2 n (θ n )e 2λ * Ln (θn+M Ln Ln) × λ * Ln L n + 1 .
Therefore the assumption that λ * Ln → +∞ as L n → 0 + is false and consequently the family (λ * L ) L is bounded from above by some positive λ > 0 whenever L is small (i.e. 0 < L ≤ L 0 ). This completes the proof of Lemma 3.1.
Remark 3.2 From Theorem 2.1, one concludes that the map (0, +∞) L → c * L can be extended by continuity to the right at L = 0 + . Furthermore, for any sequence (L n ) n of positive numbers such that L n → 0 + as n → +∞, one claims that the positive numbers λ * Ln given in (3.1) converge to < a > −1 and Lemma 3.1 implies that, up to extraction of a subsequence, λ * Ln → λ * > 0. Passing to the limit as n → +∞ in the above equation and due (3.13) together with Step 2 of the proof of Theorem 2.1, one gets whence λ * = < a > −1 H < µ > A . Since the limit does not depend on any subsequence, one concludes that the limit of λ * L , as L → 0 + , exits and The sharp lower bound of lim inf L→0 + c * L from the homogenized equation. In the following, we are going to derive the homogenized equation of (1.3), which will lead to the sharp lower bound of lim inf L→0 + c * L . However, to furnish this goal we will only consider for the sake of simplicity a particular type of nonlinearities among those satisfying (1.5). In fact, the following ideas can be generalized to a wider family of nonlinearities which satisfy (1.5), but the proof requires technical extra-arguments which will be the purpose of a forthcoming paper.
For each L > 0, let u L be a pulsating travelling front with minimal speed c * L for the reaction-diffusion equation where a L (x) = a(x/L), a is a C 2,δ (R) 1-periodic function satisfying (1.4), µ is a C 1,δ (R) positive 1-periodic function and g is a C 2 (R + ) function such that g(0) = g(1) = 0 and u → g(u)/u is decreasing in (0, +∞). Up to a shift in time, one can assume that For each L > 0, set f L (x, u) := f (x/L, u) = µ(x/L)g(u). In this setting, there holds p L ≡ 1. From standard parabolic estimates, each function u L is (at least) of class As already underlined, it follows from [1] that w L = ∂u L ∂t > 0 in R × R for each L > 0. Under the notations of the beginning of this section, it follows from (1.4) and (3.2) that k(λ, L) ≥ λ 2 α 1 + µ m for all L > 0 and λ ∈ R, where µ m = min R µ > 0. Hence, c * L ≥ 2 √ α 1 µ m for each L > 0 and lim inf L→0 + c * L ≥ 2 √ α 1 µ m > 0.
We shall now establish some estimates for the functions u L , v L and w L which are independent of L, in order to pass to the limit as L → 0 + . Notice first that standard parabolic estimates and the (t, x)-periodicity satisfied by the functions u L imply that, for each L > 0, u L (−∞, x) = 0 and u L (+∞, x) = 1 in C 2 loc (R), and w L (±∞, x) = 0 in C 1 loc (R). Let k ∈ N\{0} be given. Integrating the first equation of (3.22) by parts over R × (−kL, kL), one obtains Multiplying the first equation of (3.22) by u L and integrating by parts over R × (−kL, kL), one then gets Notice that the last integral in (3.25) converges because of (3.24) and 0 ≤ f (x/L, u L )u L ≤ f (x/L, u L ). Together with (1.4), one concludes that for each L > 0, the first integral in (3.25) converges and Multiply the first equation of (3.22) by ∂u L ∂t and integrate by parts over R × (−kL, kL). Since where F (y, s) = s 0 f (y, τ )dτ . It follows from the above estimates that for each compact subset K of R, where C(K) is a positive constant depending only on K.
In particular, for each compact K of R and for each L > 0, ||w L || L 2 (R×K) ≤ C(K). Now, differentiate the first equation of (3.22) with respect to t (actually, from the regularity of f , the function w L is of class C 2 with respect to x). There holds Multiply the above equation by w L and integrate by parts over R × (−kL, kL). From (1.4) and (3.26), it follows that where η is the positive constant defined by Then, for each compact K ⊂ R, there exists a constant C (K) > 0 depending only on K such that Let (L n ) n∈N be a sequence of real numbers in (0, 1) such that L n → 0 and c * Ln → lim inf L→0 + c * L > 0 as n → +∞. It follows from (3.27) and the bounds 0 < u Ln < 1 that there exists u 0 in H 1 loc (R×R) such that, up to extraction of a subsequence, u Ln → u 0 strongly in L 2 loc (R × R) and almost everywhere in R × R, and ∂u Ln ∂t , ∂u Ln ∂x ∂u 0 ∂t , ∂u 0 ∂x weakly in L 2 loc (R × R) as n → +∞.
By uniqueness of the limit, one gets v 0 =< a > H ∂u 0 ∂x . Passing to the limit as n → +∞ in the first equation of (3.22) with L = L n implies that u 0 is a weak solution of the equation From parabolic regularity, the function u 0 is then a classical solution of the homogenous equation . On the other hand, it follows from the second equation of (3.22) and (3.27) that that satisfies U 0 ≥ 0 in R and Standard elliptic estimates imply that U 0 converges as s → ±∞ in C 2 loc (R) to two constants U ± 0 ∈ [0, 1] such that < µ > A g(U ± 0 ) = 0, that is g(U ± 0 ) = 0. The monotonicity of U 0 and the assumption on g imply that U − 0 = 0 and U + 0 = 1. In other words, U 0 is a usual travelling front for the homogenized equation (3.29) with speed c and limiting conditions 0 and 1 at infinity. Since the minimal speed for this problem is equal to 2 √ < a > H < µ > A , one concludes that lim inf 4 Monotonicity of the minimal speeds c * L near the homogenization limit This section is devoted to the proof of Theorem 2.3. Before going further in the proof, we recall that for each L > 0, the minimal speed c * L is given by the variational formula where λ * L > 0 and k(λ, L) is the principal eigenvalue of the elliptic equation (3.2). Notice that k(λ, L) can be defined for all λ ∈ R and L > 0.
Lastly, when λ is changed into −λ or when L is changed into −L, then the operator in (4.1) is changed into its adjoint. But since the principal eigenvalues of the operator and its adjoint are identical, it follows that ∀ (λ, L) ∈ R 2 ,k(λ, L) =k(λ, −L) =k(−λ, L).
In particular, it follows that Therefore, for all λ ∈ R, But since this limit is equal to k(λ) = λ 2 < a > H + < µ > A from Step 2 of the proof of Theorem 2.1, one then gets that It also follows from (4.3) that (4.5) From (4.4) and (4.5), one deduces that Similarly, as (λ, L) → (λ, 0 + ), Remark 4.1 As a byproduct of the fact thatk and k are even in λ, it follows that the minimal speed of pulsating fronts propagating from right to left (as in Definition 1.2) is the same as that of fronts propagating from left to right.
In particular, if µ is constant and a is not constant (resp. if a is constant and µ is not constant), then this condition is not satisfied, whence lim L→0 +  µ(y)dy. Therefore, the speeds c * L are increasing in a right neighbourhood of L = 0 but, in this case, the variation is of the first order. Notice that the formula lim L→0 + dc * L dL = 2 √ β < a > H is coherent with the numerical calculations done by Kinezaki, Kawasaki and Shigesada in [21] (see Figure 3b with < µ > A = 0, that is A = 0 under the notations of [21]).

Proof of Theorem 2.6
As in the proofs of the previous theorems, we use the following formula for the minimal speed: where k z (λ) is defined as the unique real number such that there exists a positive L 0 -periodic function ψ satisfying: ψ + 2λ ψ + λ 2 ψ + µ z (x)ψ = k z (λ)ψ in (0, L 0 ).